CHAPTER 3. Fuzzy numbers were introduced by Hutton, B [Hu] and. studied by several Mathematicians like Kaleva [Kal], Diamond and

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1 CHAPTER 3 FUZZY NUMBERS* 3.1 Introduction: Fuzzy numbers were introduced by Hutton, B [Hu] and Rodabaugh, S. E. [Rod]. The theory of fuzzy numbers has been studied by several Mathematicians like Kaleva [Kal], Diamond and Kloeden [Dial,, Puri and lialescu [P; R], Goetschel and Voxman [G; V], Ma, Ming; Friedman, Meahem; Kadel, Abraham [M;F;K], Dib, K.A. [Dib] etc. In this chapter, the conventional definition of fuzzy real numbers are modified to give fuzzy analogue to the theory. We consider the set of fuzzy numbers as defined by Puri and Ralescu [P;R], define an equivalence relation therein and consider the equivalence classes as "the fuzzy numbers". 3.2 Equivalence relation between two fuzzy numbers Definition [P; R] A fuzzy sub set tx of R is called a fuzzy number if it satisfies *Some results contained in this chapter were published in the Joural of Fuzzy Mathematics [B; S] I.

2 the following conditions: 1. a is an upper semi continuous map 2. a[,] is non empty for all a, 0 < a is a bounded subset of R 4. a is convex. Definition Let a and j3 be two fuzzy numbers, then we define a - j3, if (a-(3)(c) = 1, c=o and (a- P)(c) = (a- p) (-c), c f 0. Theorem Proof : The above relation - is an equivalence relation. 1. Reflexivity (a - a) To prove that (a - a)(c) = 1, if c = 0 (a - (Y.)(c) = (a - a)(-c), if c + 0. (a- a)(o) = Su (a(a) A a(b)) *a- B = Sug (a(a) A a(b) a=

3 = Sup a(a) = I a Now to prove that (a - a)(c) = (a - a)(-c), c + 0 (a - a)(-c) = (a + -a)(-c) = Sup (a(a) A - a(b)) -c=a+h = Sup (a(-a) A - a(-b)).c =.a. h = Sup (a(-a) A a(b)) c =: ;I + h = Sup (a(-b) A a(a)) u=h+;l = Sup (a(a) A - a(b)) c=a+b 2. Symmetry (a - P 3 P -- a) To prove that (a - P)(O) = 1 and ((x - P)(c) = (a - P)(-c), c f 0 = - a)(o) = 1 and (P - a)(c) = (P - a)(-c), c + 0 (a - P)(O) = 1 = Sup (a(a) A -p(b)) O=a+b

4 = Sup (a(b) A P(a)), by symmetry O=h-a = Sup (P (a) A -a(-b)) O=a-h = Sup (13 (a) A-a(b)) O=a+h = (P - am. Given that (a -P)(c) = (a.-p)(-c) (1 (a -P)(-c) = -(a -P)(c) = (P - a)(c) (2) (a -P>(c> = -(a -P)(-C) = (P - a)(-c) (3) From (1), (2) and (3), it follows that (P - a) (c) = (0 - a ) (-c), c Transitivity Toprovethata-fi and p-y-a-y First prove that (a - y)(o) = 1 We have Sup (a(a) A P(c)) = 1 and Sup (P(c) ~y(b)) = 1 O=a-c 0-c h

5 Next to prove that (a - y) (c) = (a - y) (-c), c ;t 0 = Sup (a(a) A y(b)) ; t E R c=(a-11 t ( t h) [since 13 is normal] 5 Sup [(SUP (a(4ap(t))) A (Sup(P(t)Ay(b)))] C=C +C c =a-t c =t-h I? I 2 = Sup [(a - PMc, )A@ - )I c=c +C I 2 = Sup [(a - PXc, )A(P - y)(c, )] C=. C C 1 2 = Sup [(a -p)(c,)~(p -y)(c, )I.C=C i c I 2

6 5 (a - Y)(-c) ~.e., (a- Y)(c) 5 (a- Y)(-c) Similarly we can prove that (a- Y)(-c) 5 (a- Y)(c) From (4) and (9, it follows that (a - y)(c) = (a - 7)-c) c + 0 Definition Define the fuzzy number 0 by O(0) = 1 O(c) = O(-c),'dc Note a-(3 iff a- p -, 0 Definition Let a and p be two fuzzy numbers. If the mid point of ala1 2 mid point of 'da > 0, then we say that a 5 (3.

7 Definition A fuzzy number a is called non-negative if the mid point of Proposition Proof: The relation 5 is anti-symmetric on fuzzy numbers a s J3 s mid point of a[,] < mid point of Pral I We take a[,] == [a,, a:] and PI.] = [pa1, 021. mid point of apal 5 mid point of Pral 1 I 2 1 i.e.; -(a, + a, ) 5: -(pa' + pa2) 2 2 I 2 it.; a, + a, 5; p,' + p,' I I 2 2 i.e.; a, - p, 5 -a, + f3, I I 2 2 i.e.; -(a, - Pa ) 2 aa - pa (6) If p 5 a then mid point of Pral < mid point of a[,, mid point of j31nl mid point of

8 I 2 i.e.; pa1 + paz 5; a, + a, 2 I I i.e.; (j,-a; <:-@,+a, i.e.; 2 2 I -(pa- a, ) 2: -(-pa1 + a, ) 2 i.e.; a, - P; 2 -(a,' - pa') 2 2 i.e.; -(a,' - pa') a, - p, By equations (6) and (7), we have I 2 2 ( a - pa'> = a, - Pa i.e.; (a - P)(c) = (a - P)(-c). mid point of a,ll 12 mid point of plil and mid point of pill 12 mid point of a[ll. Hence they are equal. So (a - p)(o) := I Therefore a - D. Note Embed the set of real numbers R into R, by defining r(a)= 1, ifa= r =0, ifagr. Note that r is an upper semi continuous map from R + [O, 11.

9 Proposition Addition is compatible with equivalence - i.e; if al - PI and a2 -- P:!, then al + a2 - CJl+ P2. Proof: Given that (a1- P,)(O) = 1 (a1 - PI)(c) = - P I(-c), c f 0 and (a2 - P2)(c) = (a2 - P:)(-c), c # 0. We have to prove that [(a, +a,)-(pi +p2)k0) = 1 and [(a, +.,)-(PI +P, )kc) =[(a, +a2) -(PI +P2)k-c) [(a, +a, )-(PI + P2):1(0) = [(a, - PI )+(a,'p2 = 1, since (ai - PI)(0) = 1 and (a2- P2)(0) = 1. = Sup [(a, P,)(a)A(u, - P2 -c = a+ h = Sup [(a, - PI )(-a)a(ul - P2 )(-b)l -c - a+ h

10 = Sup [(a, - P, )(a)a(a, - P, )(b)], by hypothesis..c = -a - h = Sup [(a, - P )(a)a(a, - P, )(b)l c= l+b = [(a, -P,)+(a2 -P,)I(c) = [(a,+a,)-(pi +Dl)] (c). Since addition is commutative and associative in fuzzy numbers. Proposition If a, and y are non-negative fuzzy numbers, then multiplication is compatible with equivalence - i.e.; if a - P, then ay - Py. Proof: Given that ci - p. i.e.; (a - P)(O) =: 1 (a - P)(c) = (a - S N-c> We have to prove that (ar- Pr)(O) = 1. (ay - l%)(o) = ((a - P)Y)(O) (by distributivity of multiplication over addition in fuzzy numbers) = Sup ((a-l3)(a) A y(b)) 0 ;.ah

11 such that y(b) = 1 and since 0 = 0. b. Also we prove that == 1. Since ( a - 13 )(0) = 1 and there exists b (ay - PY)(c) = - PY)(-c) (ay - PY)(c) = ((01 - P)Y)(C) Proposition If a,, az, and pz are non-negative fuzzy numbers,then multiplication is compatible with equivalence -. i.e.; if al - PI and az-, PZ, then aluz - PIP> Proof: a,-pi-0 and az-pz -0

12 Proposition Proof: Case I: Case 11: If a is a non-negative fuzzy number, then a I 2 I 1 I 2 Let aral = [a,, a, 1. tn al 2 0 and (a, + a, ) I 3 a, + a',' 2 0. (a2)raj = [min((aal)', u,'a~, (a:)'), rna~((a,')~, aala,z, (cx,~)~)] If a,' 2 0, then m ~d point of (a2),,, 2 0 and (a2)! 2 0 a a2 > 0. If a,' i 0, (say -a,,'), then I 2 -a, +a, 2 0, since a 2 0. I I 2 i.e.; -a, >-a, =* (a,) I ala,

13 I 2 I 2 ' 2 1.e.; a, I a % a, a, I (a;) I (a, ) I a, a, I (aa ). Hence min((aa')2, aala,,(a,) ) - (a,'12 max((aal)*, aaia,z, = (a,212 There fore (a2)[,] = [(a,,')', (aa2)] > 2. a 2 0. Proposition number. Proof: The infimum of a sequence of fuzzy numbers is also a fuzzy Let {a,) be a cauchy sequence of hzzy numbers. Also assume {a,} is monotonically decreasing. i.e.; al > a* >.... Then {a,,[a1} is a cauchy sequence. So that there exists a C,E K,C, for each a E 1 such that dh(ancal,c,) -t 0 as n + co and this C, satisfies the properties of Theorem So there exists a fuzzy number a such that a,,, = C,, Va E I. Also we can prove that this a,,] = [inf a,,', iilf tx,, ] 2 I I 2 = [a,, a,'], where an[a] = 1%, a na I. As sequence {a,} is monotonically decreasing, we have

14 Hence (anl') is a monotonically decreasing bounded below sequence I in R, which converges to its greatest lower bound al. Similarly, sequence an12 converges to its greatest lower bound a12 1 I I 2 and sequence -(a,,' + a,;?) converges to (a, + a, ) I 2 Thus we have a[ll = [all, a.12] and mid point of = (a, + a, ). 2 I 2 So a, = [a,, a, 1. Now to prove that a is the greatest lower bound of {a,,); suppose that a is not the greatest lower bound. Then there exists a lower bound P such that rj > a. I rj>a ~ ~I1>al, ~ j ~ and ~ > a ~ ~ 1 I I -(pa' + ~ 2) 2 2 > - (a, + a,'). But the infanl =al I I 2 2 inf anl = al 1 I 2 inf -(ana '+al)=-(a,,+a,) 2 2

15 which is a contradiction. Hence the result. Notation The set of equivalence classes of fuzzy numbers is denoted by - R and the set of equivalence classes of all non-negative fuzzy numbers is denoted by R' The equivalence class containing a fuzzy number a is denoted by [a]. The equivalence class containing the fuzzy number 0 is denoted by O where 0 is defined by O(0) = 1, O(c) = 0(-c), Vc. Thus we have the following: Note [a] = [PI if and only if [a]- [PI = 8.

An Implicit Method for Solving Fuzzy Partial Differential Equation with Nonlocal Boundary Conditions

An Implicit Method for Solving Fuzzy Partial Differential Equation with Nonlocal Boundary Conditions American Journal of Engineering Research (AJER) 4 American Journal of Engineering Research (AJER) e-issn : 3-847 p-issn : 3-936 Volume-3, Issue-6, pp-5-9 www.ajer.org Research Paper Open Access An Implicit