Some Coding Theorems on New Generalized Fuzzy Entropy of Order α and Type β

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1 ppl. Math. If. Sci. Lett. 5, No. 2, ) 63 pplied Mathematics & Iformatio Scieces Letters Iteratioal Joural Some Codig Theorems o New Geeralized Fuzzy Etropy of Order ad Type shiq Hussai Bhat ad M.. K Baig P. G. epartmet of Statistics Uiversity of Kashmir, Sriagar-90006, Idia Received: 7 Feb. 206, Revised: 6 ug. 206, ccepted: 6 ug. 206 Published olie: May 207 bstract: I this commuicatio we itroduce a ew geeralized fuzzy iformatio measure H ) of order ad type for a fuzzy set ad establishes the validity of the measure as a fuzzy etropy. lso we defie a ew geeralized fuzzy average code-word legth L ) of order ad type for a fuzzy set ad its relatioship with geeralized fuzzy iformatio measures H ) have bee discussed. Usig L P), some codig theorems for discrete oiseless chael has bee proved. The measures defied i this commuicatio are ot oly ew but some kow measures are the particular cases of our proposed measures that already exist i the literature of fuzzy iformatio theory. Keywords: Fuzzy set, Membership fuctio, Shaos etropy, Reyis etropy, Fuzzy etropy, Code-word legth, Kraft iequality, Codig theorem, Holders iequality ad Optimal code legth. MS classificatio 947, 9424 Itroductio: Fuzziess ad ucertaity are the basic ature of huma thikig ad may real world objectives. Fuzziess is foud i our decisio, i our laguage ad i the way of process iformatio. The mai objective of iformatio is to remove ucertaity ad fuzziess. I fact, we measure iformatio supplied by the amout of probabilistic ucertaity removed i a experimet ad the measure of ucertaity removed is also called as a measure of iformatio, while measure of fuzziess is the measure of vagueess ad ambiguity of ucertaities. The cocept of etropy has bee widely used i differet areas, e.g. commuicatio theory, statistical mechaics, fiace patter recogitio, ad eural etwork etc. Fuzzy set theory developed by Lotfi.. Zadeh 29 has foud wide applicatios i may areas of sciece ad techology, e.g. clusterig, image processig, decisio makig etc. because of its capability to model o-statistical imprecisio or vague cocepts. The importace of fuzzy sets comes from the fact that it ca deal with imprecise ad iexact iformatio, may fuzzy measures have bee discussed ad derived by Kapur, Lowe 5, Nguye ad Walker 9, Parkash 24, Pal ad Bezdek 22, Zadeh 29 etc. pplicatio of fuzzy measures to egieerig, fuzzy traffic cotrol, fuzzy aircraft cotrol, medicies, computer sciece ad decisio makig etc, have already bee established. The basic oiseless codig theorems see for istace; the papers, czel, Kapur 0, Kha et al 3, Reyi 25, Va er Lubbe 28, give the lower boud for the mea code-word legth of a uiquely decipherable code i terms of Shaos 26 etropy. Kapur 2 has established relatioships betwee probability etropy ad codig. But there are situatios where probabilistic measures of etropy do ot work, to tackle such situatios, istead of takig the probability, the idea of fuzziess ca be explored. Let a uiverse of discourse X {x,x 2,...,x } the a fuzzy subset of uiverse X is defied as: {x i, µ x i )) : x i X, µ x i ) 0,} Where µ x i ) : X 0, is a membership fuctio ad gives the degree of belogigess of the elemet x i to the set ad is defied as follows: 0 If x i / ad there is o ambiguity, µ x i ) If x i ad there is o ambiguity, 0.5 If x i or x i / ad there is maximum ambiguity, Correspodig author ashiqhb4@gmail.com c 207 NSP Natural Scieces Publishig Cor.

2 64. H Bhat, M.. K Baig: Some codig theorems o ew geeralized fuzzy etropy of order ad type I fact µ x i ) associates with each x i,x gives a grade of membership fuctio i the set. Whe µ x i ) takes values oly 0 or, there is o ucertaity about it ad a set is said to be a crisp i.e. o-fuzzy) set. Some otios related to fuzzy sets which we shall eed i our discussio Zadeh 29. Cotaimet: If B µ x i ) µ B x i ) x i X Equality : If B µ x i ) µ B x i ) x i X Complemet: If c is complemet of µ cx i ) µ x i )) x i X Uio: If B is uio of ad B µ B x i ) Max{µ x i ), µ B x i )} x i X Itersectio: If B is itersectio of ad B µ B x i )Mi{µ x i ), µ B x i )} x i X Product: If B is product of ad B µ B x i ) µ x i ) µ B x i ) x i X Sum: If +B is sum of ad B µ +B x i ) µ x i )+ µ B x i ) µ x i ) µ B x i ) x i X 2 Basic Cocepts: Let X be a discrete radom variable takig values x,x 2,...,x with respective probabilities p, p 2,..., p p i 0 i,2,3,..., ad p i. Shao26 gives the followig measure of iformatio ad call it etropy. HP) p i log p i ) The measure ) serves as a suitable measure of etropy. Let p, p 2, p 3,, p be the probabilities of codewords to be trasmitted ad let their legths l,l 2,,l satisfy Kraft 4 iequality, l i 2) For uiquely decipherable codes, Shao26 showed that for all codes satisfyig 2),the lower boud of the mea codeword legth, L p i l i 3) lies betwee HP) ad HP)+.Where is the size of code alphabet. If x,x 2,...,x are members of the uiverse of discourse, with respective membership fuctios µ x ), µ x 2 ),..., µ x ),the all µ x ), µ x 2 ),..., µ x ) lies betwee 0 ad but these are ot probabilities because their sum is ot uity. µ x i ), gives the elemet x i the degree of belogigess to the set The fuctio µ x i ) associates with each x i R a grade of membership to the set ad is kow as membership fuctio. eote F.S x x 2... x µ x ) µ x 2 )... µ x ),0 µ x i ) x i X 4) We call the scheme 4) as a fiite fuzzy iformatio scheme. Every fiite scheme describes a state of ucertaity, correspodig to Shaos 26 probabilistic etropy, e-luca ad Termii 7 suggested the followig measure of fuzzy etropy. H) µ x i )log µ x i )+ µ x i ))log µ x i )) 5) The measure 5) serves as a very suitable measure of fuzzy etropy for the fiite fuzzy iformatio scheme 4) e-luca ad Termii 7 itroduced a set of four properties ad these properties are widely accepted as for defiig ew fuzzy etropy. I fuzzy set theory, the etropy is a measure of fuzziess which expresses the amout of average ambiguity i makig a decisio whether a elemet belogs to a set or ot. So, a measure of average fuzziess H) i a fuzzy set should have the followig properties to be valid fuzzy etropy:.sharpess: H) is miimum if ad oly if is a crisp set, i,e,. µ x i )0 or ; x i,,2,..., 2.Maximality: H) is maximum if ad oly if is most fuzzy set, i.e., µ x i ) 2 ; x i,,2,..., 3.Resolutio: H ) H), where is sharpeed versio of. 4.Symmetry: H) H c ), where c is the complemet of, i.e., µ cx i ) µ x i ); x i,,2,..., Later o Bhadari ad Pal 4 made a survey o iformatio measures o fuzzy sets ad gave some fuzzy iformatio measures, aalogous to Reyis 25 iformatio measure; they have suggested the followig fuzzy iformatio measure of order as: H ) log µ x i )+ µ x i )) ;, 0 6) ad aalogous to Pal ad Pals 23 expoetial etropy, they have suggested the followig fuzzy iformatio measure: H e ) e log µ x i )e µ x i ) + µ x i ))e µ x i ) alogous to Havrda ad Charvats 8 iformatio measure, Kapur suggests the followig fuzzy iformatio measure: H ) µ x i )+ µ x i )) ;, 0 8) Correspodig to Boekee ad Lubbe 5 R-orm iformatio measure, Hooda 9 proposed ad characterize the followig fuzzy iformatio measure: H R ) R µ R x i )+ µ x i )) R) R ;R,R 0 9) R 7) c 207 NSP Natural Scieces Publishig Cor.

3 ppl. Math. If. Sci. Lett. 5, No. 2, ) / 65 Correspodig to Campbells 6 measure of etropy, Parkash ad Sharma 20 proposed ad characterize the followig fuzzy iformatio measure: H ) / log { µ x i )+ µ x i )) } ;, 0 0) Correspodig to Sharma ad Taeja 27 measure of etropy of degree, ) Kapur has take the followig measure of fuzzy etropy: H /, ) {µ x i )+ µ x i )) } {µ x i)+ µ x i )) } ) Where,, or, I the ext sectio we propose a ew geeralized fuzzy iformatio measure of order ad type aalogous to shiq ad Baigs 3 geeralized etropy of order ad type ad show that it is a valid fuzzy iformatio measure. 3 New Geeralized Fuzzy Iformatio Measure ad Its Properties: Let be the fuzzy set defied o a discrete uiverse of discourse takig values x,x 2,,x havig the membership values µ x ), µ x 2 ),..., µ x ) respectively. We defie a ew geeralized fuzzy iformatio measure of order ad type aalogous to shiq ad Baigs 3 geeralized etropy of order ad type as: H) log µ x i)+ µ x i )) ),0< <,0< I order to prove that 2) i.e., H ) is a valid fuzzy iformatio measure, we shall show that four properties P-P4) i.e. sharpess, maximality, resolutio ad symmetry are satisfied. P.Sharpess): H ) is miimum if ad oly if is a crisp set, i,e, 2) H )0 if ad oly if µ x i )0 or,2,..., Proof: Let H)0 log log µ x i)+ µ x i )) ) 0 µ x i)+ µ x i )) ) 0 3) We kow that log x 0, if x. Usig this result i equatio 3), we get µ x i)+ µ x i )) ) Or equivaletly, we ca write, µ x i)+ µ x i )) ) 4) Now 0 < <,0 < 4), will hold whe either µ x i ) or µ x i )0,,2,..., Next, coversely, if is a crisp set, the either µ x i ) or µ x i )0,,2,..., It implies µ x i)+ µ x i )) ), 0< <,0< log log µ x i)+ µ x i )) ) 0 µ x i)+ µ x i )) ) H )0 Hece, H ) 0, if ad oly if is o fuzzy set or crisp set. P2.Maximality): H ) is maximum if ad oly if is most fuzzy set. Proof: We have 0 H) log µ x i)+ µ x i )) ) Now differetiatig equatio 5) with respect to µ x i ), we get H) µ x i ) 2 µ x i ) µ x i )) ) µ x i)+ µ x i )) ) Let 0 µ x i )<0.5, the H) > 0, 0< <,0<. µ x i ) Hece, H ) is a icreasig fuctio of µ x i ), wheever, 0 µ x i )<0.5 Similarly, for 0.5< µ x i ) we have H) < 0, 0< <,0<. µ x i ) Hece, H ) is decreasig fuctio of µ x i ), wheever, 0.5< µ x i ) d for µ x i )0.5 H) 0, 0< <,0<. µ x i ) Thus H ) is a cocave fuctio which has a global maximum at µ x i ) 0.5. Hece H ) is maximum if ad oly if is the most fuzzy set, i.e. µ x i )0.5,,2,...,. 5) c 207 NSP Natural Scieces Publishig Cor.

4 66. H Bhat, M.. K Baig: Some codig theorems o ew geeralized fuzzy etropy of order ad type P3.Resolutio): H ) H ), where is sharped versio of. Proof: Sice H ) is icreasig fuctio of µ x i ) i the iterval0,0.5) ad is decreasig fuctio of µ x i ) i the iterval0.5,, therefore ad µ )x i ) µ x i ) H ) H ),i0,0.5), µ )x i ) µ x i ) H ) H ),i0.5,, Takig the above equatios together, we get H ) H ). P4.Symmetry): H) H c ), where c complemet of. is the Proof: It may be oted that from the defiitio of H ) ad µ cx i ) µ x i )), we coclude that H )H c ). Hece, H ) satisfies all the properties of fuzzy etropy, therefore H ) is a valid measure of fuzzy etropy. It is easy to see that as ad 2), reduces to 5). I the ext sectio ew geeralized fuzzy code-word mea legth aalogous to shiq ad Baigs 3 ew geeralized code-word legth of order ad type is cosidered ad bouds have bee obtaied i terms of ew geeralized fuzzy etropy measure H ) aalogous to shiq ad Baigs 3 ew geeralized etropy of order ad type.the mai aim of these results is that it geeralizes some well-kow fuzzy measures already existig i the literature. Geeralized fuzzy codig theorems by cosiderig differet fuzzy iformatio measures were ivestigated by several authors see for istace, the papers: M.. K Baig ad Mohd Javid ar 6, 7 ad 8, shiq ad Baig 2, Parkash ad P. K. Sharma 20 ad 2, Bhadari ad Pal 4, Kapur. 4 Bouds for New Geeralized Fuzzy Iformatio Measure: Cosider shiq ad Baigs 3 geeralized etropy of order ad type as: H P) log p i 6) Where 0< <,0<,p i 0 i,2,3,..., ad p i. Further cosider shiq ad Baigs 3 ew geeralized code-word legth of order ad type correspodig to 6) ad is give by L P) log p i l i ),0< <,0<. 7) Where is the size of code alphabet. alogous to 6) ad 7), we propose the followig fuzzy measures respectively. ad H) log L) log Where,0< <,0< Remarks for 6) µ x i)+ µ x i )) ) µ x i)+ µ x i )) ) l i 8) ) 9).Whe 6) reduces to Reyis etropy,i.e., H P) log p i 2.Whe ad 6) reduces to Shaos 26 etropy i.e., HP) p i log p i 3.Whe, ad p i,2,3,..., the 6) reduces to maximum etropy i.e., Remarks for 7) H )log.for 7) reduces to code-word legth correspodig to Reyi s etropy i.e., L P) log p i l i ) 2.For ad 7) reduces to optimal code-word legth correspodig to Shao 26 etropy i.e., L p i l i 3.For ad l l 2 l the 7) reduces to. i.e., L c 207 NSP Natural Scieces Publishig Cor.

5 ppl. Math. If. Sci. Lett. 5, No. 2, ) / 67 Now we foud the bouds of 9) i terms of 8) uder the coditio, Or, we ca write µ x i)+ µ x i )) ) l i 20) f µ x i ), µ cx i )) l i 2) Where f µ x i ), µ cx i )) µ x i)+ µ x i )) ) Which is geeralized fuzzy Kraft 4 iequality, where is the size of code alphabet, it is easy to see that for, the iequality 2) reduces to Kraft 4 iequality. Theorem 4.. For all itegers > ) the code word legths l,l 2,...,l satisfies the coditio 2), the the geeralized code-word legth 9) satisfies the iequality L ) H ).Where,0< <,0<. 22) Where equality holds good if l i log f µ x i ), µ cx i )) Where f µ x i ), µ cx i )) µ x i)+ µ x i )) ) Proof. By Holder s Iequality we have x i y i x p i 23) ) p q yi) q 24) For all x i,y i > 0,i,2,3,..., ad p + q, p < 0),q<0 or q< 0), p < 0. We see the equality holds iff there exists a positive costat c such that, Makig the substitutio x p i cy q i 25) x i f µ x i ), µ cx i ))) l i y i f µ x i ), µ cx i ))) p ad q Usig these values i 24), ad after suitable simplificatio we get, f µ x i ),µ c x i )) l i { f µ x i ),µ c x i ))} l i { f µ x i ),µ c x i ))} Now usig the iequality 2) we get, ) { ) f µ xi, µ c ))} l x i i Or equatio 27) ca be writte as { ) f µ xi, µ c ))} x i ) 26) { f µ xi ), µ c x i ))} 27) ) { ) f µ xi, µ c ))} l x i i Takig logarithms to both sides with base to equatio 28) we get log { ) f µ xi, µ c ))} x i log 28) ) { ) f µ xi, µ c ))} l x i i 29) s 0<,multiply equatio 29) both sides by > 0, we get log { ) f µ xi, µ c ))} x i log ) { ) f µ xi, µ c ))} l x i i 30) Takig f µ x i ), µ cx i )) µ x i)+ µ x i )) ),we get log µ ) ) )) xi + µ xi log µ ) ) )) xi + µ xi ) i l Or equivaletly, we ca write L ) H ),0 < <,0 <. Hece the result. From equatio 23) we have l i log f µ x i ), µ cx i )) Or equivaletly, we ca write l i f µ x i ), µ cx i )) 3) 32) Raisig both sides to the power to equatio 32) ad after simplificatio, we get l i ) f µ x i ), µ cx i )) 33) Multiply equatio 33) both sides by { f µ x i ), µ cx i ))} ad the summig over i,2,...,, both sides to the resultat expressio ad after simplificatio, we get ) { ) f µ xi, µ c ))} l x i i { ) f µ xi, µ c ))} x i 34) c 207 NSP Natural Scieces Publishig Cor.

6 68. H Bhat, M.. K Baig: Some codig theorems o ew geeralized fuzzy etropy of order ad type Takig logarithms both sides with base to equatio 34), the multiply both sides by, we get ) log { ) f µ xi, µ c ))} l x i i log { ) f µ xi, µ c ))} x i 35) Or equivaletly, we ca write L )H ).Hece the result Where f µ x i ), µ cx i )) µ x i)+ µ x i )) ) Theorem 4.2. For every code with legths l,l 2,...,l satisfies the coditio 2), L P) ca be made to satisfy the iequality, L )<H P)+, where 0< <,0<. 36) Proof. From the theorem 4. we have Holds if ad oly if L )H ) l i f µ,0< <,0< x i ), µ cx i )) Or equivaletly we ca write l i log f µ x i ), µ cx i )) We choose the code-word legths l i,,2,..., i such a way that they satisfy the iequality, ) log f µ xi, µ c )) ) x i l i < log f µ xi, µ c )) x i + 37) Cosider the iterval ) δ i log f µ xi, µ c )) ) x i, log f µ xi, µ c )) x i + of legth uity.i every δ i, there lies exactly oe positive iteger l i, such that, ) 0<log f µ xi, µ c )) ) x i l i < log f µ xi, µ c )) x i + 38) We will first show that the sequece l,l 2,...,l, thus defied satisfies the iequality 2), which is geeralized fuzzy Kraft 4 iequality. From the left iequality of 38), we have log f µ x i ), µ cx i )) l i Or equivaletly, we ca write l i f µ x i ), µ cx i )) 39) Multiply equatio 39) both sides by { f µ x i ), µ cx i ))} ad the summig over i,2,...,, o both sides to the result that we obtai, we get the required result 2),which is geeralized fuzzy Kraft 4 iequality. Now the last iequality of 38) gives l i < log f µ x i ), µ cx i ))) + Or equivaletly, we ca write l i < f µ x i ), µ cx i ))) 40) s 0 < <, the ) > 0, ad ) > 0, raisig both sides to the power ) > 0, to equatio 40),ad after suitable simplificatio, we get l i ) < ) f µ x i ), µ cx i ))) 4) Multiply equatio 4) both sides by { f µ x i ), µ cx i ))} ad the summig over i, 2,...,, o both sides to the resulted expressio, ad after makig suitable operatios we get, ) { ) f µ xi, µ c ))} l x i i < { f µ xi ), µ c x i ))} 42) Takig logarithms to both sides with base to equatio 42), we get, log ) { ) f µ xi, µ c ))} l x i i < log { f µ xi ), µ c x i ))} + ) 43) s 0< <,0< the )> 0 ad > 0, ) multiply both sides equatio 43) by > 0, we get ) log { ) f µ xi, µ c ))} l x i i < log { ) f µ xi, µ c ))} x i + 44) Or equivaletly, we ca write L )<H )+.Hece the result for 0< <,0<. Thus from above two codig theorems we have show that H ) L )<H P)+,0< <,0< 5 Coclusio: I this paper we defie a ew geeralized fuzzy etropy measure of order ad type. ad show that this is a valid measure of fuzzy etropy. This measure also geeralizes some well-kow fuzzy measures already existig i the literature. lso geeralized fuzzy average codeword legth is cosidered ad bouds have bee obtaied i terms of ew geeralized fuzzy etropy measure of order ad type. c 207 NSP Natural Scieces Publishig Cor.

7 ppl. Math. If. Sci. Lett. 5, No. 2, ) / 69 Refereces czel J. O Shaos iequality optimal codig ad characterizatio of Shaos ad Reyis etropies. Istitute Novoal e lta Mathematics Symposia Maths 5, ). 2 shiq H. B ad M.. K. Baig. Codig Theorems o Geeralized Useful Fuzzy Iaccuracy Measure. Iteratioal Joural of Moder Mathematical Scieces 4, ). 3 shiq H. B ad M.. K Baig. Some codig theorems o geeralized Reyis etropy of order ad type. Iteratioal Joural of pplied Mathematics ad Iformatio Scieces Letters 5, ). 4 Bhadari. ad N. R. Pal. Some ew iformatio measures for fuzzy sets. Iformatio Sciece 67, ). 5 Boekee. E ad Lubbe J. C.. The R-orm iformatio measures. Iformatio ad Cotrol 45, ). 6 Campbell L. L. codig theorem ad Reyis etropy. Iformatio ad cotrol 8, ). 7 e Luca ad Termii S. efiitio of No-probabilistic Etropy i the Settig of fuzzy set theory. Iformatio ad Cotrol 20, ). 8 Harvda J. H ad Charvat F. Quatificatio method of classificatio process the cocept of structural etropy. Kyberetika 3, ). 9 Hooda. S. O geeralized measures of fuzzy etropy. Mathematica Slovaca 54, ). 0 Kapur J. N. geeralizatio of Campbells oiseless codig theorem. Joural of Bihar Mathematical Society 0, ). Kapur J. N. Measures of Fuzzy Iformatio. Mathematical Sciece Trust Society, New elhi, 997). 2 Kapur J. N. Etropy ad Codig. Mathematical Sciece Trust Society, New elhi, 998). 3 Kha. B., utar R. ad hmad H. Noiseless Codig Theorems for Geeralized No-additive Etropy. Tam Kog Joural of Mathematics 2, ). 4 Kraft L. J. device for quatizig groupig ad codig amplitude modulates pulses, M.S Thesis, epartmet of Electrical Egieerig, MIT, Cambridge, 949). 5 Lowe R. Fuzzy Set TheoryBasic Cocepts, Techiques ad Bibliography. Kluwer cademic Publicatio. pplied Itelligece 3, ). 6 M.. K Baig ad Mohd Javid ar. Some Codig theorems o Fuzzy etropy Fuctio epedig upo Parameter R ad V. IOSR Joural of Mathematics 9, ). 7 M.. K Baig ad Mohd Javid ar. Fuzzy codig theorems o geeralized fuzzy cost measure. sia Joural of Fuzzy ad pplied Mathematics 2, ). 8 M.. K Baig ad Mohd Javid ar. Some ew geeralizatio of fuzzy average codeword legth ad their Bouds merica Joural of pplied Mathematics ad Statistics 2, ). 9 Nguye H. T ad Walker E.. first course i fuzzy logic. C.R.C Press Ic. 997). 20 Om Parkash ad P. K. Sharma. ew class of fuzzy codig theorems. Caribb. J. Math. Computer Scieces 2, ). 2 Om Parkash ad P. K. Sharma. Noiseless codig theorems correspodig to fuzzy etropies. Southeast sia Bulleti of Mathematics 27, ). 22 Pal ad Bezdek. Measurig Fuzzy Ucertaity. IEEE Tras. of fuzzy systems 2, ). 23 Pal N. R ad Pal S. K. Object backgroud segmetatio usig ew defiitio of etropy. Proc. Ist. Elec. Eg. 36, ). 24 Parkash O. ew parametric measure of fuzzy etropy. Iformatio Processig ad Maagemet of Ucertaity 2, ). 25 Reyi. O measure of etropy ad iformatio. Proceedig Fourth Berkely Symposium o Mathematical Statistics ad probability Uiversity of Califoria Press, ). 26 Shao C. E. mathematical theory of commuicatio. Bell System Techical Joural 27, , ). 27 Sharma B. ad Taeja I. J. Etropies of type, ad other geeralized measures of iformatio theory. Mathematika 22, ). 28 Va er Lubbe J. C.. O certai codig theorems for the iformatio of order ad type. I. Tras., 8th Prague cof. If. Theory,. Reidell, ordrecth ). 29 Zadeh L.. Fuzzy sets. Iformatio ad cotrol 8, ). shiq Hussai Bhat is a Ph. Scholar i epartmet Of Statistics Uiversity Of Kashmir Sriagar Idia).He received the B. degree from Govt. egree College Bemia Sriagar i 200 ad received the Masters i Statistics from epartmet Of Statistics Uiversity Of Kashmir Sriagar i Jue 203). From March 204 he is a research scholar of Ph. i the epartmet Of Statistics Uiversity Of Kashmir Sriagar Idia). His research iterest icludes Iformatio Theory, Fuzzy Iformatio Measures ad Codig Theory. M.. K. Baig completed his Graduatio, Masters ad octorial degree from.m.u ligarh Idia). He was gold Medalist both i Graduatio ad Post-Graduatio, curretly he is workig i Uiversity of Kashmir Sriagar Idia), as Sr. ssistat Professor. He has supervised Six 6) Ph. ad Fiftee 5) M.Phil Scholars. He served as Head of the Statistics epartmet Uiversity of Kashmir Sriagar Idia). His research iterest icludes Iformatio Theory, Fuzzy Measures, Iformatic Reliability. He has published more tha 40 research papers i peer reviewed Jourals. c 207 NSP Natural Scieces Publishig Cor.