MOD Planes: A New Dimension to Modulo Theory. W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache

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2 MOD Plaes: A New Dimesio to Modulo Theory W. B. Vasatha Kadasamy latheral K Floreti Smaradache 2015

3 This book ca be ordered from: EuropaNova ASBL Clos du Parasse, 3E 1000, Bruxelles Belgium UL: Copyright 2014 by EuropaNova ASBL ad the Authors Peer reviewers: Professor Paul P. Wag, Ph D, Departmet of Electrical & Computer Egieerig, Pratt School of Egieerig, Duke Uiversity, Durham, NC 27708, USA Dr.S.Osma, Meofia Uiversity, Shebi Elkom, Egypt. Prof. Valeri Kroumov, Dept. of Electrical ad Electroic Egieerig, Faculty of Egieerig, Okayama Uiversity of Sciece, 1-1, idai-cho, Kita-ku, Okayama May books ca be dowloaded from the followig Digital Library of Sciece: SBN-13: EAN: Prited i the Uited States of America 2

4 CONTENTS Preface 5 Chapter Oe NTODUCTON 7 Chapter Two THE MOD EAL PLANES 9 Chapter Three MOD FUZZY EAL PLANE 73 3

5 Chapter Four MOD NEUTOSOPHC PLANES AND MOD FUZZY NEUTOSOPHC PLANES 103 Chapter Five THE MOD COMPLEX PLANES 147 Chapter Six MOD DUAL NUMBE PLANES 181 FUTHE EADNG 215 NDEX 218 ABOUT THE AUTHOS 221 4

6 PEFACE this book for the first time authors study mod plaes usig modulo itervals [0, m); 2 m. These plaes ulike the real plae have oly oe quadrat so the study is carried out i a compact space but ifiite i dimesio. We have give seve mod plaes viz real mod plaes (mod real plae) fiite complex mod plae, eutrosophic mod plae, fuzzy mod plae, (or mod fuzzy plae), mod dual umber plae, mod special dual like umber plae ad mod special quasi dual umber plae. These mod plaes ulike real plae or complex plae or eutrosophic plae or dual umber plae or special dual like umber plae or special quasi dual umber plae are ifiite i umbers. Further for the first time we give a plae structure to the fuzzy product set [0, 1) [0, 1); where 1 is ot icluded; this is defied as the mod fuzzy plae. Several properties are derived. 5

7 This study is ew, iterestig ad iovative. May ope problems are proposed. Authors are sure these mod plaes will give a ew paradigm i mathematics. We wish to ackowledge Dr. K Kadasamy for his sustaied support ad ecouragemet i the writig of this book. W.B.VASANTHA KANDASAMY LANTHENAL K FLOENTN SMAANDACHE 6

8 Chapter Oe NTODUCTON this book authors defie the ew otio of several types of MOD plaes, usig reals, complex umbers, eutrosophic umbers, dual umbers ad so o. Such study is very ew ad i this study we show we have a MOD trasformatio from to the MOD real plae ad so o. We give the refereces eeded for this study. the first place we call itervals of the form [0, m) where m N to be the MOD iterval or small iterval. The term MOD iterval is used maily to sigify the small iterval [0, m) that ca represet (, ). f m = 1 we call [0, 1) to be the MOD fuzzy iterval. We defie () = {(a, b) a, b [0, m)} to be the MOD real plae. f m = 1 we call (1) as the MOD fuzzy plae. The defiitios are made ad basic structures are give ad these MOD plaes exteded to MOD polyomials ad so o. For study of itervals [0, m) refer [20-1]. For eutrosophic umbers please refer [3, 4].

9 8 MOD Plaes For the cocept of fiite complex umbers refer [15]. The otio of dual umber ca be had from [16]. The ew otio of special quasi dual umbers refer [18]. The cocept of special dual like umbers ad their properties refer [17]. For decimal polyomials [5-10]. We have oly metioed about the books of referece. Several ope problems are suggested i this book for the researchers. t is importat that the authors ame these structures as MOD itervals or MOD plaes maily because they are small whe compared to the real lie or real plae. The term MOD is used maily to show it is derived usig MOD operatio.

10 Chapter Two THE MOD EAL PLANES this chapter authors for the first time itroduce a ew otio called MOD real plae. We call this ewly defied plae as MOD real plae maily because it is a very small plae i size but ejoys certai features which is very special ad very differet from the geeral real plae. ecall the authors have defied the otio of semi ope squares of modulo itegers [23]. Several algebraic properties about them were studied [23]. Here we take [0, m) the semi ope iterval, 1 m < ad fid [0, m) [0, m); this represets a semi ope square which we choose to call as the MOD real plae. This is give the followig represetatio. Figure 2.1

11 10 MOD Plaes The poit m o the lie is ot icluded. Further the dotted lie clearly shows that the boudary is ot icluded. Further the X-axis is termed as X to show it is the MOD X - axis ad Y the MOD Y-axis of the MOD real plae (m). So this MOD real plae is deoted by (m). fact we have ifiite umber of MOD plaes as 1 m <. Thus throughout this book (m) will deote the MOD real plae where m shows the MOD iterval [0, m) is used. We will give examples of MOD plaes. Example 2.1: Let (5) be the MOD real plae relative to the iterval [0, 5). Figure 2.2 Ay poit P of (x,y) is such that x, y [0, 5). Example 2.2: Let (6) be the MOD real plae relative to iterval [0, 6).

12 The MOD eal Plaes 11 Figure 2.3 P(4.5, 3.1) is a poit o the MOD plae (6). Clearly P(8.5,7) (6) however P(8.5 6, 7 6) = (2.5, 1) (6). Thus ay poit i the real plae say ( 17.2, 9.701) ca be marked as (0.8, 3.701). Thus every poit i has a correspodig poit o (1, 6) i the real plae is marked as (1, 0) ad so o. That is why we call this MOD plae, for ay poit i the real plae be it i ay quadrat we ca fid its place ad positio i the MOD plae (m) which has oly oe quadrat. (9, 17.01) i is mapped o to (3, 0.99) i the MOD plae (6). Let ( 10.3, 12.7), ow the correspodig poit i (6) is (1.7, 5.3) (6). Thus for a poit i there exist oe poit i (m).

13 12 MOD Plaes t is importat to keep o record for several poits (or ifiite umber of poits) i, we may have oly oe poit i (m). We see (14, 9), (8, 15) ad (20, 21) so o i are mapped to (2, 3) oly i the MOD plae (6). That is why as the real plae ca be accommodated i (m); we call (m) the MOD real plae or small real plae. Because of the fact Z caot be ordered as a group or rig we caot always fid the modulo distace betwee ay 2 poits i (m) ad eve if it exist it is ot well defied. Example 2.3: Let (7) be the MOD plae o the iterval [0, 7). We see if (3, 2) is poit i (7) it ca be mapped o to ifiite umber of poits i. For (10, 9), (17, 9), (24, 9), (31,9),, (7+3, 9); Z are all mapped o to (3,2). Also all poits are of the form (3, 9), (3, 16), (3, 23), (3, 30), (3, 37),, (3, 7 + 2) ( Z) i are mapped o to (3, 2) i (7). Thus to each poit i (7) there exist ifiite umber of poits i. O the cotrary to these ifiite set of poits i we have oly oe poit mapped i (7). Also the poits (10, 9), (17, 16), (10, 16), (17, 9),(10, 23) (17, 23), (24, 16), (24, 9) (24, 23) ad so o is also a ifiite collectio which is mapped o to (3, 2). Likewise to each poit i (m) we have ifiitely may poits associated with it i.

14 The MOD eal Plaes 13 Let ( 3, 1) the ( 3, 1) is mapped o to (4, 6) i (7). view of this we have the followig theorem. THEOEM 2.1: Let be the real plae ad (m) be the MOD real plae of the iterval [0, m). To ifiite umber of poits i we have a uique poit i (m). The proof is direct as is evidet from the examples. Next we study about the cocept of distace i (m). We see i (13) the modulo distace betwee P(2, 1) ad Q(12, 12) i the MOD real plae of (13) is zero. PQ 2 = (2 12) 2 + (1 12) 2 = (2 + 1) 2 + (1 + 1) 2 = = 0 mod 13. So a distace cocept caot be defied i a MOD plae oly the modulo distace. For cosider the MOD real plae (13). Let P(2, 1) ad Q(12, 12) be i (13) they are plotted i the followig MOD plae.

15 14 MOD Plaes y (12,12) 2 1 (2,1) (5,1) Figure 2.4 x Oe may thik that is the largest distace but o the cotrary it is zero. Cosider S(5, 1) ad 0 = (0, 0) (13). The distace 0 S 2 = (5 0) 2 + (1 0) 2 = (mod 13). So this distace is also zero i (13).

16 The MOD eal Plaes 15 So a atural ope cojecture is to fid or defie o (m) a special MOD distace. Example 2.4: Let (8) be the MOD real plae of [0, 8). 8 y 7 (6.9,7) 6 5 (4.25,5) (1,2) (2,2) (1,1) (1.5,3) Figure 2.5 x We see we have MOD real poits marked i the MOD real plae. However we are ot i a positio to place ay distace cocept. We see how far takig the square root or cube root or th root of a umber i [0, ) works out.

17 16 MOD Plaes For the preset for ay x [0,m) we defie case of reals. Also for x 2, it is mod m if x [0, m). x to be as i the We will see if 3.21 [0, 8) (3.21) 2 = [0, 8); 3.21 = We see for 4 [0,8); 4 2 = 0 (mod 8) but 4 = 2. However fidig for every x [0,m), x 1 is a challegig questio i the MOD real plae. Next atural questio will be; how may such MOD real plaes exists? The aswer is ifiite for every o zero iteger Z + we have a MOD real plae associated with it. Next we show how the MOD real plae (m) is actually got by mappig all the other three quadrats ito the first quadrat plae which is ot as large as the real plae but o the smaller scale yet it is ifiite i structure. That is if (m) is the MOD real plae o [0, m) ad the real plae the ay (x, y) is mapped o to (t = x (mod m), s = y (mod m)) if x ad y are i the first quadrat. f (x, y) is i the secod quadrat the x < 0 ad y > 0 so that (x, y) is mapped to (m t x (mod m), s y (mod m)). O similar lies the poits i other quadrats are also mapped to (m).

18 The MOD eal Plaes 17 We will first illustrate this by a example. Example 2.5: Let (10) be the MOD real plae related to the iterval [0, 10). Let P = (27, ), the first quadrat of. The P is mapped to (7, ) i (10). Let Q = ( , 16.9) be the elemet i the secod quadrat of. This poit is mapped to (6.2799, 6.9) i the MOD real plae (10). Let M = ( , ) be i the 3 rd quadrat of. M is mapped o to ( , ) i the MOD real plae (10). Let N = ( , ) be a poit i the 4 th quadrat of the real plae. N is mapped o to (5.0051, 7.079) i the MOD real plae (10). This is the way mappigs are made. Thus all the three quadrats of the real plae is mapped o to the MOD real plae. Next work will be to fid how fuctios i the MOD real plae behave. f we have a fuctio i real plae the we study its appearace i the MOD real plae. Example 2.6: Let (11) be the MOD real plae o the semi ope iterval [0, 11). Let y = x be the fuctio i the real plae, it is give i Figure 2.6. The same fuctio is give i the MOD plae i Figure 2.7.

19 18 MOD Plaes y x Figure y Figure 2.7 x

20 The MOD eal Plaes 19 So this fuctio y = x is the same i the first quadrat of the real plae ad the MOD real plae. Let us cosider the fuctio y = 2x, how does the fuctio gets mapped i the real plae ad the MOD real plae. We kow y = 2x i the real plae is give i the followig: y x Figure 2.8 y = 2x i the MOD real plae (11) is as follows:

21 20 MOD Plaes 11 y Figure 2.9 x whe whe ad whe x = 5.5, y = = 0 (mod 11), x = 6, y = 2 6 = 1 (mod 11) x = 7, y = 2 7 = 3 (mod 11). Thus i both the represetatios the fuctio y = 2x is distict. Thus fuctios defied o MOD real plae behaves differetly from the fuctios defied i the real plae. We will illustrate this situatio by some examples.

22 The MOD eal Plaes 21 Example 2.7: Let (12) be the MOD real plae associated with the iterval. 12 y Figure 2.10 x Let us cosider the fuctio y = 2x. We see two MOD parallel lies ad the poits are also the same as far as the Y coordiate is cocered. Now cosider y = 4x i the real ad the MOD real plae (12). the real plae, y = 4x graph is as follows:

23 22 MOD Plaes y Figure 2.11 x the MOD plae (12), y = 4x is as follows:

24 The MOD eal Plaes y Figure 2.12 x Thus we see y = 4x has the above represetatio i the MOD real plae associated with [0, 12). Such study is iterestig, we have ot give difficult or complicated fuctios, that will be doe i due course of time. Now we have said that there exist ifiite umber of MOD real plaes (m) associated with [0,m); m Z +. We see the smallest MOD real plae is (1) associated with the iterval [0, 1), ifact it is also kow as the fuzzy MOD real plae described i the followig;

25 24 MOD Plaes 1 Y 0 1 Figure 2.13 X The fuctio y = x ca ot have a represetatio as y = 1. x is zero. Let y = 0.5x, the the represetatio of this i the fuzzy MOD real plae is give i Figure Now we fid the graph of y = 0.2x i the fuzzy MOD real plae. We see i case of y = 0.5x the maximum value y ca get is

26 The MOD eal Plaes 25 1 Y Y 1 Figure 2.14 X Figure 2.15 X

27 26 MOD Plaes Likewise for the fuctio y = 0.2x. The maximum value y ca get is So these fuctios behave i a uique way i the fuzzy MOD real plae. We will show how these two fuctios behave i a MOD real plae which is ot fuzzy. This will be illustrated by examples. Example 2.8: Let us cosider the MOD real plae (2) for the iterval [0, 2). Cosider the fuctio y = Y Figure 2.16 X

28 The MOD eal Plaes 27 Example 2.9: Let us cosider the MOD real plae (3). Let y = 0.5x be the MOD fuctio i (3). 3 Y Figure 2.17 X Cosider the fuctio y = 0.5x. Whe x = 1 the y = 0.5 ad whe x = 2, y = 1. Whe x = 2.9 the y = = So the greatest value is boud by 1.5. Example 2.10: Let us cosider the MOD real plae (4) built usig [0, 4).

29 28 MOD Plaes Cosider the fuctio y = 0.5x. The graph of y = 0.5x is as follows: 4 Y Figure 2.18 X Example 2.11: Let us cosider the MOD real plae (20) usig the iterval [0, 20). Let y = 0.5x, the MOD fuctio; the graph of this i the real MOD plae (20) is give i Figure all cases we saw i a MOD real plae (m) we see the fuctio y = 0.5x for the highest value of x say (m 1).99.9; m the y value is oly

30 The MOD eal Plaes Y Figure 2.19 Whe we compare this with the real plae the fuctio y = 0.5 x is give Figure Thus we see they exted to ifiity. Next we aalyse the fuctio y = x. This fuctio behaves i a ice way or it is alike i both the real plae as i well as the MOD real plae. Next we cosider the fuctio y = 2x ad see how it behaves i each of the MOD plaes ad the real plae. fact i the real plae it has the uique represetatio. X

31 30 MOD Plaes 20 Y Figure 2.20 X However i the MOD real plae they have a differet represetatio represeted by two lies. Clearly y = 2x is ot defied i the MOD plae (2). Just to have some way of aalyzig the fuctio y = 2x i MOD real plaes. The graph of y = 2x i the real plae is give i Figure We give the graph of the fuctio y = 2x oly i two MOD plaes (6) ad (7) over the itervals [0, 6) ad [0, 7) respectively; the graphs of the MOD fuctio y = 2x is give i Figure 3.22 ad Figure 2.23 respectively.

32 The MOD eal Plaes y y Figure 2.21 x Figure 2.22 (6) MOD plae o the iterval [0, 6). x

33 32 MOD Plaes 7 y Figure 2.23 x (7) the MOD real plae o the iterval [0, 7). 3 y Figure 2.24 x Clearly the fuctio y = 2x ca be defied oly i the MOD real plae (m); m 3.

34 The MOD eal Plaes 33 We just see how y = 2x has its graph i the MOD real plae (3) as give i Figure Thus we see y = 2x is a two lie graph represeted i that form. We see this is true i all MOD real plaes (m). Next we study the fuctio y = 3x i the real plae ad the MOD real plae (m) o the iterval [0, m); clearly y = 3x is ot defied i the MOD plae (3). Example 2.12: Let m (5) be the MOD real plae. We see how the graph of the fuctio y = 3x looks. 5 y Figure 2.25 x We see we have three discoected lie graphs represets the fuctio y = 3x i the MOD plae (5). Cosider the MOD real plae (6) i the followig example where 3/6.

35 34 MOD Plaes We will see how these graphs look like. Example 2.13: Cosider the MOD real plae (6) 6 y Figure 2.26 x aother example we will cosider (8); (3, 8) =1. We see whe 3/6 the graphs of the fuctio y = 3x behave i a chaotic way. Thus we first see how the fuctio y = 3x behaves o the MOD real plae (8). This graph is give i Figure (3, 8) = 1 we get the about graph for y = 3x i the MOD plae (8) usig [0, 8). Example 2.14: Cosider the MOD plae (9). We see this has 3 lies which are disjoit but are parallel to each other. This graph is give i Figure 2.28.

36 The MOD eal Plaes 35 8 y y 9 Figure 2.27 x Figure 2.28 x

37 36 MOD Plaes We see this has 3 lies which are disjoit but are parallel to each other. Example 2.15: Cosider the MOD real plae (11). Cosider the graph of the MOD fuctio y = 3x. 11 y Figure 2.29 x Here also we get 3 parallel lies associated with the graph for the MOD fuctio y = 3x. Next we cosider the fuctio y = 0.4x i the MOD real plaes (1), (2), (3), (4) ad (8). Cosider the MOD fuzzy real plae.

38 The MOD eal Plaes 37 1 Y Y 2 Figure 2.30 X Figure 2.31 X

39 38 MOD Plaes Cosider the MOD real plae (2) give i Figure Cosider the MOD real plae (3) give i Figure Y Figure 2.32 X We see the graph of y = 0.4x is a sigle lie. The maximum value is bouded by 1.2. Cosider the MOD real plae (4) give i Figure The graph is a straight lie ad the maximum value is bouded by 1.6. Cosider the MOD real plae (8) give i Figure 2.34.

40 The MOD eal Plaes 39 Y y Figure 2.33 X Figure 2.34 x

41 40 MOD Plaes This is also a straight lie reachig a maximum value for this is bouded by 3.2. Let y = 4x. We ca have this fuctio to be defied i the MOD real plaes (5), (6) ad so o. Now we will study the fuctio y = 4x i the MOD real plae (5). 5 y Figure 2.35 x The graph y = 4x i the MOD real plae y = 4x gives 4 parallel lies as idicated i the graph. Cosider y = 4x i the MOD real plae (8) give i Figure 2.36.

42 The MOD eal Plaes 41 8 y Figure 2.36 x this case also we get four parallel lies for the graph y = 4x i the MOD real plae. We study the fuctio y = 4x i the MOD plae (16) which is give i Figure This fuctio y = 4x which is a sigle lie graph i the real plae has four parallel lie represetatio i the MOD real plae (16). Next we study y = 5x. This fuctio ca be defied oly i (m), m 6.

43 42 MOD Plaes y x Figure 2.37 For if m 5 this fuctio does ot exist for those MOD real plaes. We study the fuctio y = 5x i the MOD plae (6) ad the graph is give i Figure We get five parallel lies for the graph of the fuctio y = 5x i the MOD real plae (6). Now we study the fuctio y = 5x i the MOD real plae (10), the graph is give i Figure Thus we get i this case also 5 parallel lies as show i the figure.

44 The MOD eal Plaes 43 6 y y 10 Figure 2.38 x Figure 2.39 x

45 44 MOD Plaes view of all these examples we cocetrate o the study of these fuctios of the form y = ts, t a positive iteger ca be defied oly over the MOD real plae (m), but m > t. We propose the followig problems. Cojecture 2.1: f (m) is a MOD real plae, Let y = tx (t a iteger) be the fuctio m > t. Ca we say we have t umber of parallel lies as the graph of y = tx i the MOD real plae (m)? Note: Clearly if m t the the very fuctio remais udefied. Let us ow cosider the fuctios of the form y = tx + s. We see i the first case y = tx + s ca be defied over the MOD real plae (m) if ad oly if m > s ad t. We will describe them by some examples for appropriate values of m, s ad t. Example 2.16: Let (9) be the MOD real plae. Cosider y = 5x + 2 the fuctio. We fid the represetatio of this fuctio i the MOD real plae (9)[x]. The associated graph is give i Figure We get for this fuctio y = 5x + 2 also i the MOD real plae (9) six parallel disjoit lies with differet startig poits. Let us cosider the MOD fuctio y = 5x + 4 i the MOD real plae (9). The associated graph is give i Figure We get 6 distict lies for the fuctio y= 5x + 4. Let us cosider the fuctio y = 5x + 4 o the MOD real plae (11). The associated graph is give i Figure 2.42.

46 The MOD eal Plaes x y Figure x y Figure 2.41

47 46 MOD Plaes 11 y Figure 2.42 Here also we get oly six lies give by the figure. Thus we propose oe more cojecture. Cojecture 2.2: Let (m) be a MOD real plae, y = tx + s be a fuctio defied o (m); m > t ad s. Ca we say the graph of y = tx + s will have t or t+1 lies i the MOD real plae (m)? Will they be parallel lies? f they are parallel lies we have to study whether they correspod to the same MOD slope, such study is both iovative ad iterestig. Let us ow fid the graph of the fuctio y = x 2 i the MOD plae (m); m > 1. x

48 The MOD eal Plaes 47 Let us cosider the fuctio i the MOD real plae (4). 4 Y Figure 2.43 X We get two parallel lies ad the fuctio y = x 2 is ot cotiuous i the MOD real plae (4). Let us cosider y = x 2 i the MOD real plae (2). The associated graph is give i Figure We ca oly say the fuctio y = x 2 becomes zero i the iterval x =1.41 to x = 1.42 but we caot fid the poit as (1.41) 2 = ad = = =

49 48 MOD Plaes 2 Y Figure 2.44 X = = = = = = = = = = = So approximately the zero lies i the iterval [ , ] How ever = 2. Thus whe x = we get y = x 2 = 0. We get for y = x 2 two parallel lies.

50 The MOD eal Plaes 49 Thus the fuctio is ot a cotiuous fuctio. Now we study the fuctios of the form y = tx 2 i (m); m > t. Let us cosider y =3t 2 i the real MOD plae be ad (4). 4 Y Let x = 1.2, y = 3 (1.2) Figure 2.45 X x y = 3 (1.44) = 5.32 (mod 4) = f x =1.1 the y = 3 (1.1) 2 x y =

51 50 MOD Plaes = So zero lies betwee x = 1.1 ad x =1.2. To be more closer zero lies betwee x = 1.1 ad x = 1.16 or betwee x = 1.1 ad x = or betwee x = ad or betwee x = ad or betwee x = ad or betwee x = ad x = That is zero lies betwee ad So zero lies betwee ad fact betwee ad Approximately x = is such that y = 3x 2 = 0. Aother zero lies betwee x = 2.58 ad x = 2.6. fact zero lies betwee x = 2.58 ad x = fact zero lies betwee x = 2.58 ad x = fact zero lies betwee x = ad x = fact betwee x = ad fact zero lies betwee x = ad fact zero lies i betwee x = ad fact zero lies betwee x = ad fact zero lies betwee x = ad fact zero lies betwee x = ad fact at x = y = 3x 2 = 0 mod (4). We are yet to study completely the properties of y = 3x 2 i (4) ad i (m); m > 4.

52 The MOD eal Plaes 51 t is importat to ote i the MOD real plae y = x 2 has zeros other tha x = 0. fact the y = 3x 2 i the real plae attais y = 0 if ad oly if x = 0. But y = 3x 2 i (4) has zeros whe x = 0, x = 2, x = , ad x = are all such that y = 3x 2 = 0 mod 4. Thus we have see four zeros. Now a zero lies betwee x = ad x = Clearly x = is such that y = 3x 2 0 (mod 4). Thus y = 3x 2 has 4 zeros or more. We see y = 3x 2 is a cotiuous fuctio i the real plae where as y = 3x 2 is a o cotiuous fuctio i the MOD plae (m); m > 4 with for several values x; y = 0. Such study is iovative ad iterestig. We will see more examples of them. Let y = 4x 2 be the fuctio defied o the MOD real plae (5) associated with the iterval [0, 5). terested reader ca fid the zeros of y = 4x 2 that is those poits for which y = 0 for x 0. Now we leave the followig ope cojecture. Cojecture 2.3: Let (m) be the MOD plae. Let y = tx 2 where m > t. (i) Fid those x for which y = 0. (ii) Prove the graph is ot cotiuous. (iii) How may lie graphs we have for y = tx 2 o the MOD real plae (m)?

53 52 MOD Plaes (iv) Are they parallel? (v) Fid the slopes of these lies. (vi) Are these lies with the same slope? We will give some examples of fidig the slope of the lies of fuctios defied i the MOD real plae (4). Example 2.17: Let y = 3x be the MOD fuctio i the MOD plae (4). The associated graph is give i Figure Y Figure 2.46 We see there are three disjoit lies ad the agle which it makes with the MOD axis X is ta 70. Example 2.18: Let us cosider the MOD real plae (6). X

54 The MOD eal Plaes 53 Let y = 4x be the fuctio to fid the graph ad slope of the graph relative to (6). 6 y Figure 2.47 x The agle made by that graph is 75. We see there are four parallel lies ad we see the slope of them are approximately 4. Example 2.19: Let (6) be the MOD real plae. y = 5x be the fuctio. We fid the graph y = 5x o (6). The associated graph is give i Figure We see there are five parallel lies all of them have the same slope ta 80. Now we fid graphs of the form y = tx + s i the MOD real plae (m), m > t ad s.

55 54 MOD Plaes 6 y x Figure 2.48 Example 2.20: Let (5) be the MOD real plae. Cosider the fuctio y = 4x + 2 o (5). The associated graph is give i Figure Example 2.21: Let us cosider the MOD real plae (6) ad the fuctio (or equatio) y = 4x + 2 i the MOD real plae. The graph of it is give i Figure All the four lies have the same slope.

56 The MOD eal Plaes 55 5 y y Figure 2.49 x Figure 2.50 x

57 56 MOD Plaes Next we cosider the equatios of the form y = tx i the MOD real plae (m), m > t. We cosider y = x 2 i (2). The graph is as follows: 2 Y Figure 2.51 X y = 0 for some value i betwee x = 1.4 ad 1.5. We see y = x 2 is ot a cotiuous lie. Let us cosider y = x 3 i (2). The associated graph is give i Figure We see for x i betwee (1.25, 1.26); y takes the value zero. So this fuctio is also ot cotiuous i the MOD real plae (2). Let y = x 3 a fuctio i the MOD real plae (3). The graph is give i Figure 2.53.

58 The MOD eal Plaes 57 2 Y Y Figure 2.52 X Figure 2.53 X

59 58 MOD Plaes Zero lies betwee (1.4422, ). We see this yields may lies which are ot parallel. Let us cosider y = x 3 o the MOD plae (4). The graph is as follows: 4 Y Figure 2.54 f x = 0 the y = 0, f x = 1.7 the y = f x = 1 the y = 1, f x = 2 the y = 0 f x = 1.2 the y = 1.728, f x = 2.2 the y = f x = 1.4 the y = 2.744, f x = 2.3 the y = f x = 1.5 the y = 3.375, f x = 2.5 the y = f x = 1.6 the y = 4.096, f x = 2.6 the y = f x = 2.7 the y = f x = 2.8 the y = f x = 2.9 the y = f x = 3 the y = 3, f x = 3.2 the y = f x = 3.3 the y = f x = 3.31 the y = f x = 3.4 the y = X

60 The MOD eal Plaes 59 f x = 3.5 the y = f x = 3.42 the y = f x = 3.64 the y = f x = 3.7 the y = f x = 3.8 the y = f x = 3.9 the y = f x = 3.92 the y = f x = 0 the y = 0, f x (1.5, 1.6) the y = 0 f x = 2 the y = 0, For x (2.2, 2.6), y = 0 For x (2.5, 2.6), y = 0 For x (2.7, 2.74), y = 0 For x (2.8, 2.9), y = 0 For x (3.1, 3.2), y = 0 ad so o. y = x 3 has over 12 zeros i (4). Zero lies betwee to 1.6 ad at x = 2, y = 0 ad so o. Cosider y = x 3 at the MOD real plae (6) give i Figure betwee 1.8 ad 1.9 we have a zero ad so o. We see whether they are curves or straight lies. t is left as a ope cojecture for researchers to trace the curves y = tx p where t > 1 i the MOD real plae (m); m > t. (p > 2). We see this is a difficult problem ad eeds a complete aalysis. Study of the curve y = tx p + s p > 2, t, s < m over the MOD real plae (m). s the fuctio cotiuous i (m) is a ope cojecture. Next we see we ca i the first place defie fuctios f(x) = x 2 + x + 2 or ay polyomials. To this ed we make the followig defiitio.

61 60 MOD Plaes 6 y Figure 2.55 x DEFNTON 2.1: Let (m) be the MOD plae. Let [0, m)[x] = [x] where = {a a [0, m)} be the pseudo rig with 1, (m > 1). Ay polyomial i [x] ca be defied o the MOD plae i (m); (m > 1). So to each MOD plae (m) we have the polyomial pseudo rig [x] where = [0,m) ad [x] = i aix a i [0, m)} associated with it ay i= 0 polyomial p(x) i [x] ca be treated as a MOD fuctio i [x] or as a MOD polyomial i [x]. A atural questio will be, ca we defie differetiatio ad itegratio of the MOD polyomials i [x] which is associated with the MOD plae (m).

62 The MOD eal Plaes 61 We kow the MOD polyomials i geeral are ot cotiuous o the iterval [0, m) or i the MOD real plae (m). We assume all these fuctios are cotiuous o the small itervals [a, b) [0,m). We call these sub itervals as MOD itervals ad further defie these polyomials to be MOD cotiuous o the MOD real plae. Thus all MOD fuctios are MOD cotiuous o the MOD itervals i the MOD plae. We will illustrate this situatio by some examples. Example 2.22: Let (5) be the MOD plae o the MOD iterval [0,5). Let p(x) = 3x [0, 5)[x], we see p(x) is a MOD cotiuous fuctio o MOD subitervals of [0,5). So if we defie MOD differetiatio of p(x) as dp(x) dx = d dx (3x2 + 2) = x. We kow p (x) = 3x the MOD fuctio is ot cotiuous o the whole iterval [0, 5) oly o subitervals of [0, 5) that is why MOD fuctio ad they are MOD differetiable. Cosider the fuctio q(x) = 4x 5 + 3x [0,5)[x] be MOD polyomial i [x]. ( = [0,5)). d(q(x)) = 0 + 6x = x. dx We make the followig observatios. That is the differece betwee usual differetiatio of polyomials ad MOD differetiatio of MOD polyomials. (i) MOD polyomials are cotiuous oly i MOD subitervals of [0, m).

63 62 MOD Plaes (ii) f p(x) is i [x] ( = ([0,m))) is a th degree polyomial i [x] the the derivative of p(x) eed ot be a polyomial of degree 1. This is very clear from the case of q(x) = 4x 5 + 3x Further this q(x), the MOD polyomial is oly MOD cotiuous for q (1) = 0, q(0) = 3 ad so o. dq(x) dx = x. Cosider r (x) = 2x 7 + 2x 5 + 3x + 3 [x] ( = [0,5)) be the MOD polyomial the MOD derivative of r(x) is dr(x) = 4x dx We ca as i case of usual polyomial fid derivatives of higher orders of a MOD polyomials. Let us cosider r (x) = 3x 8 + 4x 5 + 3x + 2 a MOD polyomial i the MOD iterval pseudo rig (x) (where = [0, 5)). d(r(x)) = 4x = r (x) dx 2 d (r(x)) = 3x 6 2 dx 3 d (r(x)) = 3x 5 4 d (r(x)) ad = dx dx Thus the MOD differetiatio behaves i a differet way. However for r(x) [x] the r(x) ca be derived 8 times but r(x) as a MOD polyomial ca be MOD derived oly four times ad 4 d (r(x)) = 0. 4 dx This is yet aother differece betwee the usual differetiatio ad MOD differetiatio of polyomials.

64 The MOD eal Plaes 63 The study i this directio is iovative ad iterestig. Likewise we ca defie the MOD itegratio of polyomials. This is also very differet from usual itegratio. Now fially we itroduce the simple cocept of MOD stadardizatio of umbers or MODizatio of umbers. For every positive iteger m to get the MOD real plae we have to defie MODizatio or MOD stadardizatio of umbers to m which is as follows: f p is a real umber positive or egative, if p < m we take p as it is. f p = m the MODize it as 0. f p > m ad positive divide p by m the remaiig value is take as the MODized umber with respect to m or MOD stadardizatio of p relative (with respect) to m. f p is egative divide p by m the remaider say is t ad t is egative the we take m t as the MODized umber. We will illustrate this situatio for the readers. Let m = 5; the MODizatio of the real plae to get the MOD real plae usig 5 is as follows. Let p reals, if p < 5 take it as it is. f p = 5 or a multiple of p the MODize it as zero. f p > 5 the fid p 5 = t + s 5 p > 0 if p < 0 the as 5 s. the p is MODized as s if Thus for istace if p = to MODize it i the MOD real plae (5); p 5 = = 1.7 [0, 5). 5 Let p = p = 2.89 so we have 2.89 to be [0,5). Let p = the p [0,5). f p = 25 the p = 0.

65 64 MOD Plaes This is the way we MOD stadardize the umbers. MOD stadardized umbers will be kow as MOD umbers. Thus [0, m) ca be also reamed as MOD iterval. Let us cosider the MOD stadardizatio of umbers to the MOD plae (12). Let p = the p [0,12). Let p = the MOD stadardized value is [0,12). f p = the the MOD stadardized value is i [0,12). f p = the MOD stadardized value is 4.85 i [0,12). This is the way real umbers are MOD stadardized to a MOD real plae. We like wise ca MOD stadardize ay fuctio i the real umbers. For istace if y = 7x + 8 defied i the real plae the the MOD stadardized fuctio i the MOD real plae (6) is y = x + 2. All fuctios y = f (x) defied i, ca be MOD stadardized i the MOD plae (m). Also all polyomials i [x], -reals ca be made ito a MOD stadardized polyomials i the MOD real plae (m). We will illustrate this situatio by some simple examples. Let p (x) = 21x x 2 5x + 3 [x]. The MOD stadardized polyomial i (7) is as follows, p (x) = 0x 7 + 0x 2 + 2x + 3. So the first importat observatio is if degree of p(x) is s the the MOD stadardized polyomial i (m)[x] eed ot be

66 The MOD eal Plaes 65 of degree s it ca be of degree < s or a costat polyomial or eve 0. For if p(x) = 14x 8 7x x 2 35x + 7 [x] the the MOD stadardized polyomial p (x) i (7)[x] is 0. The same p(x) i (5) is p (x) = 4x 8 + 3x 4 + x view of all these we have the followig theorems. THEOEM 2.2: Let p be a real umber. p = 0 ca occur after MOD stadardizatio of p i [0, m) if ad oly if m/p or m = p where p is the MODized umber of p. The proof is direct hece left as a exercise to the reader. THEOEM 2.3: Let y = f(x) be a fuctio defied i. The MOD stadardized fuctio y of y ca be a zero fuctio or a costat fuctio or a fuctio other tha zero or a costat. Proof is direct ad hece left as a exercise to the reader. We have give examples of all these situatios. However we will give some more examples for the easy uderstadig of these ew cocepts by the reader. Example 2.23: Let y = 8x be a fuctio defied i the real plae [x]. Cosider the MODizatio of y to y i the MOD real plae (4); the y = 2 is a costat fuctio i the MOD real plae. Example 2.24: Let y 2 = 8x 3 + 4x be the fuctio defied i 2 the real plae. y defied i the MOD real plae (2) after MODizatio is zero. Example 2.25: Let y = 9x 4 + 5x be the fuctio defied i the real plae [x].

67 66 MOD Plaes Let y be the MODizatio of y i the plae (3). y = 2x Clearly whe MODizatio takes place, the polyomial or the fuctio is totally chaged. view of all these we have the followig theorem. THEOEM 2.4: Let p(x) [x] be the polyomial i the polyomial rig. Let (m) be the MOD real plae. Every polyomial p(x) i [x] ca be MODized ito a polyomial p (x) whose coefficiets are from (m). This mappig is ot oe to oe but oly may to oe. So such mappigs are possible. Thus we see [x] to (m)[x] is give by p (x) for every p(x). Several p(x) ca be mapped o to p (x). t ca be a ifiite umber of p(x) mapped o to a sigle p (x). We will just illustrate this situatio by a example. Let p(x) = 5x + 15 [x]. the MOD real plae (5) we see p (x) = 0. f p(x) = 15x x [x] the also p (x) = 0. f p(x) = 20x [x] the also p (x) = 0. Thus p(x) = 5(q(x)) [x] is such that p (x) = 0 i the MOD real plae (5). Thus we see we have ifiite umber of polyomials i [x] mapped o to the MOD polyomial i the MOD real plae. We call (m)[x] as the MOD real polyomial plae.

68 The MOD eal Plaes 67 We see we have ifiite umber of MOD real polyomial plaes. Clearly these MOD real polyomial plae is oly a pseudo rig which has zero divisors. However [x] is a commutative rig with o zero divisors. Thus a itegral domai is mapped o to the pseudo MOD rig which has zero divisors. We see is a field but (m) (m 2) is oly a pseudo MOD rig which has zero divisors. Study i all these directios are carried out i the forth comig books. Now we suggest the followig problems. Problems (1) Fid ay other iterestig property associated with the MOD real plae. (2) Defie the fuctio y = 7x i the real MOD plae (9). (i) How may lies are associated with y = 7x o (9) (ii) Fid all x i [0, 9) such that y = 7x = 0 (mod 9). (iii) What are the slopes of those lies? (iv) Does it have ay relatio with 7? (3) Show by some techiques it is impossible to defie distace betwee ay pair of poits i the MOD real plae (m) usig the iterval [0, m). (4) Ca co ordiate geometry be imitated or defied o the MOD real plaes? (5) Ca you prove / disprove that MOD real plae is ot a Euclidea plae?

69 68 MOD Plaes (6) Prove or disprove the MOD real plae ca ever have the coordiate geometry properties associated with it. (7) Ca a circle x 2 + y 2 = 2 be represeted as a circle i the MOD real plae (5)? (8) Ca a parabola y 2 = 4x i the real plae be represeted as a parabola i the MOD real plae (6)? 2 2 x y (9) Let + = 1 be the ellipse i the real plae. 4 9 What is the correspodig curve i the MOD real plae (11)? 2 2 x y (10) Show the curve + = 1 i the real plae caot be 4 9 defied i the MOD real plae (4). s it defied? 2 2 x y (11) Show + = 1 i the real plae caot be defied 4 9 o the real MOD plae (36). (12) Fid the correspodig equatio of MOD real plae (19). (13) Fid the correspodig equatio of (p), p a prime p > x y + = 1 i the x y + = 1 i 4 9 (14) Prove or disprove oe caot defie the fuctio 3x 2 + 4y 2 = 4 i the MOD real plae (2). (15) What is MOD stadardizatio? (16) MOD stadardize the fuctio y 2 = 38x i the MOD real plae (7).

70 The MOD eal Plaes 69 (17) Prove or disprove the MOD stadardizatio of the fuctio y = 8x x +16 is a zero fuctio i (2) ad (4). (18) Prove a th degree polyomial i [x] after MOD stadardizatio eed ot be i geeral a th degree polyomial i (m). (19) Let p(x) = 10x 9 + 5x 6 + 4x 3 + 3x + 7 [x]; what is the degree of the polyomial after stadardizatio i i (i) [0,2)[x] = aix a i [0,2)}. i= 0 i (ii) [0,5)[x] = aix a i [0,5)}. i= 0 i (iii) [0,10)[x] = aix a i [0,10)}. i= 0 (20) Obtai some special ad iterestig features about MODizatio of umbers. (21) Obtai some special features ejoyed by MOD real plaes (m) (m eve). (22) Obtai some special features ejoyed by MOD real plaes (m) (m a prime). (23) Obtai some special features ejoyed by pseudo MOD real rigs of (m). (24) Obtai some special features ejoyed by pseudo MOD polyomials i (m)[x]. (25) Fid the properties ejoyed by differetiatio of MOD polyomials i (m)[x]. (26) Fid the special properties ejoyed by itegratio of MOD polyomials i (m)[x].

71 70 MOD Plaes (27) Let p (x) = 8x 9 + 5x 6 + 2x + 7x + 1 (9)[x]. Fid all higher derivatives of p(x). tegrate p(x) i (9)[x]. (28) Let (7)[x] be the MOD real polyomial pseudo rig. (i) Show p(x) (7)[x] is such that p (x) = 0. (ii) Characterize those p(x) (7)[x] which has oly 3 derivatives where deg p(x) = 12. (iii) Let p(x) (7)[x] show p(x) is ot always itegrable. (iv) Fid methods to solve p(x) (7)[x]. (29) Let (12)[x] be the MOD real polyomial. Study questios (i) to (iv) of problem (28) for this (12)[x]. (30) Let (15)[x] be the MOD real polyomial. Study questios (i) to (iv) of problem (28) for this (15)[x]. (31) Let y = 8x (9) be the MOD real polyomial. Draw the graph of y = 8x (32) Let y = 4x (8) be the fuctio i MOD real plae. Draw the graph ad fid the slope of y. (33) Let y= 5x (7) be the fuctio i the MOD real plae. (i) s y = 5x cotiuous i the MOD real plae (7)? (ii) s y = 5x have oly 5 parallel lies that represets the MOD curve? (iii) Ca y = 5x have all the curves with the same slope? (34) s [x] to (10)[x] a oe to oe map? (35) Ca [x] to (10)[x] be a may to oe map?

72 The MOD eal Plaes 71 (36) Ca [x] to (10)[x] be a oe to may map? (37) Let p [x] (13)[x] fid the derivative of p(x) ad itegral of p(x). (38) Let p(x) = 8x x 2 + x + 8 (8). (i) Fid p (x). (ii) Fid p(x)dx. (39) Let p(x) = 7x 8 + 4x 3 + 5x +1 (8)[x]. Fid p (x) ad p(x)dx. (40) Let p(x) = 12x x (9)[x]. Fid p (x) ad p(x) dx. (41) Let us cosider y = 8x defied i the MOD real plae (18). (i) Fid the graph of y = 8x i the MOD real plae. (ii) s the fuctio cotiuous i the real MOD plae? (iii) Fid the slopes of the curve. (iv) Fid all zeros of y = 8x (42) Let y = 8x + 1 be a fuctio defied o the MOD real plae (5). Study questios (i) to (iv) of problem 41 for this fuctio y 2 = 8x + 1. (43) Let y = 4x be a fuctio i the MOD real plae (9). (i) s the fuctio a parabola i the MOD real plae? (ii) Does y = 4x a cotiuous fuctio i the MOD real plae (9)? (iii) Fid the slope of the curve y = 4x i (9). (iv) Study questios (i) to (iv) of problem 41 for this fuctio y = 4x i (9). 2 2 x y (44) Let + = 1 be the fuctio defied i the MOD 9 4 real plae (13).

73 72 MOD Plaes (i) Study the ature of the curve. (ii) What is the equatio of the trasformed curve? (iii) Obtai the umber of zeros of the curve. (iv) s it a lie or a curve? (45) Let y = x 2 be the fuctio defied i the MOD real plae (15). (i) Study questios (iii) ad (iv) of problem 44 for this fuctio. (ii) Fid all the zeros of the fuctio y = x 2 i (15). (46) Obtai the zeros of the fuctio y = x 2 i (12). (47) Study the fuctio y = 8x i the MOD real plae (10). (48) Does there exists a fuctio i the MOD plae (7) which is cotiuous o the semi ope iterval [0, 7)? (49) Does there exist a fuctio i the MOD real plae (12) which is cotiuous i the semi ope iterval [0, 12)? (50) Prove or disprove distace cocept caot defied o the MOD real plae? (51) Ca we defie a ellipse i a MOD real plae? (52) Ca the otio of rectagular hyperbola exist i a MOD real plae?

74 Chapter Three MOD FUZZY EAL PLANE chapter we have defied the otio of MOD real plae (m), m 1. We have metioed that if m = 1 we call the MOD real plae as the MOD fuzzy real plae ad deote it by (1). We will study the properties exhibited by the MOD fuzzy real plae (1). We will also call this as fuzzy MOD real plae. We study how fuctios behave i this plae. First of all we have o power to defie y = tx, t 1 for such fuctios do ot take its values from the iterval [0, 1). The liear equatio or fuctio y = tx + s is defied i the MOD fuzzy plae oly if 0 < t, s < 1. We ca have s = 0 is possible with 0 < t < 1. Here we proceed oto describe them by examples. Example 3.1: Let m (1) be the MOD fuzzy real plae. Cosider y = 0.2x. The graph of y = 0.2x i m (1) is give i Figure 3.1.

75 74 MOD Plaes Figure Figure 3.2

76 MOD Fuzzy eal Plae 75 This graph is a cotiuous curve. But bouded o y as 0.2. Example 3.2: Let us cosider the fuctio y = 0.5x i the MOD fuzzy real plae (1). The graph of it is give Figure 3.2 We see i the first place we ca map the etire real plae o the MOD fuzzy real plae (1). Ay iteger positive or egative is mapped to zero. Ay (t, s) = ( , 25) is mapped o to ( , 0) ad (l, m) = ( , 0.321) is mapped o to (0.5479, 0.321). Thus the real plae ca be mapped to the MOD fuzzy plae (1). This mappig is ifiitely may to oe elemet. The total real plae is mapped to the MOD fuzzy plae. All iteger pairs both positive or both egative ad oe positive ad other egative are mapped oto (0, 0). Oly positive decimal pairs to itself ad all egative decimal pairs (x, y) to (1 x, 1 y) ad mixed decimal pairs are mapped such that the iteger part is marked to zero ad the decimal part marked to itself if positive otherwise to 1 that decimal value. Some more illustratio are doe for the better grasp of the readers. Let P (0.984, 0.031). This poit is mapped oto the MOD fuzzy real plae as P (0.984, 0.969) (1). Likewise if P ( 3.81, 19.06). This poit P is mapped oto the MOD fuzzy real plae as P (0.19, 0.94) (1).

77 76 MOD Plaes Let P (4.316, 8.19), P is mapped oto (0.316, 0.81) (1). P ( 9.182, 6.01), P is mapped oto (0.818, 0.01) (1). P(8, 27) is mapped to (0, 0) (1). P( 9, 81) is mapped to (0, 0) (1). This is the way trasformatio from the real plae to the MOD fuzzy plae (1) is carried out. The trasformatio is uique ad is called as the MOD fuzzy trasformatio of to (1) deoted by η 1. For istace η 1 ((0.78, 4.69)) = (0.78, 0.31). η 1 : (1) is defied i a uique way as follows: η 1 ((x, y)) = (0,0) if x, y Z + decimalpair if x, y \ Z 1 decimalpair x, y \ Z The other combiatios ca be easily defied. Now we have to study what sort of MOD calculus is possible i this case. Throughout this book η 1 will deote the MOD fuzzy trasformatio of to (1). Now we recall [6-10]. The cocept of MOD fuzzy polyomial ca be defied usig the real polyomials. Let [x] be the real polyomial ad (1)[x] be the MOD fuzzy polyomials.

78 MOD Fuzzy eal Plae 77 Let p(x) [x]; we map usig η 1 : [x] (1)[x]. Let p(x) = i aix a i. i= 0 η 1 (a i ) = 0 if a i Z η(a i ) = decimal if a i + \ Z + η(a i ) = 1 decimal part if a i \ Z η(a i ) = 0 if a i Z. Thus we ca have for every collectio of real polyomials a MOD fuzzy polyomials. The problems with this situatio is p (x) (1)[x]. dp (x) dx = 0 for some We will illustrate these situatio by some examples. p (x) = 0.7x x x dp (x) dx = x x x + 0 = 11.2x x x = 0.2x x. 2 d p (x) 2 dx = 3.x = 0.2 so

79 78 MOD Plaes 3 d p (x) 3 dx = 0. Thus it is uusual the third derivative of a sixteeth degree polyomial is 0. Let p (x) dp (x) dx = 0.34x x x is i (1)[x]. = x x x + 0 = 1.70x x x 2 d p (x) 2 dx = x x = 2.1x x d p (x) 3 dx = x = 0.3 x d p (x) 4 dx = x + 0 = 0.6 x 5 d p (x) 5 dx = 0.6. For this fuctio p (x) which is the five degree polyomial, i this case the 5 th derivative is a costat.

80 MOD Fuzzy eal Plae 79 t is importat to keep o record that eve while derivig implicitly the MOD trasformatio fuctio η 1 is used. We have avoided the explicit appearace of η 1 i every derivative. p (x) = 0.3x x dp (x) dx = η 1 (0.3 5)x 4 + η 1 (0.8 4) x = η 1 (1.5)x 4 + η 1 (3.2)x 3 = 0.5x x 3 2 d p (x) 2 dx = η 1 (0.5 4)x 3 + η 1 (0.2 3)x 2 = η 1 (2.0)x 3 + η 1 (0.6)x 2 = x 2. 3 d p (x) 3 dx = η 1 (0.6 2)x = η 1 (1.2)x = 0.2x 4 d p (x) 4 dx = η 1 (0.2) = 0.2. Thus is the way the role of η 1 takes place i the derivatives. So i the case of MOD fuzzy polyomials the derivatives are ot atural. We have the MOD fuzzy trasformatio fuctio plays a major role i this case without the use of this MOD fuzzy trasformatio η 1 we caot differetiate the fuctios.

81 80 MOD Plaes Now ca we itegrate the MOD fuzzy polyomials? The aswer is o for we kow if p (x) (1)[x] = i aix a i s are positive decimals i (1)} i= 0 Now suppose p (x) = 0.8x x x The p (x) dx = 0.8x 9 dx + 0.9x 8 dx x 3 dx + 0.7dx = x 0.9x 0.2x x + c For 1 10 or 1 9 or 1 4 is udefied i (1) so p (x) dx is udefied wheever p (x) (1)[x]. Thus we caot itegrate ay p (x) (1)[x] of degree greater tha or equal to oe. So how to over come this problem or ca we defie polyomials differetly. The authors suggest two ways of defiig polyomials ad also we have to defie differet types of itegratios differetly. We i the first place defie d [x] = i aix a i } i p d (x) = x x x to be the decimal polyomial rig. We ca fid product, sum ad work with. We ca also differetiate ad itegrate it.

82 MOD Fuzzy eal Plae 81 Let p d (x) be give as above; d dp (x) = x x dx x Now we ca also itegrate pd(x)dx = x 8.14x 0.984x x + C This is the way itegratio is carried out i the decimal rig d [x]. Now while defiig the MOD polyomials we ca defie the MOD trasformatio i two ways. η 1 : d [x] (1)[x] d by η 1 (p(x) = i aix ) = i i η1 (a i )x. i This is similar to or same as η 1 for o operatio is performed o the variable x. Aother trasformatio is defied as follows: η : d [x] (1)[x] = { a i x i i [0, 1), a i [0, 1)} * 1 * * η 1 (p d (x) : i aix ) = i 1 η (a )x η (i) 1 i.

83 82 MOD Plaes η 1 : (1), the usual MOD trasformatio; defied as the MOD fuzzy decimal trasformatio. * η 1 is We will illustrate both the situatios by some examples. Let p d (x) = 6.731x x d [x]. η 1 (p d (x)) = 0.731x x (1)[x] = q(x). d dq(x) dx = η 1 (8.501)x η 1 (1.42) x 0.42 = ( )x x This is the way derivatives are foud i Now if p d (x) d [x], the d (1)[x]. * η 1 (p d (x)) = 0.731x x = q*(x). Now we see this q*(x) differetiated for * (1)[x] caot be easily dq *(x) dx = x x = q*(x) (therefore = ad 0.58 = 0.42). So the power will remai the same; so i case of MOD fuzzy * decimal polyomials i (1)[x] we follow a differet type of differetiatio ad itegratio which is as follows: This differetiatio will be called as MOD differetiatio.

84 MOD Fuzzy eal Plae 83 We see if p(x) = x we decide to MOD differetiate it i the followig way. We dp(x) dx(0.1) = dx dx(0.1) = 0.312x We say the fuctio is MOD differetiated usig 0.1. Let p d (x) = 0.312x x x d dp (x) dx(0.1) = ( ) x ( ) * (1)[x]. x ( )x = ( )x ( )x ( ) = ( )x ( )x The advatage of MOD decimal differetiatio is that i usual differetiatio power of x is reduced by 1 at each stage but i case of MOD decimal represetatio the MOD power of x ca be reduced by ay a [0, 1). This is the advatage for we ca differetiate a fuctio p*(x) (1)[x] i ifiite umber of ways for every a (0, 1) * gives a differet value for dp *(x) dx(a). We will illustrate this situatio by a example or two. Let p*(x) = 0.58x x x x x * (1)[x]. We MOD decimal differetiate p*(x) with respect to 0.2

85 84 MOD Plaes dp *(x) dx(0.2) = ( )x ( )x (0.62)(0.5)x ( )x ( )x We ow differetiate with respect to 0.03; dp *(x) dx(0.03) = ( )x ( )x ( )x x ( )x Clearly ad are distict the oly commo feature about dp *(x) dp *(x) them is that both ad are such that every dx(0.2) dx(0.03) respective term has the same coefficiet but the MOD degree of the polyomial is differet i both cases for each of the coefficiets. ecall whe the degree of x lies i the iterval (0, 1) we call it as MOD degree of the polyomials. The highest MOD degree ad lowest MOD degree is similar to usual polyomials. Now we MOD decimal differetiate p*(x) with respect to 0.5. dp *(x) dx(0.5) = ( )x ( )x ( ) x ( ) x

86 MOD Fuzzy eal Plae 85 This is also differet from ad. We see the degree of the MOD decimal polyomial i all the three MOD differetiate are distict. This is a iterestig ad a importat feature about the MOD fuzzy differetiatio of the MOD fuzzy polyomials. f oe is iterested i fidig itegratio the oly MOD * decimal itegratio o (1)[x] is valid all other differetiatio fails to yield ay form of meaigful results. So we ca itegrate i similar way as we have differetiated usig MOD decimal differetiatio techiques. Let p*(x) = 0.92x x x x 0.7 We itegrate with respect to * (1)[x]. p*(x)dx(0.1) = 0.92x x x x x C --- We itegrate with MOD decimal with respect to 0.2. p*(x)dx(0.2) = 0.92x x x x x C

87 86 MOD Plaes = x x x x x C --- Clearly ad are differet ad we see we ca itegrate the MOD decimal polyomial i ifiite umber of ways but they have o meaig as or etc; have o meaig i [0, 1) as o [0, 1) has iverse. Now we will itegrate with respect to the MOD decimal 0.9 (0, 1). p*(x)dx(0.9) = 0.92x x x x x C = x x x x x C --- is i * (1)[x]., ad are distict ad ifact we ca get ifiite umber of MOD decimal itegral values of p*(x) (1)[x]. * So it is a ope cojecture to defie itegrals i fuzzy MOD plaes.

88 MOD Fuzzy eal Plae 87 The ext challegig problem is to fid methods of solvig equatios i (1)[x]. * f p*(x) = 0 how to solve for the roots i * (1)[x]. Equally difficult is the solvig of equatios p d (x) = 0 i (1)[x]. d We will first illustrate this situatio by some examples. Let p*(x) = (x ) (x ) = x x x The roots are x 0.3 = 0.6 ad x 0.1 = 0.5. For p*(0.6) = ( ) ( ) = 0. Likewise p*(0.5) = ( ) ( ) = 0. This is the way roots ca be determied. t is left as a ope cojecture to solve the MOD decimal polyomials i (1)[x]. * Let us cosider p*(x) = (x ) 2 * (1)[x]. p*(x) = 0.4x dp *(x) dx(0.3) = + dx(0.3) 0.5 d(0.4x 0.04) = x = 0.2x 0.2 = q*(x).

89 88 MOD Plaes x 0.5 = 0.8 is a root but q*(0.8) 0. Thus if p*(x) has a multiple root to be a (0, 1) the a is ot dp *(x) a root of, a (0, 1). dx(a) So the classical property of calculus of differetial is ot i geeral true i case MOD decimal differetiatio. Several such properties are left as ope problems for the iterested reader. * (1)[x] is oly a pseudo MOD rig of decimal polyomials. For if q*(x) = 0.73x x x ad p*(x) = 0.53x x x are i the we see * (1)[x]; p*(x) + q*(x) = (0.73x x x ) + (0.53x x x ) = (0.73x x 0.25 ) + (0.921x x 0.7 ) x x ( ) = 0.26x x x x (1)[x]. We see with respect to additio closure axiom is true. fact p*(x) + q*(x) = q*(x) + p*(x). That is additio is commutative. Further for p*(x), q*(x), r*(x) We see (p*(x) + q*(x)) + r*(x) * * (1)[x]

90 MOD Fuzzy eal Plae 89 = p*(x) + (q*(x) + r*(x)). That is the property of additio of MOD decimal polyomials is associative. Fially 0 [0, 1) acts as the additive idetity of the MOD decimal polyomial. Further to each p*(x) * (1)[x] * we see there exists a uique s*(x) (1)[x] such that p*(x) + s*(x) = 0. That is s*(x) is the uique MOD decimal polyomial iverse of p*(x) ad vice versa. * Thus ( (1)[x], +) is a ifiite MOD decimal polyomial group of ifiite order. Now we see how the operatio acts o * (1)[x]. Before we defie we just show for ay p*(x) the additive iverse of it i (1)[x]. * Let p*(x) = 0.378x x x x (1)[x]. The additive iverse of p*(x) is p*(x) = 0.622x x x x 0.5 * (1)[x] such that p*(x) + ( p*(x)) = 0. Let p*(x) = 0.2x x x ad q*(x) = 0.9x x x 0.7 * be i (1)[x]. To defie product for these two MOD decimal polyomials. *

91 90 MOD Plaes p*(x) q*(x) = (0.2x x x ) (0.9x x x ) = 0.18x x x x x x x x x x x x x = 0.843x x x x x x x x (1)[x]. t is easily verified p*(x) q*(x) = q*(x) p*(x) for all p*(x), q*(x) i (1)[x]. We ext show whether p*(x) (q*(x) r*(x)) is equal to (p*(x) q*(x)) r*(x). Let p*(x) = 0.7x x q*(x) = 0.8x ad r*(x) = 0.2x x (1)[x]. We fid [p*(x) q*(x)] r*(x)] = [(0.7x x ) (0.8x ) (0.2x x ) = (0.56x x x x x ) (0.2x x ) = (0.112x x x x x x x x x x x x x x x x ) * * *

92 MOD Fuzzy eal Plae 91 = 0.04x x x x x x x x x Cosider p*(x) [q*(x) r*(x)] p*(x) [(0.8x ) (0.2x x )] = [0.7x x ] [0.16x x x x ] = 0.112x x x x x x x x x x x x x x x x = 0.040x x x x x x x x x ad are ot distict so the operatio is associative. Let p*(x) = 0.9x 0.7, q*(x) = 0.6x 0.9 ad r*(x) = 0.12x 0.3 (1)[x]. p*(x) [q*(x) r*(x)] = p*(x) [0.072x 0.2 ] = 0.9x x 0.2 * = x 0.9 [p*(x) q*(x)] r*(x)

93 92 MOD Plaes ad are equal. = (0.9x x 0.9 ) 0.12x 0.3 = 0.54x x 0.3 = x 0.9 a (b c) = a (b c) for a, b, c [0, 1). So operatio o * (1)[x] is always associative. fact p*(x) [q*(x) + r*(x)] p*(x) q*(x) + p*(x) r*(x). Take p*(x) = 0.5x 0.8 q*(x) = 0.7x 0.4 ad r*(x) = 0.3x 0.4 be the MOD decimal polyomials i (1)[x]. * Cosider p*(x) [q*(x) + r*(x)] = p*(x) [0.7x x 0.4 ] = p*(x) 0 = 0. Cosider p*(x) q*(x) + p*(x) r*(x) = 0.5x x x x 0.4 = 0.35x x 0.2 = 0.5x 0.2 ad are distict so the distributive law is ot i geeral. Let us cosider s d (1)[x]. (1)[x] a rig or oly a pseudo rig? d d (1)[x] = i aix a i [0, 1)}. i= 0 Let p d (x) = 0.3x x

94 MOD Fuzzy eal Plae 93 ad q d (x) = 0.7x x d (1)[x]. p d (x) + q d (x) = 0.3x x x x (1)[x]. We see + is a commutative operatio with 0 as the additive idetity. For every polyomial p d (x) q d (x) (1)[x] such that p d (x) + q d (x) = (0). d d d (1)[x] we have a uique Let p d (x) = 0.71x x x (1)[x]. q d (x) = 0.29x x x such that p d (x) + q d (x) = 0. Clearly q d (x) = p d (x). Thus abelia. d d (1)[x] is d (1)[x]uder + is a group of ifiite order which is Let p d (x) = 0.7x x ad q d (x) = 0.1x x d (1)[x] we see p d (x) q d (x) = (0.7x x ) (0.1x x + 0.4) = 0.07x x x x x x x x d (1)[x], this is the way operatio is performed o d (1)[x] We see for p d (x) = 0.3x x q d (x) = 0.4x ad r d (x) = 0.1x 6 d (1)[x].

95 94 MOD Plaes We fid [p d (x) q d (x)] r d (x) Now [p d (x) q d (x)] r d (x) = [(0.3x x + 0.1) [0.4x )] r d (x) = [0.12x x x x x ] 0.1x 6 = 0.012x x x x x x 7 Cosider p d (x) (q d (x) r d (x)) = p d (x) [0.4x x 6 ] = p d (x) [0.04x x 6 ] = (0.3x x + 0.1) (0.04x x 6 ) = 0.012x x x x x x 6 ad are idetical ifact the operatio is associative o (1)[x]. d Now let us cosider p d (x) = 0.3x , q d (x) = 0.5x ad r d (x) = x x 5 p d (x) (q d (x) + r d (x)) d (1)[x]. = p d (x) [0.5x x x ]

96 MOD Fuzzy eal Plae 95 p d (x) q d (x) + p d (x) r d (x) = (0.3x ) [0.3x x 8 ] = 0.09x x x x x 8 = (0.3x ) (0.5x ) + (0.3x ) ( x x 5 ) = 0.15x x x x x x x x 5 d Clearly ad are distict so we see ( (1)[x], +, ) is ot a rig oly a pseudo MOD polyomial rig as the distributive law is ot true. Thus both d (1)[x] ad * (1)[x] are oly pseudo rigs. fact study of properties o MOD fuzzy plae would be a substatial work for ay researcher. d * Thus both (1)[x] ad (1)[x] ejoy differet properties relative to differetiatio ad itegratio of polyomials. * For i (1)[x] usual itegratio ad differetiatio caot be defied; ifact we have ifiite umber of differetials ad * itegrals for a give MOD decimal polyomials i (1)[x] ever exists. fact for every a (0, 1) we have a MOD derivative ad a MOD itegral ot possible. We call this structure as MOD for two reasos oe they are made up of small values ad secodly they do ot have the usual itegrals or derivatives of the fuctios.

97 96 MOD Plaes Defiig equatios of circles, ellipse, hyperbola, parabola or for that matter ay coic o the MOD fuzzy plae happes to be a challegig problem for ay researcher. fact cotiuous curves i the real plae happes to be discotiuous parallel lies i the MOD fuzzy plae. fact they have zeros i certai cases. Sometimes a cotiuous lie ad so o. We suggest some problems. Problem 1. Obtai ay special ad iterestig feature about MOD fuzzy plae. 2. What is the strikig property ejoyed by the MOD fuzzy plae? 3. s the MOD fuzzy plae orderable? 4. Fid the curve y = 0.5x 2 i the MOD fuzzy plae. s the fuctio cotiuous i (1)? 5. Draw the fuctio y = 0.7x +0.4 i the MOD fuzzy plae (1). 6. Draw the fuctio y = 0.8x x o the MOD fuzzy plae (1). (i) s this fuctio cotiuous or discotiuous? (ii) Does the fuctio have zeros? (iii) s this fuctio derivable? 7. Plot the fuctio y 2 = 0.4x o the MOD fuzzy plae (1). 8. What are the advatages of usig fuctios i the fuzzy MOD plae?

98 MOD Fuzzy eal Plae s it possible to defie the distace cocept o (1)? 10. Plot the fuctio 0.7x y 2 = 0.2 o the MOD fuzzy plae (1). (i) Will this be a circle i (1)? Justify your claim. (ii) s this fuctio a lie? 11. Let 0.5x x 2 = 0.3 be a fuctio i the MOD fuzzy plae (1). (i) Ca we say the resultat is a curve? (ii) Will the graph be a straight lie or a curve? (iii) s this fuctio a cotiuous curve or a lie? 12. Ca (1), the fuzzy MOD plae be cosidered as the MOD form of the coordiate plae? 13. Ca we have the otio of coics i the MOD fuzzy plae (1)? 14. s it possible to have more tha oe MOD fuzzy trasformatio η 1 from to (1)? d 15. Let [x] be the real polyomials η : [x] to (1)[x] be a map. How may such MOD fuzzy trasformatios exist? 16. Let d [x] be the real polyomials η 1 : d * [x] (1)[x] be a MOD fuzzy real trasformatio. How may such MOD real trasformatios ca be defied? 17. Let p(x) = 0.3x x x (1)[x]. (i) Fid dp(x) ad p(x) dx. dx (ii) s the itegratio the reverse process of differetiatio or vice versa? (iii) Does itegrals i d (1)[x] exist? d

99 98 MOD Plaes 18. Let p(x) = 0.9x x x (i) Fid dp(x) dx(0.2). (ii) Fid p(x) dx (0.2), does it exist? (iii) s d p(x)dx(0.2) = p(x) possible? dx(0.2) * (1)[x]. 19. Let p(x) = 0.27x x x 0.2 d (1)[x]. Study questios (i) to (iii) of problem 18 for this p(x). 20. Let p(x) = 0.312x x x d (1)[x].Take α = Study questios (i) to (iii) of problem 18 for this p(x) with α =0.007 (0, 1). 21. Let (1)[x] be the MOD decimal fuzzy polyomial plae. d d (i) Prove o (1)[x] we ca defie ifiite umber of derivatives ad o itegrals. (ii) Let p(x) = 0.72x x x (1)[x]; does there exist α (0, 1) such that d dp(x) dx( α ) =0 ad p(x)dx (α) 0? (iii) Fid p(x) dx (0.32), does it exist? (iv) Fid dp(x) dx(0.32). * 22. Let (1)[x] be the MOD fuzzy decimal polyomials. (i) Study questio (i) to (iv) problem 21 for p(x) with respect α = (ii) Compare the results. 23. Obtai some special features ejoyed by d (1)[x].

100 MOD Fuzzy eal Plae Obtai the special features associated with 25. s 26. s d (1)[x] a group with respect to +? d (1)[x]a semigroup with respect to? 27. s (1) a rig? * (1)[x]. 28. Does the MOD fuzzy plae (1) cotai zero divisors? 29. Ca the MOD fuzzy plae (1) cotai idempotets? 30. Ca the MOD fuzzy plae (1) cotai S-zero divisors? 31. s the MOD fuzzy plae (1) be a semigroup with zero divisors? 32. Compare the MOD fuzzy plae (1) with * (1)[x]. d (1)[x] ad 33. Ca 34. Let * (1)[x] cotai zero divisors? (1)[x] be the MOD fuzzy plae. * Let y = 0.3x x + 1 Draw the graph of y o * (1)[x]. * (1) 35. Obtai some special features ejoyed by d [x]. 36. Compare d (1)[x] with * (1)[x]. 37. What ca be the practical uses of itegratio ad differetiatio usig α (0, 1)? 38. Ca we defie for every α + \ {0} the ew type of differetiatio o [x]? What is the problem faced while defiig it?

101 100 MOD Plaes 39. Draw the graph of the fuctio y = 0.3x i the MOD fuzzy plae (1)? 40. Obtai a method to solve polyomial equatios i 41. Show if α is a multiple root of p(x) geeral is ot a root of p (x). d (1)[x]. d (1)[x] the α, i 42. Show that there exists a lot of differece betwee MOD fuzzy differetiatio ad usual differetiatio. 43. Show that there exists differece betwee usual itegratio ad MOD fuzzy itegratio. 44. What are the advatages of usig derivatives with every α (0, 1) i p(x) (1)[x]? * 45. What is the justificatio i usig the derivatives of p(x) the power of x decreases by 1 ad powers of x icreases by 1 i itegrals? 46. What is the draw back i replacig 1 by ay real? 47. s every fuctio which is cotiuous i the real plae cotiuous i the MOD fuzzy plae (1)? 48. Ca we give represetatio for trigoometric fuctios i the MOD fuzzy plae? Justify your claim. 49. Obtai ay iterestig features ejoyed by the MOD fuzzy plae (1). 50. What are the special properties associated with d [x]? 51. What are the special features ejoyed by the MOD fuzzy decimal polyomials (1)[x]? *

102 MOD Fuzzy eal Plae 101 * 52. f p(x) (1)[x] does there exist distict α, β (0, 1) dp (x) dp (x) (α β) but = dx( α ) dx( β )? 53. Ca we say if α i, α j (0, 1) are distict the p(x) dx (α i ) p(x) dx (α j ) for every p(x) (1)[x] α i α j? 54. s it possible to defie step fuctios i the MOD fuzzy plae (1)? 55. Ca we say for every p(x) * (1)[x] there exists a * q(x) (1)[x] such that dp(x) dx( α i ) = dq(x) dx( α j) α i α j, α i, α j (0, 1)? 56. What is the graph of y = 0.001x 4 i the MOD fuzzy plae (1)? 57. What is the graph of y = 0.01x 4 i the MOD fuzzy plae (1)? Compare the graphs i problems 56 ad Fid the graph of y = 0.1x 4 i the MOD fuzzy plae. 59. Does the graph y = 0.5x 5 defied i (1) have 5 zeros? Justify. 60. y = 0.5x 2, y = 0.5x 3, y = 0.5x ad y = 0.5x 4 be four distict fuctios o the MOD fuzzy plae. Compare the MOD fuzzy graphs of the four fuctios? 61. Let f(x) = 0.9x x x x (1)[x]. * (i) Fid df (x) dx(0.8) * ad f(x) dx (0.8).

103 102 MOD Plaes (ii) Fid (iii) Fid (iv) Fid df (x) dx(0.6) df (x) dx(0.5) df (x) dx(0.3) ad f(x) dx (0.6). ad f(x) dx (0.5). ad f(x) dx (0.3). df (x) (v) Fid ad f(x) dx (0.120). dx(0.120) Show oe of the itegrals are defied. 62. s it possible to defie the otio of circle i a MOD fuzzy plae? 63. Ca distace cocept be adopted o a MOD fuzzy plae? 64. s a MOD fuzzy plae Euclidea? 65. Ca a metric be defied over the MOD fuzzy plae? 66. How to solve equatios i * (1)[x]? 67. f p(x) = 0.33x x 0.1 * (1)[x]. (i) s p(x) solvable? (ii) Ca p(x) have more tha two roots? (iii) Does the roots of p(x) belog to [0, 1)? 68. Ca we say 0.5 th degree polyomials ca have five roots? 69. Ca a MOD quadratic polyomial be defied i 70. f p(x) = 0.7x x is a special quadratic i 0.3 that is * (1)[x]? * (1)[x] ca we say p(x) 2 (0.3) 0.7x x ?

104 MOD Neutrosophic Plaes 103 Chapter Four MOD NEUTOSOPHC PLANES AND MOD FUZZY NEUTOSOPHC PLANES this chapter we itroduce the otio of MOD eutrosophic plaes ad MOD fuzzy eutrosophic plaes. This is very importat as the cocept of idetermiacy plays a vital role i all types of problems. ecall is called the idetermiacy defied ad developed by Floreti Smaradache i [3, 4] ad 2 = which will deote the eutrosophic or the idetermiacy cocept. For more about this cocept refer [3, 4]. We deote the MOD eutrosophic plae by m (m) = {a + b a, b [0, m)} = {(a, b) a is the real part ad b is the eutrosophic part}, m 1. The MOD eutrosophic plae is represeted as

105 104 MOD Plaes Figure 4.1 Whe m = 1 that is (1) is defied as the MOD eutrosophic fuzzy plae. We will give examples of them ad work with them. Example 4.1: Let (5) be the MOD eutrosophic plae associated with the MOD iterval [0, 5) Figure 4.2

106 MOD Neutrosophic Plaes 105 We see how represetatios of the MOD eutrosophic umbers are made i the MOD eutrosophic plae. ecall = {a + b a, b, 2 = } is the real eutrosophic plae. We ca get the map of ito (m) i the followig way. We kow m). (m) is built usig the itervals [0, m) ad [0, We show how the MOD eutrosophic trasformatio works. η : (m) ad η () =, η (t) = t if t < m η (t) = r where r m + s = t m if t > m. η (t) = 0 if t = m. η (t) = t if t < m; = r if t > m ad t m = s + r m = 0 if t = m. This is the way by the MOD eutrosophic trasformatio the eutrosophic real plae is mapped o to the MOD eutrosophic plae (m). We will illustrate this by some examples. Let

107 106 MOD Plaes η ( ) = Let η ( ) = ( ) Let ; η ( ) = 0 (5). η ( 72 49) = (3 + ) η (24 71) = (4 + 4) (5). (5). (5). (5). This is the way MOD eutrosophic trasformatio from to (5) is made. Let (12) be the MOD eutrosophic plae. η ( ) = 4 η ( ) = 3 η (10 + 4) = η ( 2) = 10 η ( 4) = 8 (12). (12). (12) (12) η ( 5 20) = (7 + 4) (12) (12) ad so o. fact we have oly a uique MOD eutrosophic trasformatio from to (m).

108 MOD Neutrosophic Plaes 107 Let us ow study the behaviour of fuctios o the eutrosophic plae. Let y = (8 + 5)x be a fuctio defied o the real eutrosophic plae. Now this fuctio is defied o (6)[x] as y = (2 + 5)x. Thus if [x] is the real eutrosophic polyomial collectio the we ca map every p(x) to p (x) (m) [x] = i aix a i i= 0 (m)}. We will illustrate this situatio by some simple examples. Let p(x) = 5.3x x x [x] η : [x] (4) [x]. η (p(x)) = η (5.34x x x 2 + 3) = 1.34x x (4). Thus we ca say several p(x) is mapped o to oe. p (x) (4). Let p(x) = 246x x x 10 5x 8 + 6x 2 + 5x 17 [x]. η (p(x)) = 2x x x x 8 + 2x 2 + x + 3 (4). This is the way MOD eutrosophic trasformatios are carried out.

109 108 MOD Plaes Oe of the atural questios ca be differetial of p (x) (4)[x]. The aswer is yes. Let p (x) = (1 + 3)x 8 + ( )x x x (4). dp (x) dx = 8 (1+3)x ( )x (1.6)x = 0 + ( )x x Now we ca also have polyomials of the form (4). p(x) = ( )x 5 + ( )x 4 + (13 43)x 3 + 7x 2 + 2x + (17 26) [x]. η 1 (p(x)) = x 5 + (2 + 3) x 4 + (1 + 2)x 3 + 3x 2 + 2x + (1 + 2) (4). This is the way the MOD trasformatio of polyomials are carried out. Let (7)[x] be the MOD eutrosophic polyomial. Let p(x) = (40 + )x 18 + (43 14)x 15 + (25 + 7)x 6 + ( )x 5 + ( 3 14)x 3 + ( )x [x]. Let η : [x] (7)[x] be the MOD eutrosophic trasformatio. η (p(x)) = (5 + )x x x 6 + (5 + 5)x 5 + (4)x 3 + (5 + 3)x + (3 + 6) (7)[x].

110 MOD Neutrosophic Plaes 109 We ca thus see this trasformatio is such that ifiite umber of eutrosophic real umbers are mapped oto a sigle MOD eutrosophic elemet i (m)[x]. Let p(x) = 3.17x 10 + ( )x 8 + ( )x 5 + ( )x 3 + ( )x + ( ) [x]. We defie η : [x] every coefficiet is less tha 6. We fid the derivative of p(x). (6)[x] as η (p(x)) = p(x) as Now dp(x) dx = x ( )x 7 + 5( ) x 4 + 3( )x 2 + ( ) = 1.7x 9 + ( )x 7 + ( )x 4 + ( ) x 2 + ( ) (6). t is importat to ote that if p(x) is a polyomial i the MOD eutrosophic plae of degree say s (s > 1), the the first derivative eed ot be of degree (s 1); it ca be zero or a lesser power tha s 1. All these will be illustrated by some examples. Let p(x) = 2x x (5)[x]. Clearly degree of p(x) is 10 but the first derivative of p(x) is dp(x) dx = 20x x = 0.

111 110 MOD Plaes Thus the first derivative of the MOD polyomial p(x) (5)[x] is zero. Let p(x) = 2.5x x x We ow fid the derivative of p(x), (5)[x]. dp(x) dx = x x x = 0. Let p(x) = 4x x (5)[x]. dp(x) dx = 20x x + 0 = 4.62x (5)[x]. Thus p(x) is a degree 5 ad the first derivative is of degree oe. So the fifth degree polyomial i derivative to be of degree oe. (5)[x] has its first Let p(x) (4)[x] we ca fid p (x) which ca be zero or a costat or of degree much less tha degree p(x) 1. p(x) = 0.3x x x x (4)[x]. dp(x) dx = 1.8x x x (4)[x]. Degree of p(x) is 6 ad its first derivative of p(x) which is p (x) is of degree 5.

112 MOD Neutrosophic Plaes 111 Let p(x) = 2x 5 + 4x (4)[x]; dp(x) dx = 2x 4. Thus a fifth degree polyomial is a costat. So polyomials i (m)[x] ad i [x] behave differetly for a sth degree polyomial i [x] has its derivative to be of degree (s 1). But a sth degree polyomial i (m)[x] may have its first derivative to be zero or costat or of degree s k (k 1) is possible. Next we have the mai result which is based o the roots of the polyomial i (m)[x]. t is pertiet to keep o record that i geeral it is very difficult to solve polyomial equatios i (m)[x] (m > 1). Thus it is left as a ope cojecture to solve the followig problems. (i) (ii) (iii) (iv) Ca we say a sth degree polyomial i has s ad oly s roots? Ca we have a sth degree polyomial i to have more tha s roots? Ca sth degree polyomial i tha s roots? f α is a multiple root of s(x) say α is a root of ds(x) dx? (m)[x] (m)[x] (m)[x] has less (m)[x]; ca we All the four problems are left as ope cojectures.

113 112 MOD Plaes Let s(x) = (x + 2.1) (x +1) 2 (x + 3) (4)[x]. The roots i the usual sese are 3, 3 1 ad 1.9 [0, 4). s(x) = (x + 2.1) (x 2 + 2x + 1) (x+3) or = (x 3 + 2x 2 + x + 2.1x x) (x + 3) = (x 4 + 2x 3 + x x x + 0.2x 2 + 3x 3 + 2x 2 + 3x + 2.3x x) = x 4 + ( )x 3 + ( )x 2 + ( )x s(x) = x x x x s(3) = = = = But if s(x) = (x + 2.1) (x+1) 2 (x+3) s(3) = ( ) (3 + 1) 2 (3 + 3) = 0. Thus oe may be surprised to see 3 is a root of s(x) = (x + 2.1) (x + 1) 2 (x+3). But whe the same s(x) is expaded s(3) 0. So we ow claim this is due to the fact i geeral i (m)[x] the distributive law i ot true. (m)

114 MOD Neutrosophic Plaes 113 That is why studyig the properties of MOD eutrosophic polyomials i (m)[x] is a difficult as well as it is a ew study i case of polyomials. That is why we have proposed those ope problems. Now so we ca thik of itegratig the MOD eutrosophic polyomials i (m)[x] to see the problems we face i this adveture. Let us to be more precise take p(x) = 4.2x x x + 3 (6)[x]. p(x) dx = 4.2x 5 dx + 3.2x 4 dx + 2.5xdx + 3dx = udefied x 2.5x + + 3x + c. 5 2 Clearly i [0, 6) x 2 is ot defied as 2 does ot have iverse Thus i geeral itegratio is ot defied for every p(x) (m)[x]. So itegratio i case of MOD eutrosophic polyomials i geeral is ot a easy task. Now ca we defie o the MOD eutrosophic plae the otio of coics? This also remais as a ope problem. Now we study the algebraic structure ejoyed by (m).

115 114 MOD Plaes We will first give some examples before we proceed oto prove a few theorems. Example 4.2: Let (7) = {a + b a, b [0, 7), 2 = } be the MOD eutrosophic plae. Let ( ) ad ( ) + ( ) = = (7). t is easily verified 0 = for 0 + x = x + 0 for all x Let x = y = (7). (7) we see (7). (7) is such that x + y = 0. (7) acts as the idetity, Thus to every x (7) we have a uique y (7) such that x + y = 0 ad y is called the additive iverse of x ad vice versa. Thus ( (7), +) is a abelia group with respect to additio kow as the MOD eutrosophic plae group. DEFNTON 4.1: Let (m) be the MOD eutrosophic plae. (m) uder + (additio) is a group defied as the MOD eutrosophic plae group. Example 4.3: Let ( (12); +) be a MOD eutrosophic group. Clearly (12) is of ifiite order ad is commutative.

116 MOD Neutrosophic Plaes 115 Example 4.4: Let { (19), +} be a MOD eutrosophic group of ifiite order. Next we proceed oto defie the otio of product i (m). DEFNTON 4.2: Let (m) be the MOD eutrosophic plae relative to the MOD iterval [0, m). (m); Defie i (m) as for x = a + b ad y = c + d x y = (a + b) (c + d) = ac + bc + ad + bd is i (m). = ac + (bc + ad + bd) (mod ) Thus { (m), } is a semigroup which is commutative ad is of ifiite order. { (m), } is defied as the MOD eutrosophic plae semigroup. We will give some examples of this i the followig. Example 4.5: Let S = { (10), } be the MOD plae eutrosophic semigroup. Let x = ad y = S; x y = (4 + 8) (5 + 5) = 0.

117 116 MOD Plaes Thus S has zero divisors. Let x = 5 (10). x 2 = x. Thus the semigroup S has idempotets. The MOD eutrosophic semigroup has both zero divisors ad idempotets. Example 4.6: Let { (17), } be the MOD eutrosophic semigroup. Let x = 8.5 ad y = be i (17). x y = 0. Example 4.7: Let { (20), } = S be the MOD eutrosophic semigroup. S has zero divisors, uits ad idempotets. x = 3 ad y = 7 S is such that x y = 1 (mod 20). x = 9 S such that x 2 = 1 (mod 20). Example 4.8: Let { (25), } = S be the MOD eutrosophic semigroup. We see P = {a + b a, b {0, 5, 10,15, 20} [0, 25)} is a subsemigroup of fiite order. terested reader ca fid subsemigroups. Example 4.9: Let M ={ (127), } be the MOD eutrosophic plae semigroup. M has subsemigroups of fiite order. We see (m) is a group uder + ad semigroup uder, however { (m), +, } is ot a rig. The mai problem beig the distributive law is ot true. For a (b + c) a b + a c for all a, b, c (m).

118 MOD Neutrosophic Plaes 117 Example 4.10: Let (6) be the MOD eutrosophic plae. (6) is a MOD eutrosophic plae group ad eutrosophic plae semigroup. Let x = , y = ad z = (6). (6) is a MOD Cosider x (y + z) = ( ) [( ) + ( )] = ( ) (0) = 0 Now x y + x z = ( ) ( ) + ( ) ( ) = ( ) + ( ) = = Clearly ad are distict hece the claim. That is the distributive law i geeral is ot true i (m). DEFNTON 4.3: Let { (m), +, } be defied as the pseudo MOD eutrosophic plae rig which is commutative ad is of ifiite order. We give some more examples of this situatio.

119 118 MOD Plaes Example 4.11: Let { (7), +, } = be the MOD eutrosophic plae pseudo rig. is commutative ad is of ifiite order. Take x = y = ad z = (7). x (y + z) = ( ) ( ) Now we fid = ( ) ( ) = ( ) = x y + x z = = ( ) = Clearly ad are distict, hece the distributive law i geeral is ot true i (7). Now we see that is why we had problems i solvig equatios. ecall (m)[x] = eutrosophic polyomials. i aix a i i= 0 (m)} is the MOD The atural iterest would be the highest structure ejoyed by this MOD eutrosophic polyomials (m) [x].

120 MOD Neutrosophic Plaes 119 Example 4.12: Let (9)[x] be the MOD eutrosophic polyomials. (9)[x] uder additio is a group of ifiite order which is abelia. For take We fid p(x) = 0.8x 7 + 5x x ad q(x) = 3.4x x x be i (9)[x]. p(x) + q(x) = 0.8x 7 + 5x x x x x = 4.2x 7 + 5x x x (9)[x]. This is the way the operatio of additio is performed. Clearly + operatio is commutative for p(x) + q(x) = q(x) + p(x) ca be easily verified. Now 0 (9) [x] is such that p(x) + 0 = 0 + p(x) = p(x). We see for every p(x) i q(x) (9)[x] such that p(x) + q(x) = 0. (9)[x] there is a uique For let p(x) = 0.321x x x x x x (9)[x]. Now q(x) = 8.679x x x x x x i (9)[x] is such that p(x) + q(x) = 0. q(x) is the uique additive iverse of p(x) ad verse versa.

121 120 MOD Plaes Thus (9)[x] is a abelia group uder +. Example 4.13: Let ( (5)[x], +) be the MOD polyomial eutrosophic group. For every p(x) = 0.43x x x x the iverse of p(x) is deoted by p(x) = 4.57x x x x is such that p(x) + ( p(x)) = 0. (5)[x]; (5)[x] DEFNTON 4.4: { (m)[x], +} is defied to be the MOD polyomial group of ifiite order which is commutative. Example 4.14: Let polyomial set. follows. (12)[x] be the MOD eutrosophic Defie the operatio o Let p(x) = 6.16x x x ad q(x) = 3.1x x (12)[x]. p(x) q(x) = (6.16x x x ) (3.1x x 2 + 7) (12)[x] as = 7.096x x x x x x x x x x x 5 = 7.096x x x x x x x x x (12)[x]. This is the way product is obtaied. t is easily verified that p(x) q(x) = q(x) p(x).

122 MOD Neutrosophic Plaes 121 Further (m)[x] for ay m is a commutative semigroup of ifiite order. This semigroup is defied as the MOD eutrosophic polyomial semigroup. Next we proceed oto give examples of pseudo MOD eutrosophic rigs of polyomials. Example 4.15: Let (10)[x] be the MOD eutrosophic polyomial. ( (10)[x], +) is a group ad ( (10)[x], ) is a semigroup. Let p(x) = 0.3x x , q(x) = 0.4x x + 1 ad r(x) = 5x x Cosider p(x) (q(x) + r(x)) = (10) [x]. = (0.3x x ) [0.4x x x x ] = 0.12x x x x x x +0.3x x x x x x x x x x = 1.5x x x x x x x x x x x x Cosider p(x) q(x) + p(x) r(x) = (0.3x x ) (0.4x x + 1) + (0.3x x ) (5x x )

123 122 MOD Plaes = 0.12x x x x x x + 0.3x x x x x x x x x x 3 = 1.5x x x x x x x x t is easily verified ad are ot distict for this set of MOD polyomials. Let us cosider p(x) = 0.3x 4 + 8, q(x) = 4x 3 + 8x ad r(x) = 6x p(x) [q(x) + r(x)] (10)[x]. = (0.3x 4 + 8) [4x 3 + 8x x ] = (0.3x 4 + 8) (8x + 0.5) = 2.4x 5 + 4x x p(x) q(x) + p(x) r(x) = (0.3x 4 + 8) (4x 3 + 8x + 0.1) + (0.3x 4 + 8) (6x ) = 1.2x 7 + 2x x x x 7 + 4x x = 3x x 4 + 6x Clearly ad are distict so for this triple p(x), q(x), r(x) (10)[x];

124 MOD Neutrosophic Plaes 123 p(x) (q(x) + r(x)) p(x) q(x) + p(x) r(x). Thus ( (m)[x], +, ) is oly a MOD eutrosophic polyomial pseudo rig ad is ot a rig for the distributive law i geeral is ot true. We proceed to defie the MOD pseudo eutrosophic polyomial rig. DEFNTON 4.5: Let (m)[x] be the MOD eutrosophic polyomial, ( (m)[x], +) is a abelia group uder +. ( (m)[x], ) is a commutative semigroup. Clearly for p(x), q(x), r(x) (m)[x] we have p(x) [q(x) + r(x)] p(x) q(x) + p(x) q(x) i geeral for every triple. Hece S = ( (m)[x], +, ) is defied as the MOD eutrosophic polyomial pseudo rig. We see S is of ifiite order ad S cotais idempotets ad zero divisors ad is commutative. We will give oe to two examples of them. Example 4.16: Let { (7)[x], +, } be the MOD pseudo eutrosophic polyomial rig. Example 4.17: Let { (12)[x], +, } be the MOD pseudo eutrosophic polyomial rig. Now we see solvig equatios i (m)[x] is a very difficult task. For eve if we cosider s(x) = (x α 1 ) (x α 2 ) (x α ) x (α α ) x 1 + ± α 1 α (mod m) = s 1 (x). Thus this leads to a major problem for s (α i ) = 0 but s 1 (α i ) 0.

125 124 MOD Plaes So workig with pseudo rigs we ecouter with lots of problems. However i several places the itegrals i (m)[x] is ot defied. Differetiatio does ot i geeral follow the classical properties i (m)[x]. Thus lots of research i MOD eutrosophic polyomials pseudo rigs remais ope. Next we proceed oto defie the ew cocept of MOD fuzzy eutrosophic plae. the MOD eutrosophic plae if we put m = 1 we get the MOD eutrosophic fuzzy plae. We will deote the MOD fuzzy eutrosophic plae by (1) = {a + b a, b [0, 1), 2 = } Figure 4.3

126 MOD Neutrosophic Plaes 125 Now we see (1) is the MOD fuzzy eutrosophic plae. All properties of (m) ca be imitated for (1), however the MOD fuzzy eutrosophic polyomials do ot help i itegratio or differetiatio if m = 1. For if p(x) = 0.3x x 3 (1)[x]; dp(x) dx = 5 0.3x x 2 = 0.5x x 2. But if p(x) = 0.9x x 2 is i (1)[x] the dp(x) dx So we see p(x) is degree 10 but its derivative is zero. However p(x) dx = 0.3x 5 dx + 0.2x 3 dx is ot defied. Let x = ad y = Now x + y = t is easily verified 0 for all x (1). We see for every x that x + y = 0. For x = i x + y = 0. = (1). (1). = 0. (1) is such that x + 0 = 0 + x = x (1) there exists a uique y such (1) we have y = such that

127 126 MOD Plaes { (1), +} is a MOD fuzzy eutrosophic abelia group of ifiite order uder +. Cosider x = ad y = (1). x y = ( ) ( ) = = (1). This is the way product is defied ad { (1), } is defied as the MOD fuzzy eutrosophic semigroup of ifiite order. Let x = 0.4 ad y = 0.8 x y = 0.24 (1). (1); Now this MOD fuzzy eutrosophic semigroup has subsemigroups ad ideals. P = is a fuzzy eutrosophic MOD subsemigroup of { (1), }. Let M = {a a [0, 1)} { (1), } be a ideal; called the fuzzy MOD eutrosophic ideal of { (1), }. Thus the MOD fuzzy eutrosophic semigroup has both ideals ad subsemigroups. Let x = (1)

128 MOD Neutrosophic Plaes 127 the x 2 = ( ) ( ) = = (1). { (1), +, } is oly a pseudo MOD fuzzy eutrosophic rig for distributive law is ot true. Let x = y = ad z = (1). x (y + z) = ( ) = = 0 xy + xz = = = Clearly ad are differet that is why we call { (1), +, } as the MOD fuzzy eutrosophic pseudo rig. ideal of Clearly this pseudo rig has subrigs ad ideals. Cosider P = {a a [0, 1)} (1) of ifiite order. (1), +, }. P is a pseudo = {a a [0, )} is oly a pseudo subrig of ot a ideal of (1). (1) ad is

129 128 MOD Plaes We see MOD fuzzy eutrosophic plaes behave like MOD eutrosophic plaes except some special features like x y = (a + b) (c + d) = 0 is ot possible with a, b, c, d (0, 1) Study of these cocepts is left as a exercise to the reader. Next we proceed oto study the otio of MOD fuzzy eutrosophic polyomials of two types. Let (1)[x] = i aix a i i= 0 Let p(x) = 0.3x x ad q(x) = 0.7x x (1)}. (1)[x]; p(x) + q(x) = 0.3x x x x = 0.3x x x t is easily verified p(x) + q(x) (1)[x]. Further p(x) + q(x) = q(x) + p(x) ad 0 that 0 + p(x) = p(x) + 0 = p(x). Now we show for every p(x) uique q(x) (1)[x] is such (1)[x] there exists a (1)[x] such that p(x) + q(x) = q (x) + p(x) = 0. We call q(x) the iverse of p(x) with respect to additio ad vice versa.

130 MOD Neutrosophic Plaes 129 Thus { (1)[x], +} is a additive abelia group of ifiite order. Let p(x) = 0.9x x x x (1)[x]. We see q(x) = 0.1x x x x (1)[x] is such that p(x) + q(x) = 0. Cosider p(x), q(x) p(x) q(x) ad p(x) q(x) (1)[x] we ca fid product (1)[x]. Let p(x) =0.31x x ad q(x) = 0.9x x be i (1)[x]. p(x) q(x) = (0.31x x ) (0.9x x ) = 0.279x x x x x x x x (1)[x]. All these happes whe the coefficiets are real. Let p(x) = ( )x 10 + ( )x 3 + ( ) ad q(x) = ( )x 3 + ( ) (1)[x]. p(x) + q(x) = ( )x 10 + ( )x 3 + ( ) + ( )x 3 + ( )

131 130 MOD Plaes = ( )x 10 + ( )x 6 + ( ) is i (1)[x]. p(x) q(x) = [( )x 10 + ( )x 3 + ( )] ( )x 3 + ( ) = ( ) ( )x 13 + ( ) ( )x 6 + ( ) ( )x 3 + ( ) ( )x 10 + ( ) ( )x 3 + ( ) ( ). = ( )x 13 + ( )x 6 + ( )x 3 + ( )x 10 + ( )x 3 + ( ) = ( )x 13 + ( )x 6 + ( )x 3 + ( )x 10 + ( )x 3 + ( ) = )x 13 + ( )x 6 + ( )x 10 + ( )x 3 + ( ) is i (1)[x]. { (1)[x], } is a semigroup of ifiite order which is commutative. We see { (1)[x], +, } is ot a MOD fuzzy eutrosophic polyomial rig oly a pseudo rig; for p(x) (q(x) + r(x)) p(x) q(x) + p(x) r(x) for p(x), q(x), r(x) Let p(x) = 0.3x x , q(x) = 0.5x x ad (1)[x].

132 MOD Neutrosophic Plaes 131 r(x) = 0.9x x Cosider p(x) (q(x) + r(x)) (1)[x]. = p(x) [0.5x x x x ] = (0.3x x ) [0.9x x] = 0.27x x x x x x We fid p(x) q(x) + p(x) r(x) = 0.3x x x x x x x x = 0.15x x x x x x x x x x x x x x x x = 0.27x x x x x x x x x x x We see ad are distict hece i geeral p(x) (q(x) + r(x)) p(x) q(x) + p(x) r(x); for p(x), q(x), r(x) (1)[x]. Thus { (1)[x], +, } is a MOD fuzzy eutrosophic polyomial pseudo rig of ifiite order.

133 132 MOD Plaes We say we caot perform usual itegratio o p(x) (1)[x]. However differetiatio ca be performed for some fuzzy eutrosophic polyomials, it may be zero. i Let p(x) = 0.7x x x x x be (1)[x]. We fid the derivative of p(x). dp(x) dx = 0.4x x x is i (1)[x]. The secod derivative of p(x) is 2 d p(x) dx 2 = 0.4x x is i (1)[x]. The third derivative of p(x) is 3 d p(x) dx 3 = 0.2x. Thus a twelfth degree MOD fuzzy polyomial has its third derivative to be a polyomial of degree oe ad the forth 4 d p(x) derivative is a costat viz = 0.2 ad the fifth 4 dx 5 d p(x) derivative of p(x); = 0. Thus a 12 th degree polyomial 5 dx i (1)[x] has its fifth degree to be zero. Hece these derivatives of MOD fuzzy polyomials do ot behave like the usual derivative polyomials i [x].

134 MOD Neutrosophic Plaes 133 Fially we caot itegrate ay of these polyomials i (1)[x]. To over come this situatio we defie the ew otio of MOD fuzzy eutrosophic decimal polyomials. d (1)[x] = { a i x i a i (1) ad i [0, 1)}. d (1)[x] is defied as the decimal MOD fuzzy eutrosophic polyomials or MOD fuzzy eutrosophic decimal polyomials. Let p(x) = 0.3x x ( )x ( )x ( ) d (1)[x]. We see the powers of x are i [0, 1) ad the coefficiets of x are i (1). We ca add two polyomials i f p(x) = 0.34x x d (1)[x], as follows. ad q(x) = 0.4x x x are i the p(x) + q(x) d = 0.34x x x x x = 0.74x x x (1)[x], d (1)[x]. d This is the way + operatio is performed o (1)[x]. Further 0 acts as the additive idetity ad for every p(x) (1) [x] we have a q(x) i (1)[x] such that d d p(x) + q(x) = 0.

135 134 MOD Plaes Thus ( order. d d (1) [x], +) is a additive abelia group of ifiite Now we ca fid product of two polyomials p(x), q(x) i (1)[x]. Let p(x) = 0.3x x x ad q(x) = 0.9x x be i d (1)[x]. p(x) q(x) = (0.3x x x ) (0.9x x ) = 0.27x x x x x x x x x x = 0.18x x x x (1)[x]. d d Now ( (1)[x], ) is defied as the MOD fuzzy decimal polyomial eutrosophic semigroup of ifiite order. d However ( (1)[x], +, ) is ot the MOD fuzzy decimal polyomial eutrosophic rig as p(x) (q(x) + r(x)) p(x) q(x) + p(x) r(x) i geeral for p(x), q(x), r(x) (1)[x]. d Cosider p(x) = 0.3x x , q(x) = 0.8x x ad r(x) = 0.2x x p(x) (q(x) + r(x)) d (1) [x].

136 MOD Neutrosophic Plaes 135 = p(x) [0.8x x x x ] = 0.3x x = 0.03x x Cosider p(x) q(x) + p(x) r(x) = (0.3x x ) (0.8x x ) + (0.3x x ) (0.2x x ) = 0.24x x x x x x x x x x x x x x x x Clearly ad are distict hece the defiitio of pseudo rig. We ca differetiate ad itegrate i a very special way. We ca ot use usual differetiatio or itegratio istead we use special itegratio or differetiatio usig α (0,1). We will just illustrate this by a example or two. Let p(x) = ( )x ( )x ( ) (1)[x]. We take α = 0.3 we fid d dp(x) dx(0.3) = ( ) 0.7 x ( ) (0.5x 0.2 ) + 0 = ( )x ( ) x 0.2.

137 136 MOD Plaes Similarly p(x) dx = ( )x ( )x We see a sectio of the itegratio is udefied i such cases we say the decimal itegratio does ot exist for α = 0.3. Suppose we decimal itegrate usig α = 0.1 the p(x) dx (0.1) = ( )x ( )x ( )x We see for the same polyomial the decimal itegral is ot defied for α = 0.1 (0, 1). Let p(x) = ( )x x x ( ) d (1)[x]. We fid the decimal derivative ad the decimal itegral of p(x) relative to α = 0.6 (0.1). dp(x) dx(0.6) = ( )0.8x x x 0.6 = ( )x x x 0.6 d (1)[x]. p(x) dx(0.6) = ( )x x x

138 MOD Neutrosophic Plaes 137 ( )x C is defied. This is the way decimal itegratio ad decimal differetiatio are carried out. Study i this directio is iterestig. We propose the followig problems for the reader. Problems : 1. Derive some special features ejoyed by MOD eutrosophic plae. (m) the 2. Prove there exists ifiite umber of distict MOD eutrosophic plaes. 3. Prove solvig equatios i task. (m)[x] is ot a easy 4. Study the properties ejoyed by (5). 5. What is the algebraic structure ejoyed by (8)? 6. Study (m) whe m is a prime. 7. (m) whe m is ot a prime; study the special feature ejoyed by a o prime. 8. Let p(x) = 9x x x x 5 (20) [x]. Fid the derivative of p(x). Does 20 d p(x) dx 20 exist? 9. Ca we itegrate p(x) (20)[x] i problem 8.

139 138 MOD Plaes 10. Fid some special features ejoyed by the MOD eutrosophic semigroups ( (m), ). (i) Fid ideals i (m). (ii) Ca (iii) Ca (m) have ideals of fiite order? (m) have subsemigroups of fiite order? (iv) Ca we have subsemigroups of ifiite order which are ot ideals of (m)? 11. Let S = { (20), } be the MOD eutrosophic semigroup. (i) Study questios (i) to (iv) of problem 10 for this S. (ii) Ca S have S-zero divisors? (iii) Prove S has zero divisors. (iv) Ca S have S-idempotets? (v) Prove S has idempotets. (vi) Ca S have idempotets which are ot S-idempotets? 12. Let S 1 = { (19), } be the MOD eutrosophic semigroup. Study questios (i) to (vi) of problem 11 for this S Let S 2 = { (25), } be the MOD eutrosophic semigroup.

140 MOD Neutrosophic Plaes 139 Study questios (i) to (vi) of problem 11 for this S Obtai ay other special feature ejoyed by the MOD fuzzy eutrosophic group ( (m), +). 15. Let M = { (m), +, } be the MOD eutrosophic pseudo rig. (i) Show M has zero divisors. (ii) Prove M has subrigs of fiite order. (iii) Ca M have ideals of fiite order? (iv) Prove M has subrigs which are ot ideals. (v) Prove M has ifiite order pseudo subrigs which are ot ideals. (vi) Ca M have S-idempotets? (vii) Ca M have S-ideals? (viii) s M a S-pseudo rig? (ix) Ca M have S-zero divisors? (x) Obtai ay other special feature ejoyed by M. 16. Let N = { (29), +, } be the MOD eutrosophic pseudo rig. Study questios (i) to (x) of problem 15 for this N. 17. Let T = { (30), +, } be the MOD eutrosophic pseudo rig. Study questios (i) to (x) of problem 15 for this T.

141 140 MOD Plaes 18. Let M 1 = { (2), +, } be the MOD eutrosophic pseudo rig. Study questios (i) to (x) of problem 15 for this M Let P = { (20), +, } be the MOD eutrosophic pseudo rig. Study questios (i) to (x) of problem 15 for this P. 20. Obtai ay other special feature ejoyed by the MOD eutrosophic pseudo rig { (m), +, }. 21. Let (m)[x] be MOD eutrosophic polyomials. (i) Show all polyomials i differetiable. (ii) Show for several polyomials i itegrals does ot exist. (m)[x] are ot (m)[x] the (iii) f degree of p(x) (m)[x] is t ad t/m will the (a) derivative exist? (b) itegral exist? 22. Let (7)[x] be the MOD eutrosophic polyomial pseudo rig. Study questios (i) to (x) of problem 15 for this (7)[x]. 23. Let (24)[x] be the MOD eutrosophic polyomial pseudo rig.

142 MOD Neutrosophic Plaes 141 Study questios (i) to (x) of problem 15 for this (24)[x]. 24. Let M = polyomial. (15)[x] be the MOD eutrosophic Study questios (i) to (iii) of problem 21 for this M. 25. Let p(x) = 9x x x x x (10)[x]. (i) Fid dp(x) dx. (ii) s p(x) a itegrable fuctio? (iii) How may higher derivatives of p(x) exist? (iv) s every polyomial i (v) Ca every derivative of p(x) i (10)[x] itegrable? 26. Give some special properties ejoyed by (10)[x] exists? (m)[x]. 27. M = ( (m) [x], ) be a MOD eutrosophic polyomial semigroup. (i) Ca M have zero divisors? (ii) Ca M have S-ideals? (iii) s M a S-semigroup? (iv) Ca M have zero divisors? (v) Ca M have zero divisors which are ot S-zero

143 142 MOD Plaes divisors? (vi) Ca M have subsemigroups of ifiite order which are ot ideals? (vii) Ca M have ideals of fiite order? (viii) s deg (p(x)) deg q(x) = deg (p(x) q(x)) i geeral for p(x), q(x) i M? 28. Let N = ( (12)[x], ) be a MOD eutrosophic polyomial semigroup. Study questios (i) to (viii) of problem 27 for this N. 29. Let P = ( (17)[x], ) be a MOD eutrosophic polyomial semigroup. Study questios (i) to (viii) of problem 27 for this P. 30. Let M = ( (m)[x], ) be a MOD eutrosophic polyomial semigroup. (i) Ca M have subgroups of fiite order? (ii) Let p(x) ad q(x) M. p(x) is of degree t ad q(x) of degree s. s deg (p(x) + q(x)) = deg p(x) + deg q(x)? 31. Let M 1 = ( (20)[x], ) be a MOD eutrosophic polyomial semigroup. Study questios (i) to (ii) of problem 30 for this M Let M 2 = ( (47)[x], ) be a MOD eutrosophic polyomial semigroup.

144 MOD Neutrosophic Plaes 143 Study questios (i) to (ii) of problem 30 for this M What is the algebraic structure ejoyed by { (m)[x], +, }? 34. Prove { (m)[x], +, } caot be give the rig structure. (i) Fid a triple i distributive. (ii) Does there exist a triple i distributive? (m)[x] such that they are ot (m)[x] which is 35. Let S = { (27)[x], +, } be the MOD eutrosophic polyomial pseudo rig. Study questios (i) to (ii) of problem 34 for this S. 36. Study questios (i) to (ii) of problem 30 for S 1 = { (29)[x], +, } the MOD eutrosophic polyomial pseudo rig. 37. Let = { (m)[x], +, } the MOD eutrosophic polyomial pseudo rig. (i) Ca have zero divisors? (ii) Ca have S-zero divisors? (iii) Ca be a S-pseudo rig? (iv) Ca have fiite pseudo ideals? (v) Ca have fiite subrigs? (vi) Ca have S-uits? (vii) Ca have fiite pseudo subrigs? (viii) Will the collectio of all distributive elemets i from ay proper algebraic structure? (ix) Ca have S-idempotets?

145 144 MOD Plaes (x) Ca we say itegratio ad differetiatio of all p(x) is possible? 38. Let 1 = { (216)[x]; +, } be the MOD eutrosophic polyomial pseudo rig. Study questios (i) to (x) of problem 37 for this Let 2 = { (29)[x]; +, } be the MOD eutrosophic polyomial pseudo rig. Study questios (i) to (x) of problem 37 for this Study the special features associated with MOD fuzzy eutrosophic polyomials (1)[x]. 41. Study the special features associated with MOD fuzzy eutrosophic plae. (1) the 42. Let { (1), +} be the MOD fuzzy eutrosophic group. (i) Fid subgroups of fiite order. (ii) Fid subgroups of ifiite order. (iii) Give ay special feature ejoyed by (1). 43. Let S = { (1), } be the MOD fuzzy eutrosophic semigroup. (i) Fid ideals of S. (ii) Ca S be a S-semigroup? (iii) Fid S-ideals of S. (iv) Ca S have S-zero divisors? (v) Ca S have S-uits? (vi) Ca S have S-idempotets? (vii) Ca the cocept of uits ever possible i S? (viii) Show ideal i S are of ifiite order. (ix) Ca S have fiite order subsemigroup?

146 MOD Neutrosophic Plaes 145 (x) Obtai ay other iterestig property associated with S. 44. Let P = { (1), +, } be the MOD fuzzy eutrosophic pseudo rig. (i) Study all special features ejoyed by P. (ii) Ca P have zero divisors? (iii) s P a S-pseudo rig? (iv) Ca P have S-zero divisors? (v) Ca P have uits? (vi) Ca P have fiite subrigs? (vii) Ca P have ideals of fiite order? (viii) Ca P have S-ideals? (ix) Metio ay other strikig feature about P. 45. Let S = { (1)[x], +} be the MOD fuzzy eutrosophic polyomial group. Study questios (i) to (iii) of problem 42 for this S. 46. Let S = { (1)[x], } be the MOD fuzzy eutrosophic polyomial semigroup. Study questios (i) to (x) of problem 43 for this S. 47. Let B ={ (1)[x], +, } be the MOD fuzzy eutrosophic pseudo rig. Study questios (i) to (ix) of problem 44 for this B. d 48. Study (1)[x], the MOD eutrosophic fuzzy decimal polyomials. 49. Study for S = { d (1)[x], +} the MOD fuzzy eutrosophic decimal polyomial group questios (i) to (iii) of problem 42 for this S.

147 146 MOD Plaes 50. Let B = { d (1)[x], } be the MOD fuzzy eutrosophic decimal polyomial semigroup. Study questios (i) to (x) of problem 43 for this B. 51. Let M = { d (1)[x], +, } be the MOD fuzzy eutrosophic decimal polyomial pseudo rig. Study questios (i) to (ix) of problem 44 for this M. d 52. Show i case of (1)[x] for ay p(x) ca have ifiite umber of d (1)[x] we (i) decimal differetiatio. (ii) decimal itegratio. (iii) f α = β, α, β (0, 1) ca we have dp(x) = dp(x) dx( α) dx( β) d for some p(x) (1) [x]? (iv) Show for α β, α, β (0, 1) ca we have a d p(x) (1) [x] such that p(x) dx(α) = p(x) dx (β)? (v) Ca we have a p(x) (1)[x] for which dp(x) dx( α) = 0 d for a α (0, 1)? d (vi) Ca we have a p(x) (1)[x] for which p(x) dx(α) is udefied for a α (0, 1)? 53. What are advatages of usig decimal differetiatio ad decimal itegratio of MOD fuzzy eutrosophic decimal polyomials? 54. Obtai ay other iterestig feature about d (1)[x].

148 Chapter Five THE MOD COMPLEX PLANES this chapter we proceed oto defie, develop ad describe the cocept of MOD complex plaes. Let C be the usual complex plae we ca for every positive iteger defie a MOD complex plae. Thus we have ifiite umber of MOD complex plaes. DEFNTON 5.1: Let C (m) = {a + bi F a, b [0, m), 2 i F = m 1}. C (m) is a semi ope plae defied as the MOD complex plae. We diagrammatically describe i Figure 5.1. This plae is defied as the MOD complex plae. We will describe the above by the followig examples. 2 Example 5.1: Let C (10) = {a + bi F a, b [0, 10), i F = 9} be the MOD complex plae associated with the MOD iterval [0, 10).

149 148 MOD Plaes Figure 5.1 Example 5.2: Let C (7) = {a + bi F a, b Z 7 ad i 2 F = 6} be the MOD complex plae associated with the MOD iterval [0, 7). Example 5.3: Let C (3) = {a + bi F a, b [0, 3) ad i 2 F = 2} be the MOD complex plae associated with the MOD iterval [0, 3). Example 5.4: Let C (2) = {a + bi F a, b [0, 2) ad i 2 F = 1} be the MOD complex plae associated with the MOD iterval [0, 2). C (2) is the smallest MOD complex plae. However we caot say C (m) for a particular m is the largest MOD complex plae as oe ca go for m+2, m+3,, m+r; r a big iteger. How to get at these MOD complex plae from the complex plae C? We kow for every a + bi C is mapped by the complex MOD trasformatio; η c : C C (m); defied as

150 The MOD Complex Plaes 149 η c (a + bi) = a + bi F if 0 a, b < m ad i 2 = m 1. F η c (a + bi) = c + di F if both a, b > m is give by a c x m = + m ad d is give by b = y + d. m m η c (a + bi) = 0 if a = b = m or a = mx ad b = my. η c (a + bi) = a + ci F if 0 < a < m ad c is give by b c = x +. m m η c (a + bi) = d + bi F if 0 < b m ad a d = y +. m m η c (a + bi) = m a + (m b)i F if a ad b are egative but less tha m. η c (a + bi) = m c + (m d)i F if both a ad b are ve ad greater tha m; c ad d defied as earlier. We will first illustrate these situatios before we proceed to defie properties associated with C (m). Example 5.5: Let C (6) be the MOD complex plae. We defie η c : C C (6). For i C. η c ( i) = ( i F ) where i 2 = 5. F For i C; η c ( i) = i F. For i C; η c ( i) = i F

151 150 MOD Plaes For i C; η c ( i) = 0. For i C; η c ( i) = 0.089i F. For i C; η c ( i) = i F i C; η c ( i) = i F C (6) ad so o. This is the way the MOD complex trasformatio is carried out. Example 5.6: Let C (11) = {a + bi F a, b [0, 11), i 2 F = 10} be the MOD complex plae. η c : C C (11) is the MOD complex trasformatio ad it is defied as follows: η c (a + bi) = a + bi F if a, b [0, 11). η c ( i) = i F C (11) η c ( i) = ( i F ) η c ( i) = (8 + 3i F ) η c ( i) = ( i F ). This is the way the MOD complex trasformatio is doe. fact ifiite umber of poits i C are mapped by η c o to a sigle poit. Example 5.7: Let η c : C C (13). η c ( i) = ( i F ) η c ( i) = ( i F ) ad so o. η c ( i) = ( i F )

152 The MOD Complex Plaes 151 η c (8 48.3i) = ( i F ) ad η c (( i) = i F. This is the way η c the MOD complex trasformatio is defied from C to the MOD complex plae C (13). Now we proceed oto defie the MOD complex plae polyomial or MOD complex polyomials with coefficiets from the MOD complex plae C (m). C (m)[x] = i aix a i C (m) = {a + bi F a, b [0, m), i= 0 2 i F = m 1} is defied as the MOD complex polyomials. Now we have ifiite umber of distict MOD complex polyomials for varyig m. We will give examples of them. Example 5.8: Let C (5)[x] = i aix a i C (5) = {a + bi F a, i= 0 2 b [0, 5), i F = 4} be the MOD complex polyomial with coefficiets from C (5). Example 5.9: Let C (9)[x] = b [0, 9), i aix a i C (9) = {a + bi F a, i= 0 2 i F = 8}} be the MOD complex polyomial. Clearly C (9)[x] is differet from C (5)[x] give i example 5.8.

153 152 MOD Plaes Example 5.10: Let C (12)[x] = i aix a i C (12) = {a + i= 0 2 bi F a, b [0, 12), i F = 11}} be the MOD complex polyomial with coefficiets from C (12). We ca as i case of MOD complex trasformatio, η c : C C (m) i case of MOD complex polyomial also defie the same MOD complex trasformatio. η c : C[x] C (m)[x] as η c [x] = x ad η c : C C (m) ad η c case of MOD complex trasformatio. Thus we see the total complex polyomial C[x] is mapped o to the MOD complex polyomials. We will illustrate this situatio by some examples. Example 5.11: Let C[x] be the complex polyomial rig ad C (20)[x] be the MOD complex polyomial. We have a uique MOD trasformatio η c : C[x] C (20)[x] such that η c (x) = x ad η c (C) = C (20). Now we show how operatios o C (m) ad C (m)[x] are defied i the followig. Example 5.12: Let C (15) = {a + bi F a, b [0, 15), be the MOD complex plae. x = i F ad y = i F C (15), x + y = ( i F ) + ( i F ) = ( i F ) C (15). t is easily verified for every pair x, y C (15) x + y C (15). 2 i F = 14}

154 The MOD Complex Plaes 153 fact 0 C (15) serves as the additive idetity. For x + 0 = 0 + x = x for all x C (15). Further for every x C (15) there exist a uique y C (15) such that x + y = 0. Let x = i F C (15) we have y = i F C (15) such that x + y = 0. Thus y is the iverse of x ad x is the iverse of y. Hece (C (15), +) is a group of ifiite order ad is commutative. Example 5.13: Let C (10) = {a + bi F a, b [0, 10), i 2 F = 9} be a group of ifiite order uder additio modulo 10. Example 5.14: Let C (7) = {a + bi F a, b [0, 7), i 2 F = 6} be a group of ifiite order uder +. Example 5.15: Let C (20) = {a + bi F a, b [0, 20), be a group of ifiite order uder + modulo 20. Now we ca give some other operatio o C (m). We defie the product o C (m). Example 5.16: Let C (12) = {a + bi F a, b [0, 12), be the MOD complex plae. Let x = i F ad y = 5 + 9i F C (12). x y = ( i F ) (5 + 9i F ) 2 i F = 19} 2 i F = 11}

155 154 MOD Plaes = i F i F + 4.2i F 9i F = i F i F = i F C (12). We see for every x, y C (12); we have x y C (12). x y = y x for all x, y C (12). We have 0 C (12) is such that x 0 = 0 for all x C (12). We may or may ot have C (12) to be a group. Clearly {C (12), } is oly a semigroup. Further {C (12), } has zero divisors, uits. idempotets ad C (12) is a ifiite commutative semigroup uder product. Example 5.17: Let {C (10), } be the semigroup. Let x = 5+ 5i F ad y = 2 + 4i F C (10) is such that x y = 0. x 2 = (5 + 5i F ) (5 + 5i F ) = i F + 5i F = 0. Let x = i F ad y = 9 +8i F C (10). x y = ( i F ) (9 + 8i F )

156 The MOD Complex Plaes 155 = i F i F = i F + 5.6i F = 8.8i F C (10). Example 5.18: Let C (11) be the semigroup uder. x = i F ad y = i F C (11). x y = ( i F ) (3 + 10i F ) = i F + 8i F + 3 = i F C (11). Let x = i F ad y = i F C (11); x y = ( i F ) (10 +10i F ) = i F + 3.7i F = i F C (11). Example 5.19: Let {C (5), } be the semigroup. Let x = i F ad y = i F C (5); x y = ( i F ) ( i F ) = i F + 2.4i F = i F C (5). Thus C (5) is oly a semigroup. THEOEM 5.1: Let C (m) be the MOD complex plae. {C (m), +} be a group.

157 156 MOD Plaes The (1) {C (m), +} has subgroups of ifiite order. (2) f m is ot a prime, {C (m), +} has more umber of subgroups. The proof of the theorem is left as a exercise to the reader. THEOEM 5.2: Let {C (m), } = S be the semigroup (m, a o prime). (i) S has subsemigroups of ifiite order. (ii) S has zero divisors. (iii) S has idempotets. The proof is left as a exercise to the reader. Next we show {C (m), +, }, MOD complex plae is ot a rig for distributive law is ot true i geeral for elemets i C (m). We will just show this fact by some examples. Let x = i F, y = 6 + 7i F ad z = 4 + 3i F C (10). Cosider x (y + z) = i F (6 + 7i F i F ) = i F 0 = 0 We fid x y + x z = i F 6 + 7i F i F 4 + 3i F = i F i F i F i F = ( ) + (

158 The MOD Complex Plaes )i F = i F Clearly ad are distict. Hece i geeral x (y + z) x y + x z for x, y, z C (m). Thus we defie {C (m), +, } be the MOD complex pseudo rig. {C (m), +} to be the MOD complex group ad {C (m), } to be the MOD complex semigroup. We have several iterestig properties associated with the pseudo rig of MOD complex plae. The pseudo MOD complex rig has ideals, subsemirigs, uits, zero divisors ad idempotets. This will be illustrated by some examples. Example 5.20: Let {C (6),+, } be the MOD complex pseudo rig. {C (6), +, } has subrigs which are ot ideals of both fiite ad ifiite order. S = {[0, 6) C (6), +, } is a pseudo subrig of ifiite order which is ot a ideal of {C (6), +, }. {Z 6, +, } C (6) is a subrig of fiite order which is ot a ideal. S 1 = {{0, 2, 4}, +, } C (6) be a subrig of fiite order which is ot a ideal. Example 5.21: Let S = {C (10), +, } be the MOD complex pseudo rig. S has zero divisors, idempotets ad uits.

159 158 MOD Plaes Example 5.22: Let S = {C (19), +, } be the MOD complex pseudo rig. has zero divisors ad uits. Now we proceed oto defie MOD complex polyomial pseudo rig ad describe their properties. DEFNTON 5.2: Let C (m)[x] = a i ix a i C (m)} be the i= 0 MOD complex polyomial plae. We will illustrate this situatio by some examples. Example 5.23: Let C (10)[x] be the MOD complex polyomial pseudo rig plae. Example 5.24: Let C (19)[x] be the MOD complex polyomial plae pseudo rig. Example 5.25: Let C (43)[x] be the MOD complex polyomial pseudo rig. Example 5.26: Let C (148)[x] be the MOD complex polyomial pseudo rig. Let ( i F ) + ( i F )x 2 + ( i F ) x 3 = p(x) C (8)[x] where C (8)[x] is the MOD complex polyomial plae. Let q(x) = (4 +0.3i F ) +( i F )x + ( i F )x 2 + ( i F )x 4 C (8)[x] p(x) + q(x)

160 The MOD Complex Plaes 159 = ( i F ) + ( i F )x 2 +( i F )x 3 + ( i F ) + ( i F )x + ( i F )x 2 + ( i F )x 4 = ( i F ) + ( i F )x + ( i F )x 2 + ( i F )x 3 + ( i F ) x 4. This is the way the operatio + is performed o C (8)[x]. We see for every q(x), p(x) C (8)[x]; p(x) + q(x) C (8)[x], 0 C (8)[x] acts as the additive idetity as p(x) + 0 = 0 + p(x) = p(x) C (8)[x]. Now for every p(x) C (8)[x] there exist a uique q(x) i C (8)[x] such that p(x) + q(x) = 0 ad q(x) is the called the additive iverse of p(x) ad vice versa. Let p(x) = ( i F ) + ( i F )x + ( i F )x 2 + ( i F )x 3 C (8)[x]. We see q(x) = ( i F ) + ( i F )x + ( i F )x 2 + ( i F )x 3 C (8)[x] is such that q(x) + p(x) = p(x) + q(x) = 0. Thus {C (8)[x], +} is a group defied as the MOD complex polyomial group of ifiite order which is abelia. We will illustrate this situatio by some examples. Example 5.27: Let {C (6)[x], +} be the MOD complex polyomial group. Example 5.28: Let G = {C (37)[x], +} be the MOD complex polyomial group. Example 5.29: Let G = {C (64)[x], +} be the MOD complex polyomial group.

161 160 MOD Plaes Before we proceed to describe MOD semigroups we describe the MOD complex trasformatio from C[x] to C (m)[x]. Let C[x] be the complex polyomials; C (m)[x] be the MOD complex polyomial plae. η c : C[x] C (m)[x] is defied as η c [x] = x ad η c : C C (m) is defied i the usual way that is η c is the MOD complex trasformatio from C to C (m). Thus usig this MOD complex trasformatio we ca map the whole plae C[x] ito the MOD complex polyomials plae C (m)[x] for m N. fact just ow we have proved C (m)[x] to be a group uder +. Now we defie the operatio o C (m)[x]. For p(x) = 3.2x x 3 + 4x + 1 ad q(x) = 0.5x x i C (5)[x] we fid p(x) q(x) = (3.2x x 3 + 4x + 1) (0.5x x 2 + 2) = 1.6x x 6 + 2x x x 9 + 2x x x x x + 2 C (5)[x]. This is the way product operatio is performed. Let p(x) = (3 + 2i F )x 2 + ( i F )x + ( i F ) ad q(x) = ( i F )x 3 + ( i F ) C (5) [x]. We fid p(x) q(x) = ((3 + 2i F )x 2 + ( i F )x + ( i F )) (( i F )x 3 + ( i F ))

162 The MOD Complex Plaes 161 = ( i F i F )x 4 + ( i F + 1.2i F )x 5 + ( i F i F )x 3 + ( i F + 4i F + 4)x 2 + ( i F + 1.8i F )x + ( i F + 2.1i F ) = ( i F )x 4 + ( i F )x 5 + ( i F )x 3 + ( i F )x 2 + ( i F )x + ( i F ) C (5) [x]. This is the way product operatio is performed o C (5)[x]. {C (5)[x]} is a semigroup of ifiite order which is commutative. Example 5.30: Let S = {C (8)[x], } be the MOD complex polyomial semigroup. S has zero divisors ad uits. Example 5.31: Let P = {C (12)[x], } be the MOD complex polyomial semigroup. S has zero divisors ad uits. Example 5.32: Let M = {C (13)[x], } be the MOD complex polyomial semigroup. {C (m)[x], } = B is defied as the MOD complex polyomial semigroup. B is of ifiite order ad is commutative. Example 5.33: Let B = {C (12)[x], } be the MOD complex polyomial semigroup. {Z 12 [x], } is a subsemigroup of B ad is of ifiite order. {Z 12, } is a fiite subsemigroup of B. (12)[x] B is agai oly a subsemigroup ad is ot a ideal of B.

163 162 MOD Plaes {[0, 12), } B is oly a subsemigroup ad is ot a ideal of B. {[0, 12)[x], } is oce agai a subsemigroup of B ad is ot a ideal. p(x) = 4 + 4x 2 + 8x 3 ad q(x) = 6 + 3x 2 B are such that p(x) q(x) = 0. fact B has ifiite umber of zero divisors. Example 5.34: Let {C (29)[x], } = S be the MOD complex polyomial semigroup. S has ifiite umber of zero divisors. S has subsemigroup of both fiite ad ifiite order. Now havig see MOD complex polyomial groups ad semigroups. We ow proceed oto defie MOD complex polyomial pseudo rigs. Let S = {C (m)[x], +, } be the MOD complex polyomial uder the biary operatios + ad. Now S is defie as the MOD complex polyomial pseudo rig as for p(x), q(x), r(x) S we may ot i geeral have p(x) (q(x) + r(x)) = p(x) q(x) + p(x) r(x). Because the triples eed ot always be distributive we defie {C (m)[x], +, } as oly a pseudo rig. We will proceed oto describe with examples the complex MOD polyomial pseudo rigs. Example 5.35: Let {C (15) [x], +, } be the MOD complex polyomial pseudo rig. Let p(x) = 0.7x 8 + (2 +4i F )x 4 + 2i F,

164 The MOD Complex Plaes 163 q(x) = 10.3x 5 + ( i F )x 2 +( i F ) ad r(x) = 4.7x 5 + ( i F )x 2 + ( i F ) C (15)[x]. Cosider p(x) (q(x) + r(x)) = (0.7x 8 + (2 + 4i F )x 4 + 2i F ) (10.3x 5 + ( i F )x 2 +( i F ) + 4.7x 5 + ( i F )x 2 + ( i F )) = (0.7x 8 + (2 + 4i F )x 4 + 2i F ) 0 Now we fid p(x) q(x) + p(x) + r(x) = [0.7x 8 + (2 + 4i F )x 4 + 2i F ] (10.3x 5 + ( i F )x 2 + ( i F ) + [0.7x 8 + (2 + 4i F )x 4 + 2i F ] (4.7x 5 + ( i F )x 2 + ( i F )) = 7.21x 13 + ( i F )x 10 + ( i F ) + ( i F )x 9 + ( i F )x 6 + ( i F )x i F x 5 + ( i F )x 2 +(6.2i F + 6.6) x 13 +( i F )x 10 +( i F )x i F x 5 +(10i F + 7.6)x 2 +(8.8i F + 9.6) + ( i F )x 9 + ( i F )x 6 + (3.2i F +10.6) x 2 0 Clearly ad are distict. Hece we see {C (m)[x], +, } ca oly be a pseudo rig. t is ot a rig. Further C (m)[x] has ideals subrigs, zero divisors, uits ad idempotets. Study i this directio is iterestig as the o distributivity gives lots of challeges.

165 164 MOD Plaes For istace we caot write (x + α 1 ) (x + α 2 ) (x + α ) = x (α α ) x 1 + xix j x α 1 α 2 α. i j That is why α i may make oe side equal to zero but the other side of the equatio is ot zero i geeral this is maily because the distributive law is ot true. Example 5.36: Let {C (6)[x], +, } be the complex MOD polyomial pseudo rig. Let (x + 3.2) (x + 1.5) (x + 3) = p(x). We see p(3) = 0, p(4.5) = 0 ad p(2.8) =0. (x + 3.2) (x + 1.5) (x + 3) = (x x + 1.5x + 4.8) (x + 3) = x x x + 3x x + 4.5x = x x x = q(x). Put x = 3 q(3) = = q(2.8) = (2.8) (2.8) = = Thus q(3) 0 but p(3) = 0 ad so o. t is a ope problem how to over come this situatio.

166 The MOD Complex Plaes 165 Study i this directio is beig carried out. Now regardig the derivatives ad itegrals we defie a ew type of MOD complex polyomials. d C (m)[x] = a x i i a i i this case we call polyomials. d C (m) ad i + reals or zero} d C (m)[x] as the MOD complex decimal Let p(x) = ( i F ) x 2.1 +(1 +0.2i F )x ( i F ) ad q(x) = ( i F ) x 1.2 d + ( i F ) C (5)[x]; be the MOD complex decimal polyomial plae. p(x) q(x) = [( i F )x (1 +0.2i F )x ( i F )] [( i F )x 12 +(4 +0.2i F )] = ( i F + 0.9i F )x 3.3 +(2+0.4i F +0.6i F ) x 1.3 +(0.2 + i F i F ) x ( i F + 0.6i F )x 2.1 +( i F + 0.2i F )x 0.1 +( i F i F + 0.4) = ( i F )x 3.3 +( i F )x ( i F )x ( i F ) x ( i F )x 0.1 +( i F ) is i C (5)[x]. d This is the way operatios are performed o d C (5)[x]. Now we ca both differetiate ad itegrate MOD decimal polyomials. Here also at times they will exist ad at times they may ot exist. Example 5.37: Let { C d (6)[x], +} be the MOD decimal complex polyomial group.

167 166 MOD Plaes Example 5.38: Let { C d (7)[x], +} = G be the MOD decimal complex polyomial group. G has subgroups of both fiite ad ifiite order. Example 5.39: Let { C d (64)[x], +} = G be the MOD decimal complex polyomial group. G has subgroups. Now we defie G = { C d (m)[x], +} to be the MOD complex decimal polyomial group. Example 5.40: Let { C d (10)[x], } = S be the MOD decimal complex polyomial semigroup. S is of ifiite order. Example 5.41: Let P = { C d (15)[x], } be the MOD decimal complex polyomial semigroup. P has ideals ad subsemigroups. P has uits, zero divisors ad idempotets. Example 5.42: Let X = { C d (13)[x], } be the MOD complex polyomial semigroup. P = { C d (m)[x], } is defied as the MOD complex decimal polyomial semigroup. We have ifiite umber of such semigroups depedig o m Z + \ {1}. We will give some examples of them. Example 5.43: Let S = { C d (15)[x], } be the MOD complex decimal polyomial semigroup. S has ifiite umber of zero divisors ad few uits. Example 5.44: Let M = { C d (13)[x], } be the MOD complex polyomial semigroup. M has zero divisors, uits ad idempotets.

168 The MOD Complex Plaes 167 Example 5.45: Let S = { C d (24)[x], } be the MOD complex polyomial semigroup. M has zero divisors ad idempotets. Example 5.46: Let T = { C d (128)[x], } be the MOD complex decimal polyomial semigroup. T has zero divisors, uits ad idempotets. Now havig see group ad semigroup of MOD complex decimal polyomials, we proceed oto defie MOD complex decimal polyomials pseudo rig. { C d (m)[x], +, } is defied as the MOD complex decimal polyomial pseudo rig as the distributive laws are ot true i case of MOD complex decimal polyomial triples. That is i geeral p(x) (q(x) + r(x)) p(x) q(x) + p(x) r(x) for p(x), q(x), r (x) { C (m) [x], +, }. We will give examples of them. d Example 5.47: Let { C d (4)[x], +, } = S be the MOD complex decimal polyomial pseudo rig. Let p(x) = (2 + i F )x 8 + ( i F )x 4 + ( i F ), q(x) = (3 +0.7i F ) + ( i F )x 3 ad r(x) = ( i F ) + ( i F )x 3 p(x) [q(x) + r(x)] d C (4)[x]. = (2 + i F )x 8 + ( i F )x 4 + ( i F ) [( i F ) + ( i F )x 3 + ( i F ) + (3.4 +2i F )x 3 ]

169 168 MOD Plaes = p(x) 0 = 0 Now p(x) q(x) + p(x) r(x) = [(2 + i F )x ( i F )x ( i F )] ( i F ) + ( i F )x 3 + [(2 + i F )x 8 + ( i F )x 4 + ( i F )] ( i F ) + ( i F )x 3 = [(6 + 3i F +1.4i F ) x ( i F + 2.1i F )x ( i F i F ) + ( i F )x ( i F + 6i F + 1 3)x ( i F + 1.4i F + 3) x 3 + (2 + i F i F )x ( i F i F )x ( i F i F ) + ( i F + 2 3)x ( i F + 2i F + 3)x ( i F i F + 3)x 3 = ( i F )x ( i F )x ( i F ) + ( i F )x 3 + ( i F )x ( i F )x ( i F ) + ( i F )x ( i F )x ( i F )x 3 = ( i F ) x 3 + ( i F ) x ( i F ) x ( i F ) (3 + 2i F )x 7.2 Clearly ad are distict. d Hece C (4)[x] is a MOD complex decimal polyomial pseudo rig. We have show how the distributive law is ot true.

170 The MOD Complex Plaes 169 Example 5.48: Let B = { C d (128)[x], +, } be the MOD complex decimal polyomial pseudo rig. Example 5.49: Let S = { C d (17)[x], +, } be the MOD complex decimal polyomial pseudo rig. Example 5.50: Let P = { C d (18)[x], +, } be the MOD complex decimal polyomial pseudo rig. Now havig see examples of MOD complex decimal polyomial rigs we ow proceed oto defie differet types of differetiatio ad itegratio o these polyomials. Example 5.51: Let S = { C d (7)[x], +, } be the MOD complex decimal polyomial pseudo rig. Let p(x) = (2 + 3i F )x ( )x ( i F ) S. dp(x) dx(0.5) = (2 + 3i F) 3.2x term ot defied + 0. So we see this special type of derivatives are also ot defied. So we see the derivative with respect other values are ot useful. However the itegratio is defied i most cases with some simple modificatios. Let p(x) = 0.31x x d C (5) [x]. p(x) dx (0.3) = x 2.432x x C

171 170 MOD Plaes However we do ot have meaig for this situatio. Thus we may have seemigly calculated the itegral but is meaigless. We just give i supportive of our argumet that the divisio is meaigless i Z, [0, ), ([0, )) ad so o. Cosider Z 5 for istace we wat to fid 3 2 ; for 3, 2 Z but 3 = = 9 4 (mod 5). So what sort of cosistet divisio is to be used if we wat to have multiplicatio to be the reverse process of divisio? Let Z 7 be the rig of modulo iteger but 3, 4 Z = = = = 0.75 i ; So with this hurdle eve i modulo itegers we leave this problem as a ope problem. Thus we feel it is ot a easy task to itegrate these p(x) C (m)[x]. Let p(x) = ( i F )x 3 + ( i F )x + ( i F ) C (5) [x] p(x) dx = 4 2 ( i F)x ( i F )x ( i F )x + k

172 The MOD Complex Plaes 171 the itegral exist. p(x) dx (0.51) = ( i F)x ( i F)x ( i F)x k [x] ad C[x] we see every polyomial ca be both differetiated ad itegrated however i case C [m][x] ad d [m][x] this is ot possible. More so this is ot possible i case of C (m)[x] ad (m)[x]. Now we have studied such itegratio ad differetiatio of MOD complex polyomials ad MOD complex decimal polyomials. We ow proceed o to suggest some problems. Problems: 1. Fid some special features ejoyed by MOD complex plaes. 2. Show we have ifiite umber of MOD complex plaes. 3. Prove we have for a give MOD complex plae C (m) oly oe MOD complex trasformatio η c. 4. Obtai iterestig features ejoyed by η c. 5. Defie η c : C C (20) ad show kerel η c {φ}. 6. Prove {C (15), +} = G is a abelia group of ifiite order. (i) Fid subgroups of G. d

173 172 MOD Plaes (ii) Ca G have ifiite order subgroups? (iii) Ca G have fiite subgroups? 7. Let {C (29), +} = S be the MOD complex plae group. Study questios (i) to (iii) of problem 6 for this S. 8. Let B = {C (24), +} be the MOD complex group. Study questios (i) to (iii) of problem 6 for this B. 9. Let S = {C (12), } be the MOD complex semigroup. (i) Fid subsemigroups of S which are ot ideals. (ii) s S a S-semigroup? (iii) Fid S-ideals if ay i S. (iv) Ca ideals of S be of ifiite order? (v) Ca S have S-uits? (vi) Ca S have idempotets which are ot S-idempotets? (vii) Ca S have zero divisors ad S-zero divisors? 10. Let B = {C (19), } be the MOD complex semigroup. Study questios (i) to (vii) of problem 9 for this B. 11. Let S 1 = {C (27), } be the MOD complex semigroup. Study questios (i) to (vii) of problem 9 for this S Let M = {C (m),, +} be the MOD complex pseudo rig. (i) Prove M is commutative. (ii) Ca M have zero divisors? (iii) Ca M have pseudo ideals? (iv) Ca M have S-pseudo ideals? (v) s every ideal of M is of ifiite order?

174 The MOD Complex Plaes 173 (vi) Ca M have subrigs of fiite order? (vii) s M a S-pseudo rig? (viii) Ca M have uits? (ix) Does every elemet i M ivertible? (x) Does M have oly fiite umber of ivertible elemets? (xi) Fid ay other special feature ejoyed by M. (xii) Ca M have idempotets which are ot S- idempotets? 13. Let = {C (29),, +} be the MOD complex pseudo rig. Study questios (i) to (xii) of problem 12 for this. 14. Let B = {C (28),, +} be the MOD complex pseudo rig. Study questios (i) to (xii) of problem 12 for this B. 15. Let S = {C (64),, +} be the MOD complex pseudo rig. Study questios (i) to (xii) of problem 12 for this S. 16. Let C (m)[x] be the MOD complex plae. (i) Study all properties of C (m)[x]. (ii) Defie η c ; C[x] C (m)[x] so that η c is a MOD complex trasformatio. 17. Let S ={C (m)[x], +} be the MOD complex polyomial group. (i) Prove S is commutative. (ii) Prove C (m) S. (iii) Ca S have fiite subgroups? (iv) Obtai ay other special property ejoyed by S.

175 174 MOD Plaes 18. Let P = {C (24)[x], +} be the MOD complex polyomial group. Study questios (i) to (iv) of problem 17 for this P. 19. Let B = {C (19)[x], +} be the MOD complex polyomial group. Study questios (i) to (iv) of problem 17 for this B. 20. Let M = {C (625)[x], +} be the MOD complex polyomial group. Study questios (i) to (iv) of problem 17 for this M. 21. Let S = {C (m)[x], } be the MOD complex polyomial semigroup. (i) s S a S-semigroup? (ii) Ca S have S-ideals? (iii) Ca S have S-uits? (iv) Ca S have uits which are ot S-uits? (v) Ca S have fiite ideals? (vi) Ca S have fiite subsemigroups? (vii) Ca S have idempotets? (viii) Ca S have zero divisors? (ix) Obtai ay other special property associated with S. 22. Let S = {C (12)[x], } be the MOD complex polyomial semigroup. Study questios (i) to (ix) of problem 21 for this B. 23. Let S = {C (31)[x], } be the MOD complex polyomial semigroup.

176 The MOD Complex Plaes 175 Study questios (i) to (ix) of problem 21 for this S. 24. Let T = {C (243)[x], } be the MOD complex polyomial semigroup. Study questios (i) to (ix) of problem 21 for this T. 25. Let S = {C (m)[x], +, } be the MOD complex polyomial rig. (i) Obtai some of the ew properties associated with S. (ii) Ca S be a S-pseudo rig? (iii) Ca S have S-pseudo ideals? (iv) Ca S have S-uits? (v) Ca S have zero divisors? (vi) Ca S have ifiite umber of idempotets? (vii) Prove S has subrigs of fiite order. (viii) Ca S have ifiite umber of uits? (ix) Prove S has ideals of ifiite order oly. (x) Ca S have ifiite umber of idempotets? (xi) Prove i S; x (y + z) = x y + x z for some x, y, z S. 26. Let S 1 = {C (127)[x], +, } be the MOD complex polyomial rig. Study questios (i) to (xi) of problem 25 for this S Let P = {C (48)[x], +, } be the MOD complex polyomial rig. Study questios (i) to (xi) of problem 25 for this P. 28. Let M = {C (128)[x], +, } be the MOD complex polyomial rig. Study questios (i) to (xi) of problem 25 for this M.

177 176 MOD Plaes 29. Obtai some igeious methods of solvig MOD complex polyomials i C (20)[x]. 30. Solve p(x) = x 2 C (5)[x]. 31. Solve p(x) = x x C (7)[x]. 32. Solve p(x) = 5x x + 5 C (15)[x]. 33. Prove if p(x) C (81)[x] has a multiple root α the the derivative of p(x) may or may ot have α to be a root of p (x) i geeral. 34. Prove for (x + 0.3) 2 (x + 6.5) C (10)[x], (x + 0.3) 2 (x + 6.5) (x x ) (x + 6.5). 35. Show because of the o distributivity of a triple i C (m)[x]; α is a root of p(x) = (x + α 1 ) (x + α ) the α i geeral is ot a root of (x + α 1 ) (x + α ) expaded by multiplyig. 36. Whe ca p(x) C (m)[x] be itegrated? 37. Show every p(x) C (m)[x] eed ot have derivatives. 38. What are the problems associated with itegral ad differetial of p(x) C (m)[x]? 39. Let p(x) = 5x x C (6)[x]. (i) Fid p (x). (ii) Fid p(x) d(x). (iii) Fid all the roots of p(x). (iv) Ca p(x) have more the two roots? (v) p 1 (x) = 3.2x 6 + 4x 3 + 3x x C (6)[x]. Study questios (i) to (iv) for this p 1 (x). (vi) How may times a th degree polyomial i

178 The MOD Complex Plaes 177 C (m)[x] be derived? (vii) Show the differece betwee usual differetiatio i C[x] ad i C (m)[x]. 40. Ca we use ay other modified form of differetiatio ad itegratio for p(x) C (m)[x]? 41. Let p(x) = 14.3x x C (30)[x]. Fid (i) dp(x) dx. (ii) p(x) dx. 42. What are special features ejoyed by 43. Ca p(x) d C (m)[x]? d C (q)[x] be differetiable, where p(x) = (3 + 5i F )x (2 +0.3i F )x ( )x ( i F )? 44. Fid the itegral of p(x) give i problem Let p(x) = 0.31x ( i F ) + ( i F )x ( i F )x 2.1 C (7)[x]. (i) Fid dp(x) dx(0.3).

179 178 MOD Plaes (ii) Fid dp(x) ; does the derivative exist? dx(6.3) (iii) Fid p(x) dx (0.3). (iv) p(x) dx (6.3). 46. Let {C (10)[x], +, } = S be the MOD complex pseudo polyomial rig. (i) Prove S has ideal. (ii) Ca S have S-ideals? (iii) Ca S has S-zero divisors? (iv) Ca S have S-uits? (v) Ca S have idempotets? (vi) s S a S-pseudo rig? (vii) Ca S have S-sub pseudo rigs which are ot S-ideals? (viii) Ca S have fiite S-ideals? (ix) Fid all uits of S. (x) Obtai ay other special feature ejoyed by S. 47. Let B = {C (29)[x], +, } be the MOD complex polyomial pseudo rig. (i) Study questios (i) to (x) of problem 46. (ii) Fid 3 polyomials i B which does ot have derivatives. (iii) Give 3 polyomials i B which has derivatives. (iv) Give 2 polyomials i B which is ot itegrable. (v) Give 2 polyomials i B which are itegrable. 48. Let P = {C (24)[x], +, } be the MOD complex polyomial pseudo rig. Study questios (i) to (v) of problem 47 for this P.

180 The MOD Complex Plaes Let P = {C (4)[x], +, } be the MOD complex polyomial pseudo rig. For p(x) = ( i F )x 8 +( i F )x 4 + ( i F )x 12 C (4) [x]. (i) Fid dp(x) dx ad p(x) dx. (ii) Study questios (i) to (v) of problem 47 for this P. 50. Obtai the special features ejoyed by d C (m)[x]. 51. Show all p(x) d C (m)[x] caot be itegrated. 52. Show B = { C d (m)[x],, +} is oly a pseudo rig of MOD complex polyomials. 53. Let B = { C d (24)[x], +, } be the pseudo MOD decimal polyomial complex rig. (i) Show B has zero divisors. (ii) Ca B have S-zero divisors? (iii) Fid uits B which are ot S-uits. (iv) Ca B be a S-pseudo rig? (v) Ca B have ideals of fiite order? (vi) Does B have S-ideals? (vii) Fid all fiite subrigs of B. (viii) Show there are subrigs of ifiite order i B which are ot ideals. (ix) Fid some special features associated with B. (x) Show B ca have fiite umber of uits. (xi) Show B caot have ifiite umber of idempotets.

181 180 MOD Plaes 54. Let C = { C d (128)[x], +, } be the MOD complex decimal polyomial pseudo rig. Study questios (i) to (xi) of problem 53 for this C. 55. Let p(x) = ( i F ) x ( i F )x ( i F ) C (C give i problem 54). dp(x) (i) Fid dx(0.3). (ii) How may times p(x) should be derived so that t d p(x) = 0? t dx(0.3) (iii) Fid p(x) dx(0.3).

182 Chapter Six MOD DUAL NUMBE PLANES this chapter we itroduce the otio of MOD dual umber plae. [16] has itroduced several properties about dual umbers. ecall (g) = { a + bg / a, b are reals i ad g is such that g 2 = 0 } is defied as dual umbers. Clearly (g) is of ifiite order we ca give the dual umbers the plae represetatio. The associated graph is give i Figure 6.1. This is defied as the real dual umber plae. Several properties about these plaes have bee systematically carried out i [16]. Here we at the outset claim, we have ifiite umber of MOD dual umber plaes for a give g. Let (m) deote the usual MOD real plae associated with [0, m). (m)[g] = { a + bg a, b [0, m) with g 2 = 0}. We defie (m)[g] as the MOD dual umber plae associated with [0, m).

183 182 MOD Plaes 12 g x Figure 6.1 We will give examples of MOD dual umber plaes before we proceed o to defie the cocept of MOD dual umber trasformatio. Example 6.1: Let (3) [g] = { a + bg a, b [0,3), g 2 = 0} be the MOD dual umber plae associated with the MOD iterval [0, 3). The associated graph is give i Figure g i (3)[g] is marked i the MOD dual umber plae. Example 6.2: Let (10)[g] = { a + bg a, b [0,10), g 2 = 0} be the MOD dual umber plae. The associated graph is give i Figure 6.3.

184 MOD Dual Number Plaes 183 g 3 2 (0.5,1.5) Figure 6.2 x 10 g (2,8) (5,3) Figure 6.3 x

185 184 MOD Plaes Let 5 + 3g (10) [g]; it is represeted i the MOD dual umber plae as above. Let 2 + 8g (10) [g] it is represeted as above. This is the way MOD dual umber are represeted i the MOD dual umber plae. The elemets a + bg (m)[g] are kow as MOD dual umbers lyig i the MOD dual umber plae (m)[g]. Example 6.3: Let (7) [g] = {a + bg a, b [0,7), g 2 = 0} be the MOD dual umber plae. Ay MOD dual umbers 3 = 4.1g ad g (7) [g] are give the followig represetatio i the MOD dual umber plae. 7 g (3,4.1) 3 2 (2.5,2.6) Figure 6.4 x

186 MOD Dual Number Plaes 185 Example 6.4: Let (8)[g] = {a +bg a, b [0,8), g 2 = 0} be the MOD dual umber plae. Ay MOD dual umber i (8)[g] ca be represeted i the MOD dual umber plae. We have see the cocept of MOD dual umber plaes ad the represetatio of MOD dual umbers i that plae. Now we see for a give g with g 2 = 0 we ca have oe ad oly oe real dual umber plae but however MOD dual umber plaes are ifiite i umber for a give g with g 2 = 0. For (g) = { a + bg a, b, g 2 = 0} for the same g we have (m) [g] = { a + bg a, b [0, m) g 2 = 0} for varyig fiite iteger m Z + \{0}. Thus we have ifiite umber of MOD dual umber plaes for a give g. We ca defie η d : (g) (m)[g] by gif a =0 ad b=1. a + bgif both a ad b are less tha m. a r r + bg where b<m ad = d + m m b s a + sg whe a < mad = d + m m a r b s η d (a+bg) = r + sg whe = d + ad = e + m m m m = 0if a = mr, b = ms b r = 0 + rg if a = mt ad = d + m m a s = s + 0 if b = mt = e + m m whe egative appropriate mod ificatio is made

187 186 MOD Plaes This map η d is defied as the MOD dual umber trasformatio. We will illustrated this situatio by some examples. Example 6.5: Let (6)(g) be the MOD dual umber plae. We will defie η d : (g) (6)(g) Let g (g) η d ( g) = ( g) (6)(g). η d ( 40: g) = ( g) = ( g) (6)[g]. η d ( g) = ( ) (6)[g]. η d ( g) = g (6)[g]. η d ( g) = 0. η d ( g) = 2.7g ad so o. This is the way operatios are performed o (6)[g]. As i case of (m) or C (m) or (m)[x] or C (m)[x] we ca show (m)[g] the MOD dual umbers plae is a group uder additio +. We will just illustrate this by a example or two. Example 6.6: Let (5)(g) = {a + bg a, b [0, 5), g 2 = 0} be the MOD dual umber plae. S = { (5)(g), +} is a group. For let x = g ad y = g S x + y = ( g) + ( g)

188 MOD Dual Number Plaes 187 = g S. This is the way operatio + is performed o S. We see 0 acts as the additive idetity of S. For every p (5)(g) we have a uique q (5)(g) such that p + q = 0. Let p = g (5)(g) the q = g i (5) (g) is such that p + q = 0. Thus we see (5)(g) is a abelia group uder +. (m)(g) = {a + bg a, b [0, m), g 2 = 0} uder + is a group defied as the MOD dual umber group. We will give some examples of them. Example 6.7: Let G = { (20)(g), +} be the MOD dual umber group of ifiite order. Example 6.8: Let G = { (37)(g), +} be the MOD dual umber group of ifiite order. Example 6.9: umber group. Let G = { (128)(g), +} be the MOD dual Now we ca defie product o MOD dual umber plae (m)(g). { (m)(g), } is a semigroup which has ifiite umber of zero divisors, fiite umber of uits ad idempotets. S = { (m)g,, g 2 = 0} is defied as the MOD dual umber semigroup.

189 188 MOD Plaes We will examples of them. Example 6.10: semigroup. Let G = { (5)(g), } be a MOD dual umber Example 6.11: Let S = { (15)(g), } be the MOD dual umber semigroup. S has zero divisors ad uits. Example 6.12: Let T = { (140)(g), } be the MOD dual umber semigroup. Now havig see the defiitio ad examples it is a matter of routie for the reader to study all the related properties. Now { (m) (g), +, } is defied as the MOD dual umber pseudo rig as i this case also the distributive law is ot true i geeral. We will give examples of them. Example 6.13: Let S = { (10)(g), +, } be the MOD dual umber pseudo rig. S has zero divisors, uits ad idempotets. Example 6.14: Let S = { (43)(g), +, } be the MOD dual umber pseudo rig. S has ifiite umber of zero divisors but oly fiite umber of uits ad idempotets. Example 6.15: Let S = { (128)(g), +, } be the MOD dual umber pseudo rig. S has uits, zero divisors ad idempotets. Now we proceed oto study the MOD dual umber pseudo polyomial rig. P ={ (m)(g) [x] = { a i x i a i (m)(g)} is defied as the MOD dual umber polyomials. Now it is a matter of routie to check { (m)(g) [x], +} is defied as the MOD dual umber polyomial group. We will first give oe or two examples of them.

190 MOD Dual Number Plaes 189 Example 6.16: Let S = { (9)(g)[x], +} be the MOD dual umber polyomial group uder +. Example 6.17: Let S = { (12)(g)[x], +} be the MOD dual umber polyomial group. This has subgroups of both fiite ad ifiite order. Example 6.18: Let S = { (37)(g)[x], +} be the MOD dual umber polyomial group. Now we ca defie the product operatio o the MOD dual umber polyomials { (m)(g)[x], } = S is defied as the MOD dual umber polyomial semigroup. The study of properties of S is a matter of routie ad is left as a exercise to the reader. Example 6.19: Let S = { (3)(g)[x], } be the MOD dual umber polyomial semigroup which has ifiite umber of zero divisors. Example 6.20: Let P = { (12)(g)[x], } be the MOD dual umber polyomial semigroup. P has idempotets ad uits oly fiite i umber. Example 6.21: Let B = { (243)(g)[x], } be the MOD dual umber polyomial semigroup. B has ifiite umber of zero divisors. Study of ideals, subsemigroups etc is cosidered as a matter of routie. We ow proceed oto study MOD dual umber polyomial pseudo rig. We defie ={ (m)(g)[x], +, } to be the MOD dual umber polyomial pseudo rig. has uits zero divisors,

191 190 MOD Plaes idempotets, subrigs ad ideals. This is left as a matter of routie. However we will provide examples of them. Example 6.22: Let M = { (20)(g)[x], +, } be the MOD dual umber polyomial pseudo rig. M has ifiite umber of zero divisors. All ideas of M are of ifiite order. Example 6.23: Let B = { (4)(g)[x], +, } be the MOD dual umber polyomial pseudo rig. Let p(x) = ( g)x 7 + ( g)x 3 + (0.8 + g) ad q(x) = ( g)x 2 + ( g) B. p(x) q(x) = [( g)x 7 + ( g)x 3 + (0.8 + g)] [( g)x 2 + ( g)] = ( g + 2g)x 9 + ( g g)x 5 + ( g + 0.4g)x 2 + ( g g) x 7 + ( g g) x 3 + ( g + 3.2g) = ( g)x 9 + ( g)x 5 + ( g) x 2 + ( g)x 7 + ( g)x 3 + ( g) B. This is the way product is defied o these pseudo dual umber polyomials. We will ow proceed oto show how itegratio ad differetiatio is possible o p(x) (m)[x] oly if defied. Let p(x) = ( g)x 3 + ( g)x 2 + ( g) (6)(g) [x]

192 MOD Dual Number Plaes 191 dp(x) dx = 3( g)x 2 + 2( g)x + 0 = ( )x 2 + ( g) x. p(x) dx = 4 3 ( g)x ( g)x + + ( g)x + c. 4 3 Clearly as udefied. 1 4 ad 1 3 are ot defied this itegral is This sort of study is cosidered as a matter of routie ad is left as a exercise to the reader. Next we proceed oto defie MOD special dual like umber plaes. ecall a + bg where a ad b are reals with g 2 = g is defied as the special dual like umbers. We kow (g) = {a + bg a, b (reals) g 2 = g} is defied as the special dual like umber collectio. For more about this please refer [17]. (m)(g) = {a + bg a, b [0, m), g 2 = g} is defied as the MOD special dual like umber plae. We will illustrate this situatio by some examples. Example 6.24: Let (4)(g) = {a + bg a, b [0, 4), g 2 = g} be the MOD special dual like umber plae. The associated graph is give i Figure 6.5. We call the elemets of (4)(g) as MOD special dual like umbers.

193 192 MOD Plaes 4 g 3 2 (1,2) g 5 Figure 6.5 x (2.1,2) 1 (0.5,0.5) Figure 6.6 x

194 MOD Dual Number Plaes 193 Example 6.25: Let S = { (5)(g)} = {a + bg a, b [0, 5) ad g 2 = g} be the MOD special dual like umber plae g ad g are kow as MOD special dual like umbers of S. The associated graph is give i Figure 6.6. We will show how product ad sum are defied o MOD special dual like umbers. Example 6.26: Let S = { (10)(g)} = {a + bg a, b [0, 10) ad g 2 = g} be the MOD special dual like umbers. Let x = g ad y = g S. x + y = ( g) + ( g) = ( g) S. t is easily verified (S, +) is a group uder +, 0 acts as the additive idetity. Further for every x i S we have a uique y S such that x + y = y + x = 0 y is the iverse of x ad vice versa. Let x = g S. y = g S is such that x + y = 0 ad y is the iverse of x ad x is the iverse of y. Thus { (m)(g) g 2 = g, +} is defied as the MOD special dual like umber group. We have subgroups of both fiite ad ifiite order. We will illustrate this situatio by some examples. Example 6.27: Let S = { (23)(g) g 2 = g, +} be the MOD special dual like umber group.

195 194 MOD Plaes Example 6.28: Let S = { (24)(g) g 2 = g, +} be the MOD special dual like umber group. Let us defie product o (m)(g). For x = g ad y = g (24)(g); we fid x y = ( g) ( g) = g + 4g g = g (24)(g). We see { (m)(g), } is a semigroup defied as the MOD special dual like umber semigroup. We will give a few examples of them. Example 6.29: Let S = { (47)(g) g 2 = g, } be the MOD special dual like umber semigroup. Example 6.30: Let S = { (48)(g) g 2 = g, } be the MOD special dual like umber semigroup. We see these semigroups have uits, zero divisors, idempotets subsemigroups ad ideals. Study i this directio is a matter of routie hece left as a exercise to the reader. Now we defie S = { (m)(g), +, } to be the MOD special dual like umber pseudo rig. Cleary S is of ifiite order we call it pseudo as a (b + c) = (a b + a c) for a, b, c. We will show this by examples.

196 MOD Dual Number Plaes 195 Example 6.31: Let = { (6)(g), +, } be the MOD special dual like umber pseudo rig. Let x = g y = g ad z = ( g). x (y + z) = ( g) [ g g] = Cosider x y + x z = ( g) ( g) + ( g) ( g) = ( g g g) + ( g g g) = ( g) + ( g) = g --- Clearly ad are distict that is why we call to be the MOD special dual like umber pseudo rig. We will give a example or two of this. Example 6.32: Let S = { (17)(g) g 2 = g, +, } be the MOD special dual like umber pseudo rig. Example 6.33: Let M = { (28)(g) g 2 = g, +, } be the MOD special dual like umber pseudo rig. M has uits, zero divisors, idempotets, subrigs ad ideals. This study is cosidered as a matter of routie ad hece left as a exercise to the reader.

197 196 MOD Plaes Next we proceed oto study the MOD special dual like umber polyomials. T = { (m)(g)[x] = { a i x i a i (m) (g); g 2 = g} is defied as the MOD special dual like umber polyomials. We defie + ad operatio o T. fact T uder + operatio is a group ad uder operatio is a semigroup. We will just give some examples for all these work is cosidered as a matter of routie. Example 6.34: Let S = { (6)(g)[x], +} be the MOD special dual like umber polyomial group. Example 6.35: Let S = { (11)(g)[x]} be the MOD special dual like umber polyomial group. We defie { (m)(g) [x], +} = to be the MOD special dual like umber polyomial group. Now { (m)(g)[x], } = S is defied as the MOD special dual like umber polyomial semigroup. We give examples of them. Example 6.36: Let S = { (20)(g)[x] g 2 = g, } be the MOD special dual like umber polyomial semigroup. Let p(x) = ( g)x 5 + ( g)x + (5 + 7g) ad q(x) = ( g)x 2 + ( g) S. p(x) q(x) = [( g)x 5 + ( g)x + (5 + 7g)] [( g)x 2 + ( g)]

198 MOD Dual Number Plaes 197 = ( g +10g + 4g) x 7 + ( g + 10g + 0.5g)x 6 + ( g + 5g + 15g)x 2 + ( g + 1.2g +0.16g) x 5 +( g + 2g g)x + ( g + g+ 1.4g) = ( g)x 7 + ( g)x 6 + ( g)x 2 + ( g)x 5 +( g)x + ( g) S. This is the way the product operatio is performed o S. Example 6.37: Let P = { (43)(g)[x], } be the MOD special dual like umber semigroup. Now havig see the form of MOD special dual like umber group ad semigroup we proceed oto defie, describe ad develop the otio of MOD special dual like umber pseudo rigs. S = { (m)(g)[x], +, } is defied as the MOD special dual like umber polyomial pseudo rig. We call it pseudo rig as for p(x), q(x) ad r(x) S we see p(x) (q(x) + r(x)) p(x) q(x) + p(x) r(x). We will give oe or two examples before we proceed to defie MOD special quasi dual umbers plae. Example 6.38: Let S = { (29)(g)[x], +,, g 2 = g} be the MOD special dual like umber pseudo polyomial rig. S has zero divisors, uits, idempotets ideals ad subrigs. Study of this type is cosidered as a matter of routie ad is left as a exercise to the reader. Example 6.39: Let S = { (48)(g)[x], +,, g 2 = g} be the MOD special dual like umber polyomial rig.

199 198 MOD Plaes Now we ca defie differetiatio itegratio o p(x) S. Let p(x) = (8 + 10g)x 7 + ( g)x 5 + ( g) S. dp(x) dx = 7 (8+10g)x (4+0.5g)x 4. p(x) dx = 8 6 (8 + 10g)x ( g)x + + ( g)x + C. 8 6 We ca have derivatives or itegrals to be defied or at times ot defied i S. This is the marked differece betwee the usual polyomials i [x] ad MOD polyomials i (m)(g)[x]. Now we proceed oto defie the MOD special quasi dual umber plae. ecall (g) ={a + bg a, b (reals) g 2 = g} is defied as the special quasi dual umbers. We defie (m)(g) = {a + bg a, b [0, m) with g 2 = (m 1)g} to be the MOD special quasi dual umbers plae. We will give examples of them ad describe the plae. Further we ca have the η q to be MOD special quasi dual umber trasformatio. Example 6.40: Let (3)(g) = {a + bg a, b [0, 3), g 2 =2g} be the MOD special quasi dual umber plae. The associated graph is give i Figure g = x (3)(g). a + bg are called MOD special quasi dual umbers.

200 MOD Dual Number Plaes 199 g (2.1,0.5) Figure 6.7 x For η q : (g) (m)g defied similar to η d ; η q (g) = g. This is the way the mappig is doe. We ca as i case of other MOD dual umbers defie + ad o { (m)g g 2 = (m 1)g} { (m)(g), +, g 2 = (m 1)g} is a group defied as the MOD special quasi dual umber group. We will give examples of them. Example 6.41: Let S = { (7)(g) g 2 = 6g, +} be the MOD special quasi dual umber. Example 6.42: Let M = { (12)(g) g 2 = 11g, +} be a MOD special quasi dual umber group.

201 200 MOD Plaes Example 6.43: Let B = { (256)(g) g 2 = 255g, +} be the MOD special quasi dual umber group. Now we ca defie product o (m)(g) with g 2 = (m 1)g as follows: Let us cosider (8) g with g 2 = 7g. Let x = g ad y = g (8)(g). x y = ( g) ( g) = g + 7.5g g = g + 5.5g = 2.4g + g (8)(g). This is the way product is defied o (8) (g). Cleary (m)(g) with g 2 product. = (m 1)g is a semigroup uder Example 6.44: Let B = { (9)(g) g 2 = 8g, } be the MOD special quasi dual umber semigroup. Example 6.45: Let B = { (23)(g) g 2 = 22g, } be the MOD special quasi dual umber semigroup. All properties of this semigroup is cosidered as a matter routie ad hece left as a exercise to the reader. Now { (m)(g) with g 2 = (m 1)g, +, } is defied as the MOD special quasi dual umber pseudo rig. We say it is oly a pseudo rig as for ay x, y, z (m)(g);

202 MOD Dual Number Plaes 201 x (y + z) x y + x + z. We will give oe or two examples of them. Example 6.46: Let S = { (93)(g) g 2 = 92g, +, } be the MOD special quasi dual umber pseudo rig. S has uits, zero divisors, idempotets, subrigs ad ideals. This work is a matter of routie ad is left as a exercise to the reader. Example 6.47: Let S = { (19)(g) g 2 = 18g, +, } be the MOD special quasi dual umber pseudo rig. Example 6.48: Let B = { (128)(g) g 2 = 127g, +, } be the MOD special quasi dual umber pseudo rig. Now havig see examples them we proceed to defie MOD special quasi dual umber polyomials ad the algebraic structures o them. Let (m) (g) [x] = { a i x i a i (m)g; g 2 = (m 1)g} is defied as the MOD special quasi dual umber polyomials. Example 6.49: Let S = { (17)(g) [x] g 2 = 16g} be the MOD special quasi dual umber polyomials. Example 6.50: Let B = { (44)(g)[x] g 2 = 43g} be the MOD special quasi dual umber polyomials. We ca defie the operatio +, ad (+, } o (m)(g) [x]. This is also cosidered as a matter of routie. Example 6.51: Let B = { (10)(g)[x] g 2 = 9g, +} be the MOD special quasi dual umber polyomial group. Example 6.52: Let B = { (43)(g)[x] g 2 = 42g, +} be the MOD special quasi dual umber polyomial group.

203 202 MOD Plaes Example 6.53: Let M = { (24)(g)[x] g 2 = 23g, +} be the MOD special quasi dual umber polyomial group. Example 6.54: Let B = { (19)(g)[x] g 2 = 18g, } be the MOD special quasi dual umber polyomial semigroup. B has zero divisors, uits idempotets, ideals ad subsemigroups. Example 6.55: Let T = { (25)(g)[x] g 2 = 24g, } be the MOD special quasi dual umber polyomial semigroup. S = { (m)(g)[x], +,, g 2 = (m 1)g} is defied as the MOD special quasi dual umber polyomial pseudo rig. t is a matter of routie to check i geeral p(x) q(x) + r(x)) p(x) q(x) + p(x) r (x) for p(x), q(x), r(x) S. We will give oe or two examples of them. Example 6.56: Let S = { (19) (g) [x] g 2 = 18g, +, } be the MOD special quasi dual umber polyomial pseudo rig. S has uits, zero divisors, idempotets, ideals ad subrigs. Study i this directio is a matter of routie ad left as a exercise to the reader. We ca itegrate or differetiate ay p(x) S if ad oly if it is defied otherwise it may ot be possible. We will just give oe or two illustrates is differetiatio ad itegratio. Example 6.57: Let = { (7)(g)[x] g 2 = 6g, +, } be the MOD special quasi dual umber polyomial pseudo rig.

204 MOD Dual Number Plaes 203 Let p(x) = (5 + 3g)x 8 + ( g) x 4 + ( g) belog to (7)(g)[x]. dp(x) dx = 8(5 + 3g)x 7 + 4( g)x 3 = (5 + 3g)x 7 + ( g)x 3. For this p(x) the derivative exist. Cosider p(x)dx = 9 5 (5 + 3g)x ( g)x ( g)x + c = 9 5 (5 + 3g)x ( g)x + + ( g) x + c 2 5 = 4(5 + 3g)x 9 + 3( g) x 5 + ( g)x + c = (6 + 5g)x 9 +( g)x 5 + ( g)x + c. This is the way both itegratio ad differetiatio is carried out. The itegratio ad differetiatio may ot exist for all p(x). Let p(x) = (5 + 5g)x 7. dp(x) dx = 0. However p(x) dx = (5 + 5g)x 7 + ( g) (5 + 5g)x ( g) x = c

205 204 MOD Plaes Thus the itegral exists. However p(x) = ( g)x 6 + ( g)x 13 (7)(g)[x] p(x) dx is ot defied i this case. Example 6.58: Let = { (19)(g)[x], g 2 = 18g, +, } be the MOD special quasi dual umber polyomial pseudo rig. Let p(x) = ( g) x 18 + (10g + 3.7)x 37. The itegral of p(x) does ot exist. The derivative exist i this case. Example 6.59: Let = { (4)(g)[x], g 2 = 3g, +, } be the MOD special quasi dual umber polyomial pseudo rig. Let p(x) = (2 + 2g)x x g ad q(x) = (2 + 2g) + 2x gx 15. Clearly p(x) q(x) = 0. p(x) dx = (2 + 2g)x 2x + + 2gx + c is ot defied. q(x) dx = (2 + 2g)x x 2gx + + c this is also ot defied as 16 0 (mod 4). dp(x) dx = 20 (2 + 2g)x x 14

206 MOD Dual Number Plaes 205 = 2x d p(x) dx 2 = 2 14x 13 = 0. Thus we see a 20 th degree MOD special quasi dual umber polyomial i is such that its secod derivative itself is zero. Thus we see i case of MOD special quasi dual umbers polyomials a th degree polyomial i geeral eed ot have all the derivatives to be o zero. some cases eve the first derivatives may be zero. some case the secod derivatives may be zero ad so o. Thus all the classical properties of calculus may ot i geeral be true i case of MOD special quasi dual polyomials. Likewise the itegrals may ot be defied for every MOD special quasi dual umber polyomials. Example 6.60: Let S = { (6) (g) [x], g 2 = 5g, +, } be the MOD special quasi dual umber polyomial pseudo rig. S has ifiite umber of zero divisors. f p(x) = x 12 + x 6 + (3 + 4g) the dp(x) dx = 0. p(x) dx = 13 7 x x + + (3 + 4g)x + c 13 7 = x 13 + x 7 + (3+ 4g)x + c. We see the basic or classical property itegratio is the reverse process of differetiatio is ot true. Further d dx p(x) dx = p (x) for this p(x).

207 206 MOD Plaes d dx p(x) dx = d dx (x13 + x 7 + (3 + 4g)x + c) = 13x x 6 + (3 + 4g) = x 12 + x 6 + (3 + 4g). However cosider p(x) = x 3 + x 2 + (4 + 3g) dp(x) dx = 3x 2 + 2x + 0 p(x) dx = 4 3 x x + + (4 + 3g)x + c 4 3 Clearly as 1 4 ad 1 3 are ot defied i (6)(g). We see p(x) dx is ot defied. So d p(x) dx p(x). dx view of all these we ca state the followig result. THEOEM 6.1: Let { (m)(g)[x], +, } = be the MOD special quasi dual umber polyomial pseudo rig. p1 p2 (i) f m is ot a prime ad p(x) = a x + a x + p a 3 3x where a 1, a 2, a 3 (m)(g) ad p 1, p 2 ad p 3 divide m the p(x)dx is udefied. 1 2

208 MOD Dual Number Plaes 207 (ii) f p(x) = x tm + x sm + x rm + + x m the dp( x ) dx = 0. The proof is direct ad hece left as a exercise to the reader. THEOEM 6.2: Let = { (m)(g)[x], g 2 = (m 1)g, +, } be the MOD special quasi dual umber polyomial pseudo rig. f p(x) is a th degree MOD special quasi dual umber polyomial i the i geeral p(x) eed ot have all the ( 1) derivatives to be o zero. Proof is direct ad hece left as a exercise to the reader. However solvig polyomials i happes to be a challegig process as happes to be a pseudo rig i the first place ad secodly workig i (m)(g) is a difficult task. We suggest a few problems for the reader. Problems: 1. Obtai some special features ejoyed by the MOD dual umbers plae (m)(g) where g 2 = Prove { (9)(g), g 2 = 0, +} = G be the MOD dual umber group. (i) Ca G have fiite subgroups? (ii) Ca G have subgroups of ifiite order? 3. Let { (43)(g); g 2 = 0, +} = H be the MOD dual umber group. Study questios (i) to (ii) of problem 2 for this H.

209 208 MOD Plaes 4. Eumerate all the special features ejoyed by MOD dual umber groups. 5. Let S= { (5)(g), g 2 = 0, } be the MOD dual umber semigroup. (i) Fid S-subsemigroups if ay of S. (ii) s S a S-subsemigroup? (iii) Ca S have S-ideals? (iv) Ca ideals of S be of fiite order? (v) Ca S have S-uits? (vi) Ca S have S-idempotets? (vii) Ca S have ifiite umber of idempotets? (viii) Ca S have zero divisors? (ix) Fid ay other special feature ejoyed by S. (x) Ca S have S-zero divisors? 6. Let S 1 = { (24)g, g 2 = 0, } be the MOD dual umber semigroup. Study questios (i) to (x) of problem 5 for this S Let S = { (43)g, g 2 = 0, } be the MOD dual umber semigroup. Study questios (i) to (x) of problem 5 for this S Let S = { (27)g, g 2 = 0, +, } be the MOD dual umber pseudo rig. (i) Ca S be a S-pseudo rig? (ii) Ca S have ideals of fiite order? (iii) Ca S have S-ideals? (iv) Ca S have S-subrigs which are ot S-ideals? (v) Ca S have S-zero divisors? (vi) Ca S have ifiite umber of uits? (vii) Ca S have ifiite umber of S-idempotets? (viii) Ca S have S-zero divisors? (ix) Obtai some special features ejoyed by S. (x) Ca S have S-uits? (xi) Ca S have idempotets which are ot S-idempotets?

210 MOD Dual Number Plaes Let S 1 = { (17)g, g 2 = 0, +, } be the MOD dual umber pseudo rig. Study questios (i) to (xi) of problem 8 for this S Obtai all special features associated with the MOD dual umber polyomial pseudo rigs. 11. Prove i case of such polyomials pseudo rig the usual properties of differetiatio ad itegratio are ot i geeral true. 12. Let = { (12)(g)[x], g 2 = 0, +, } be the MOD dual umber pseudo polyomial rig. Study questios (i) to (xi) of problem 8 for this. 13. Let M = { (19)(g)[x], g 2 = 0, +, } be the MOD dual umber pseudo polyomial rig. Study questios (i) to (xi) of problem 8 for this M. 14. Let W = { (12)(g)[x], g 2 = 0, +, } be the MOD dual umber pseudo polyomial rig. (i) Study questios (i) to (xi) of problem 8 for this W. (ii) Fid 3 polyomials i W whose derivatives are zero. (iii) Fid 3 polyomials i W whose itegrals are zero. (iv) Does there exist a polyomial p(x) W which is itegrable but ot differetiable ad vice versa. 15. Let = { (m)(g), g 2 = g} be the MOD dual like umber plae. Study the special features ejoyed by M. 16. Let = { (24)(g)[x], g 2 = g, +} be the MOD dual like umber group. (i) What are the special features associated with P? (ii) Give the map η : (g) ={a + bg a, b reals, g 2 = g} (24) g; which is a MOD dual like umber trasformatio.

211 210 MOD Plaes (iii) Fid subgroups of both fiite ad ifiite order i P. 17. Let = { (43)(g) g 2 = g, +} be the MOD dual like umber group. Study questios (i) to (iii) of problem 16 for this P. 18. Let B = { (28)(g) g 2 = g, } be the MOD dual like umber semigroup. (i) Ca B be a S-semigroup? (ii) Ca B have subsemigroups which are S-subsemigroups? (iii) Ca B have S-ideals? (iv) Ca ideals i B be of fiite order? (v) Ca B have S-zero divisors? (vi) Ca B have ifiite umber of S-uits ad uits? (vii) Ca B have S-idempotets? (viii) Fid all special features ejoyed by B. 19. Let = { (43)(g) g 2 = g, } be the MOD dual like umber semigroup. Study questios (i) to (viii) of problem 18 for this N. 20. What are the special features associated with MOD dual like umber pseudo rigs? 21. Let B = { (12)(g) g 2 = g, +, } be the MOD dual like umber pseudo rig. (i) Ca B be a S-pseudo rigs? (ii) Fid all fiite subrigs of B. (iii) Ca B be have S-zero divisors? (iv) Ca B have idempotets which are ot S-idempotets? (v) Ca B have ifiite umber of uits?

212 MOD Dual Number Plaes 211 (vi) What are special features associated with B? (vii)ca B have fiite S-ideals? (viii) Fid S-subrigs which are ot ideals or S-ideals. 22. Let S = { (43)(g) g 2 = g, +, } be the MOD special dual like umber pseudo rig. Study questios (i) to (viii) of problem 21 for this S. 23. Let M = { (m)(g)[x] g 2 = g} be the MOD special dual like umber polyomials. Eumerate the special features ejoyed by M. 24. Let W = { (24)(g)[x] g 2 = g, +, } be the MOD special dual like umber polyomial pseudo rig. Study questios (i) to (viii) of problem 21 for this W. 25. Let B = { (43)(g)[x] g 2 = g, +, } be the MOD special dual like umber polyomial pseudo rig. Study questios (i) to (viii) of problem 21 for this B. 26. Let B = { (m)(g) g 2 = (m 1)g} be the MOD special quasi dual umber plae. Study the special features associated with B. 27. Let M = { (8)(g) g 2 = 7g, +} be the MOD special quasi dual umber group. (i) Fid all subgroups of fiite order M. (ii) Obtai all subgroups of ifiite order i M. (iii) Study ay other special features ejoyed by M. 28. Let B = { (24)(g) g 2 = 23g, +} be the MOD special quasi dual umber group.

213 212 MOD Plaes Study questios (i) to (viii) of problem 21 for this B. 29. Let N = { (m)(g) g 2 = (m 1)g, } be the MOD special dual umber semigroup. Obtai all the special features ejoyed by N. 30. Let B = { (6)(g) g 2 = 5g, } be the MOD special dual umber semigroup. (i) s B a S-semigroup? (ii) Ca B have S-ideals? (iii) Ca ideals of B have fiite umber of elemets i them? (iv) Ca B have ifiite umber of uits? (v) Ca B have uits which are ot S-uits? (vi) Fid all S-idempotets. (vii) Fid all S-zero divisors of B. (viii) Ca B have ifiite umber of zero divisors? (ix) Ca B have S-subsemigroups? 31. Let W = { (24)(g) g 2 = 23g, } be the MOD special quasi dual umber semigroup. Study questios (i) to (ix) of problem 30 for this W. 32. Let S = { (17)(g)[x] g 2 = 16g, } be the MOD special quasi dual umber polyomial semigroup. Study questios (i) to (ix) of problem 30 for this S. 33. Let S 1 = { (28)(g)[x] g 2 = 27g, } be the MOD special quasi dual umber polyomial semigroup. Study questios (i) to (ix) of problem 30 for this S 1.

214 MOD Dual Number Plaes Let B = { (m)(g) g 2 = (m 1)g, +, } be the MOD special quasi dual umber pseudo rig. (i) Study all special features ejoyed by B. (ii) s B a S-pseudo rig? (iii) Ca B have S-ideals? (iv) Ca B have ideals of fiite order? (v) Ca B have S-zero divisors? (vi) Ca B have uits? (vii) Ca B have idempotets? (viii) Ca B have uits which are ot S-uits? (ix) Fid all subrigs of fiite order. (x) Ca B have a S-subrigs which is ot a S-ideals? 35. Let M = { (24)(g) g 2 = 23g, +, } be the MOD special quasi dual umber pseudo rig. Study questios (i) to (x) of problem 34 for this M. 36. Let N = { (47) (g) g 2 = 46g, +, } be the MOD special quasi dual umber pseudo rig. Study questios (i) to (x) of problem 34 for this N. 37. Let S = { (m) (g)[x] g 2 = (m 1)g, +, } be the MOD special dual umber pseudo rig. (i) Study questios (i) to (x) of problem 34 for this S. (ii) Give three MOD special quasi dual umber polyomials such that their derivative is zero ad itegral does ot exist. (iii) Give three MOD special quasi dual umber polyomial such that their derivative exist ad the itegral does ot exist. 38. Let M= { (92)(g)[x] g 2 = 91g, +, } be the MOD special quasi dual umber polyomial pseudo rig.

215 214 MOD Plaes (i) Study questios (i) to (x) of problem 34 for this M. (ii) Give three MOD special quasi dual umber polyomials whose derivative is zero but the itegral exist. 39. Eumerate the special features associated with MOD special quasi dual umber polyomial pseudo rigs. 40. Prove solvig equatios i (m)(g)[x] with g 2 =(m 1)g is a difficult problem. Prove this by example. 41. Let p(x) = (3 + 4g)x 10 + ( g)x 5 + ( g) (5)(g)[x] where g 2 = 4g. Fid (i) dp(x) dx. (ii) p(x) dx. (iii) Solve p(x) for roots. (iv) Ca p(x) geerate a ideal i (5)(g)?

216 FUTHE EADNG 1. Herstei.,.N., Topics i Algebra, Wiley Easter Limited, (1975). 2. Lag, S., Algebra, Addiso Wesley, (1967). 3. Smaradache, Floreti, Defiitios Derived from Neutrosophics, Proceedigs of the First teratioal Coferece o Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability ad Statistics, Uiversity of New Mexico, Gallup, 1-3 December (2001). 4. Smaradache, Floreti, Neutrosophic Logic Geeralizatio of the tuitioistic Fuzzy Logic, Special Sessio o tuitioistic Fuzzy Sets ad elated Cocepts, teratioal EUSFLAT Coferece, Zittau, Germay, September Vasatha Kadasamy, W.B., O Muti differetiatio of decimal ad Fractioal Calculus, The Mathematics Educatio, 38 (1994) Vasatha Kadasamy, W.B., O decimal ad Fractioal Calculus, Ultra Sci. of Phy. Sci., 6 (1994)

217 216 MOD Plaes 7. Vasatha Kadasamy, W.B., Muti Fuzzy differetiatio i Fuzzy Polyomial rigs, J. of st. Math ad Comp. Sci., 9 (1996) Vasatha Kadasamy, W.B., Some Applicatios of the Decimal Derivatives, The Mathematics Educatio, 34, (2000) Vasatha Kadasamy, W.B., O Partial Differetiatio of Decimal ad Fractioal calculus, J. of st. Math ad Comp. Sci., 13 (2000) Vasatha Kadasamy, W.B., O Fuzzy Calculus over fuzzy Polyomial rigs, Vikram Mathematical Joural, 22 (2002) Vasatha Kadasamy, W. B. ad Smaradache, F., N-algebraic structures ad S-N-algebraic structures, Hexis, Phoeix, Arizoa, (2005). 12. Vasatha Kadasamy, W. B. ad Smaradache, F., Neutrosophic algebraic structures ad eutrosophic N-algebraic structures, Hexis, Phoeix, Arizoa, (2006). 13. Vasatha Kadasamy, W. B. ad Smaradache, F., Smaradache Neutrosophic algebraic structures, Hexis, Phoeix, Arizoa, (2006). 14. Vasatha Kadasamy, W.B., ad Smaradache, F., Fuzzy terval Matrices, Neutrosophic terval Matrices ad their Applicatios, Hexis, Phoeix, (2006). 15. Vasatha Kadasamy, W.B. ad Smaradache, F., Fiite Neutrosophic Complex Numbers, Zip Publishig, Ohio, (2011). 16. Vasatha Kadasamy, W.B. ad Smaradache, F., Dual Numbers, Zip Publishig, Ohio, (2012). 17. Vasatha Kadasamy, W.B. ad Smaradache, F., Special dual like umbers ad lattices, Zip Publishig, Ohio, (2012).

218 Further eadig Vasatha Kadasamy, W.B. ad Smaradache, F., Special quasi dual umbers ad Groupoids, Zip Publishig, Ohio, (2012). 19. Vasatha Kadasamy, W.B. ad Smaradache, F., Algebraic Structures usig Subsets, Educatioal Publisher c, Ohio, (2013). 20. Vasatha Kadasamy, W.B. ad Smaradache, F., Algebraic Structures usig [0, ), Educatioal Publisher c, Ohio, (2013). 21. Vasatha Kadasamy, W.B. ad Smaradache, F., Algebraic Structures o the fuzzy iterval [0, 1), Educatioal Publisher c, Ohio, (2014). 22. Vasatha Kadasamy, W.B. ad Smaradache, F., Algebraic Structures o Fuzzy Uit squares ad Neturosophic uit square, Educatioal Publisher c, Ohio, (2014). 23. Vasatha Kadasamy, W.B. ad Smaradache, F., Algebraic Structures o eal ad Neturosophic square, Educatioal Publisher c, Ohio, (2014).

219 NDEX D Decimal polyomial rig, 73-4 F Fuzzy MOD real plae, 20-5 Fuzzy real MOD plae, 20-5 M MOD complex decimal polyomial group, MOD complex decimal polyomial pseudo rig, MOD complex decimal polyomial semigroup, MOD complex decimal polyomials, MOD complex group, MOD complex plaes, MOD complex polyomial group, MOD complex polyomial pseudo rig,

220 MOD complex polyomial semigroup, MOD complex polyomial, MOD complex pseudo rig, MOD complex semigroup, MOD complex trasformatio, MOD dual umber plae group, MOD dual umber plae pseudo rig, MOD dual umber plae semigroup, MOD dual umber plae, MOD dual umber polyomial group, MOD dual umber polyomial pseudo rig, MOD dual umber polyomial semigroup, MOD dual umber polyomials, MOD dual umber trasformatio, MOD fuctio, 21-8 MOD fuzzy eutrosophic abelia group, MOD fuzzy eutrosophic decimal polyomials, MOD fuzzy eutrosophic plaes, MOD fuzzy eutrosophic polyomial group, MOD fuzzy eutrosophic polyomial pseudo rig, MOD fuzzy eutrosophic polyomial semigroup, MOD fuzzy eutrosophic polyomials, MOD fuzzy eutrosophic pseudo rig, MOD fuzzy eutrosophic semigroup, MOD fuzzy plae, 20-5, MOD fuzzy polyomials, 70-5 MOD fuzzy trasformatio, 70-5 MOD iterval pseudo rig, 54-6 MOD eutrosophic plae group, MOD eutrosophic plae pseudo rig, MOD eutrosophic plae semigroup, MOD eutrosophic plaes, MOD eutrosophic polyomial group, MOD eutrosophic polyomial semigroup, MOD eutrosophic polyomial pseudo rig, MOD eutrosophic trasformatio, MOD plae, 7 MOD polyomials, 50-9 MOD real plae, 9-58 dex 219

221 220 MOD Plaes MOD special dual like umber group, MOD special dual like umber plae, MOD special dual like umber polyomials, MOD special dual like umber pseudo rig, MOD special dual like umber semigroup, MOD special quasi dual umber group, MOD special quasi dual umber plae, MOD special quasi dual umber polyomial group, MOD special quasi dual umber polyomial pseudo rig, MOD special quasi dual umber polyomials, MOD special quasi dual umber pseudo rig, MOD special quasi dual umber semigroup, MOD stadardized fuctio, 57-9 MOD stadardized, 56-9 MOD trasformatio, 7 MODizatio, 56-9 MODized polyomial, 58-9 MODized, 56-9 P Pseudo MOD eutrosophic plae pseudo rig Pseudo MOD rig, S Semi ope square, 9 Special MOD distace, 14

222 ABOUT THE AUTHOS Dr.W.B.Vasatha Kadasamy is a Professor i the Departmet of Mathematics, dia stitute of Techology Madras, Cheai. the past decade she has guided 13 Ph.D. scholars i the differet fields of o-associative algebras, algebraic codig theory, trasportatio theory, fuzzy groups, ad applicatios of fuzzy theory of the problems faced i chemical idustries ad cemet idustries. She has to her credit 694 research papers. She has guided over 100 M.Sc. ad M.Tech. projects. She has worked i collaboratio projects with the dia Space esearch Orgaizatio ad with the Tamil Nadu State ADS Cotrol Society. She is presetly workig o a research project fuded by the Board of esearch i Nuclear Scieces, Govermet of dia. This is her 104 th book. O dia's 60th depedece Day, Dr.Vasatha was coferred the Kalpaa Chawla Award for Courage ad Darig Eterprise by the State Govermet of Tamil Nadu i recogitio of her sustaied fight for social justice i the dia stitute of Techology (T) Madras ad for her cotributio to mathematics. The award, istituted i the memory of dia-america astroaut Kalpaa Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prize-moey for ay dia award) ad a gold medal. She ca be cotacted at vasathakadasamy@gmail.com Web Site: or Dr. K. latheral is the editor of The Maths Tiger, Quarterly Joural of Maths. She ca be cotacted at ilatheral@gmail.com Dr. Floreti Smaradache is a Professor of Mathematics at the Uiversity of New Mexico i USA. He published over 75 books ad 200 articles ad otes i mathematics, physics, philosophy, psychology, rebus, literature. mathematics his research is i umber theory, o-euclidea geometry, sythetic geometry, algebraic structures, statistics, eutrosophic logic ad set (geeralizatios of fuzzy logic ad set respectively), eutrosophic probability (geeralizatio of classical ad imprecise probability). Also, small cotributios to uclear ad particle physics, iformatio fusio, eutrosophy (a geeralizatio of dialectics), law of sesatios ad stimuli, etc. He got the 2010 Telesio-Galilei Academy of Sciece Gold Medal, Adjuct Professor (equivalet to Doctor Hooris Causa) of Beijig Jiaotog Uiversity i 2011, ad 2011 omaia Academy Award for Techical Sciece (the highest i the coutry). Dr. W. B. Vasatha Kadasamy ad Dr. Floreti Smaradache got the 2012 New Mexico-Arizoa ad 2011 New Mexico Book Award for Algebraic Structures. He ca be cotacted at smarad@um.edu

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