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### FuzzyRelations.pptx

• 1. BANGALORE UNIVERSITY UNIVERSITY VISVESVARAYA COLLEGE OF ENGINEERING K R Circle, Bangalore 560001 Department of Computer Science and Engineering COMPUTER NETWORKING Seminar On FUZZY RELATIONS Submitted By: VAISHALI BAGEWADIKAR 20GACS4017 Dept of CSE, CN Branch. 2021-2022 Under the Guidance of: Dr. PRATIBHAVANI .P.M Associate Professor Dept of CSE,UVCE Bangalore
• 2. Agenda • Crisp sets • Fuzzy sets • Cartesian product • Crisp relation • Fuzzy Relations
• 3. Basics • Crisp set • A set defined using a characteristic function that assigns a value of either 0 or 1 to each element of the universe, discriminating between members and non-members of the crisp set under consideration. • Example : light is ON or OFF • In the context of fuzzy sets theory, crisp sets are referred as “classical” or “ordinary” sets • Fuzzy sets are generalization of crisp sets where the degree of inclusiveness of an element may be anything from 0 to 1. Not just 0 or 1. • Example: weather is very cold.
• 4. • Definition Fuzzy set A fuzzy set F on a given universe of discourse U is defined as a collection of ordered pairs (x, μF (x)) where x ∊ U, and for all x ∊ U, 0.0 ≤ μF (x) ≤ 1.0. F = {(x, μF (x))} | x ∊ U, 0.0 ≤ μF (x) ≤ 1.0}
• 5. Definition of Cartesian product Let A and B be two sets. Then the Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs (a, b) such that a ∊ A, and b ∊ B. A × B = {(a, b) | a ∊ A, and b ∊ B} Since (a, b) ≠ (b, a) we have in general A × B ≠ B × A. Hence the operation of Cartesian product is not commutative
• 7. Definition -Membership Function • Given an element x and a set S, the membership of x with respect to S, denoted as μ S (x), is defined as : • µS (x) = 1, if x ∈ S • µS (x) = 0, if x ∉ S
• 8. Definition -Crisp relation • Given two crisp sets A and B, a crisp relation R between A and B is a subset of A × B. and R ⊆ A × B • Consider the sets A = {1, 2, 3}, B = {1, 2, 3, 4} relation R = {(a, b) | b = a + 1, a ∊ A, and b ∊B}. Then R = {(1, 2), (2, 3), (3, 4)}. • Here R ⊂ A × B. • A crisp relation between sets A and B is expressed with the help of a relation matrix T.
• 9. Example The rows and the columns of the relation matrix T correspond to the members of A and B respectively. A = {1, 2, 3}, B = {1, 2, 3, 4} relation R = {(a, b) | b = a + 1, a ∊ A, and b ∊ B} R = {(1, 2), (2, 3), (3, 4)}. Relation matrix for R is given below
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• 21. Cardinality of Fuzzy Relations
• 22. Operations On Fuzzy Relation μ
• 24. Properties of Fuzzy Relations • The properties of fuzzy sets hold good for fuzzy relations as well.  Commutativity  Associativity  Distributivity  Involution  Idempotency  DeMorgan’s Law  Excluded Middle Laws.
• 27. = max [ min (0.6, 1),min (0.3,0.8)] = max [0.6, 0.3] = 0.6 =max [min (0.6,0.5),min (0.3,0.4)] =max [min (0.6,0.5),min (0.3,0.4)] = max (0.5, 0.3) = 0.5 max [min (0.6,0.3),min (0.3,0.7)] MT(x1,z3)=max [min (0.6,0.3),min (0.3,0.7)] = max [0.3, 0.3] = 0.3 MT(x2,z1)=max [min (0.2,1),min (0.9,0.8)] MT(x2,z1)=max [min (0.2,1),min (0.9,0.8)] = max [ 0.2, 0.8] = 0.8 =max [min (0.2,0.5),min (0.3,0.4)] MT(x2,z2)=max [min (0.2,0.5),min (0.3,0.4)] = max [0.2, 0.4] = 0.4 max [min (0.2,0.3),min (0.9,0.7)] MT(x2,z3)=max [min (0.2,0.3),min (0.9,0.7)] = max (0.2, 0.7) = 0.7 T = RoS = [0.6 0.5 0.3 0.8 0.4 0.7]
• 29. Example(Max-Prod) T = R . S MT(x1,z1) = max [MR(x1,y1).MS(y1,z1)] = max [MR(x1,y1).MS(y1,z1)] MR(x1,y2).MS(y2,z1)] MR(x1,y2).MS(y2,z1)]= max (0.6, 0.24) = 0.6 MT(x1,z2)=max [MR(x1,y1).MS(y1,z2)] =max [MR(x1,y1).MS(y1,z2)] [MR(x1,y2).MS(y2,z2)][MR(x1,y2).MS(y2,z2)] = max [0.30, 0.12] = 0.30
• 30. Example(Max-Prod) MT(x1,z3)=max [0.18,0.21]=0.21 MT(x2,z1)=max [0.2,0.72]=0.72 MT(x2,z2)=max [0.10,0.36]=0.36 MT(x2,z3)=max [0.06,0.63]=0.63 z1 z2 z3 T= R.S = x1 [ 0.6 0.3 0.21 ] x2 [ 0.72 0.36 0.63]
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