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lOMoARcPSD|28342969 Fuzzy Logic IMportantLecture Notes Notes book V.V.IMP Master in Science in IT Ai And Soft Computing (University of Mumbai) Studocu is not sponsored or endorsed by any college or university Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) lOMoARcPSD|28342969 1.2 FUzZY LOGIC Fuzzy set theory proposed in 1965 by Lotfi A. Zadeh (1965) is a generalization of classical set theory. Fuzzy Logic representations founded on Fuzzy set theory try to capture the way humans represent and reason with real-world knowledge in the face of uncetainty. Uncertainty could arise due to generality, vagueness, ambiguity, chance, or incomplete knowledge. A fuzzy set can be defined mathematically by assigning to universe of discourse, a value representing in each possible individual the its grade of membership in the fuzzy set. This grade with the concept corresponds to the degree to which that individual is similar or compatible represented by the fuzzy set. In other words, fuzzy sets support a flexible sense of membershipof elements to a set. to a set and hence, theory, an element either belongs to or does not belong (between 0 and I) such sets are termed crisp sets. But in a fuzzy set, many degrees of membership associated with a fuzzy set A such that the are allowed. Thus, a membership function MA(x) is the interval [0, 1]. function maps every element of the universe of discourse X to universe of discourse), the fuzzy set "tall" For example, for a set of students in a class (the to are tall with a degree of membership equal (fuzzy set A) has as its members students who of membership equal to 0.75 (4aC) are of medium height with a degree who students 1), 1(4Ax) to 0 (MAx) 0), to cite a few are dwarfs with a degree of membership equal who those and = 0.75) values between student of the class could be graded to hold membership cases. In this way, every on their height. 0 and 1 in the fuzzy set A, depending sets to express gradual transitions from membership The capability of fuzzy = 0) and vice versa has a broad utility. It not only (4A(T) non-membership to < 1) (0 HA(x) measurement of uncertainties, but also In classical set = = of meaningful and powerful representation of vague concepts expressed in natural language. a meaningful representation difference, and as union, intersection, subsethood, product, equality, such Operations sets to varying degree of on fuzzy sets. Fuzy relations associate crisp defined also are disjunction of such as union, intersection, subsethood, and composition membership and support operations provides for provides for a relations. Just as fuzzy logic. crisp rise theory has influenced symbolic logic, fuzzy set theory has given in symbolic logic, truth values True or False alone are accorded set While Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) to to lOMoARcPSD|28342969 propositions, in fuzzy logic multivalued truth values such as true, absolutely true, fairky true, rules (which are false, absolutely false, partly false, and so forth are supported. Fuzzy inference rule based systems computational procedures used for evaluating linguistic descriptions) and fuzzy in real-world problems. (which are a set of fuzzy F-THEN rules) have found wide applications especially promoted by Fuzzy logic has found extensive patronage in consumer products control systems, pattern recognition the Japanese companies and have found wide use in applications, and decision making, to name a few. Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) lOMoARcPSD|28342969 Product The of two fuzzy sets product of two fuzzy defined as sets A and B is a new fuzzy set A B whose membership function H (x) H(r) 4;() 6.29) = Example = {71, 0.2), (r2, 0.8). (xz, 0.4)} B Since A = max = Unions Min = Intersection { , 0.4) (2, 0), (rz, 0.1)} B = {(71, 0.08) (p, 0) (3, 0.04)} Hi.B(a) = i a)"z(az) 0.2 HaB(2) = 0.4 0.08 Hj(z)Hz (z2) 0 is 0= Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) = 0.8 lOMoARcPSD|28342969 HaB(x3) = H(x3)Hz() = 0.4 0.1 = 0.04 Equality Two fuzzy sets and B are said to be equal (A = Ë) if uz(x) = H(t) Example {,0.2)(*2,0.8)} B =((.0.6)(*,,0.8) C ={(.0.2)(x,,0.8)} Since Pa() Hz(x) although but Since Ha(y) = HE (%) = 0.2 and Product of a H x2) = H(x2) = 0.8 fuzzy Multiplying a fuzzy set with membership function set A a by crisp number a crisp number a results in a new fuzzy set P a ) = a-Hjt) Example A= For {(z,0.4), (z2,0.6), (*z,0.8)} a = 0.3 {(.0.12), (rz 0.18), (x3.0.24)} Ha)= a Hz(x1) a since, = = 0.3 0.4 0.12 PaA2)= a Hj(x2) 0.3 0.6 0.18 Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) product a. A with t: (6.3 lOMoARcPSD|28342969 Haa) aPj(x3) = 0.3 0.8 = 0.24 Power of a Juzzy set n a w e r of a fuzzy set A 1S a new fuzzy set A" whose membership function is given by HA (x) = (4z («)" n.idine fuzzy a Is called set to its Dilation (6.32) second power is called Concentration (CON) and taking the square root (DIL). Example {(x.0.4), (x2.0.2), (x3.0.7)} a= 2 For (A = {(.0.16), (F2,0.04). (x3,0.49 Hence Ha:() Since = (4z(z))* (0.4 = = Ha(2)= (4z(x)* (0.2 (Hz73)) (0.7) H(a3) = 0.16 = = 0.04 = = 0.49 Difference The uerence of two fuzzy sets and B is a new fuzzy set A -B defined as A-B=( Example A (6.33) {x,0.2), (rz,0.5). (x3,0.6)); B= {(%.0.1), (rz 0.4), (73,0.5)) B = x.0.9), (x2.0.6), (x3,0.5)} = {(x,0.2)(7z,0.5)(x.0.5)) Disjunctive sum disjunctive sur Sum of two fuzzy sets and B is a new fuzzy set A B defined as (6.34) Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) lOMoARcPSD|28342969 176 Neural Netoorks, Fuzzy Logic, and GeneticAigorithms-ymthesis and Applications Example {.0.4)(x,,0.8)(x,0.6) B = (z,0.2)(x,.0.6)(,0.9)1 A Now, = {lz,.0.6)%*,,0.2)(x3,0.4)) B° {(,0.8)(r2,0.4)(x3,0.1)} AnB = {(7,,0.2)(z,.0.2)(x3,0.4)} An ={G;,0.4)(7g,0.4)(a,0.)) AB 6.3.3 = (7,0.4)(T2,0.4) x3,0.4)} Properties of Fuzzy Sets Fuzzy sets follow some of the properties satisfied by crisp sets. In fact, crisp sets can be thoui of as special instances of fuzzy sets. Any fuzzy set membership of any element belonging to the null set belonging to the reference set is 1. is a subset of the reference set X. Also, te is 0 and the membership of any elkemet The properties satisfied by fuzzy sets are Commutativity: (6.39 Associativity: U ) =(U B) u An (Bn ) =(Ã n B) n Distributivitry: Au(Bn ) = ( B) n(ãu ) An(Bu ) =(Ä n B)udn ) Idempotence: An Identity: Transitivitry: If Involution: De Morgan's laws: AUX= X C , then s AY = An By=(Au B°) AU B = (Ä n B°) Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) (6.60 lOMoARcPSD|28342969 6.5.2 Operations Let on Fuzzy Relations and S be fuzzy relations on Xx Y. Union HRUF(x,y)) = max (j4g(,). 4z(x,y)) (6.54) Intersection HRos(x,) = min(4^l(x,), Ag(x,y) Complement Hre (x,y) =l-Hr(zy) Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) (6.55) (6.56) lOMoARcPSD|28342969 6.5 FUZZY RELATIONS fuzzy n-tuples (1, X2,.., Xn) Fuzzy relation is a set defined on the Cartesian product of crisp X1, X... X whnere the within sets of membership within the may have varying degrees of the relation between the tuples. membership values indicate the strength relatine relation. The Example Let R be the fuzzy relation between two sets X1 and X2 where X1 is the set of diseases and X is the set of symptoms. X X2 The = {typhoid, viral fever, = {running nose, fuzzy 6.5.1 high common cold} temperature, relation R may be defined shivering} as Running High Shivering nose 0.8 Typhoid 0.1 temperature 0.9 Viral fever 0.2 0.9 0.7 Common cold 0.9 0.4 0.6 Fuzzy Cartesian Product Let A be fuzzy o n the universe. defined on the universe X and B be a fuzzy set defined on the u the Cartesian product between the fuzzy sets A and B indicated as AxB and resu fuzzy relation R is given by a set Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) in a lOMoARcPSD|28342969 Fuzzy Set Theory where R has its here R R=AxBc XxY given by HRx,) = HixBl7,y) Example = (652) membership function = Let A 183 min (6.53) ((x), Hg (y) {(X1. 0.2), (T2, 0.7), (x3, 0.4)} and = sets defined ((V. 0.5). (ya. 0.6)) be two the universes of discourse X = {x1, x2, xal and Y= {y1, yat respectively. Then fuzzy the fuzzy relation on r resulting out of the fuzzy Cartesian product A xB is given by y2 0.2 0.2 R AxB= *2 0.5 0.6 30.4 0.4 since, ) min ("z(x),Hz(y) R(1.2)=min(0.2,0.6) =0.2 R( = R(x2,y) =min(0.7,0.5) = 0.5 min (0.7,0.6) = 0.6 = 0.4 R(2,y2) = R(x3.y)= min(0.4,0.5) R(x3, y2) min (0.4,0.6) = = = min (0.2,0.5) 0.4 Downloaded by 22MTCSE004 SHUBHA LAXMI PATRA (22mtcse004.shubhalaxmipatra@giet.edu) = 0.2