CHAPTER 3 FUZZY RELATION and COMPOSITION
Crisp relation! Definition (Product set) Let A and B be two non-empty sets, the prod uct set or Cartesian product A B is defined as follows, A B = {(a, b) a A, b B } Example A = {a 1, a 2, a 3 }, B = {b 1, b 2 } A B = {(a 1, b 1 ), (a 1, b 2 ), (a 2, b 1 ), (a 2, b 2 ), (a 3, b 1 ), (a 3, b 2 )} b 2 (a 1, b 2 ) (a 2, b 2 ) (a 3, b 2 ) b 1 (a 1, b 1 ) (a 2, b 1 ) (a 3, b 1 ) a 1 a 2 a 3
Crisp relation n n Binary Relation If A and B are two sets and there is a specific property between elements x of A and y of B, this property can be described using the ordered pair (x, y). A set of such (x, y) pairs, x A and y B, is called a relation R. R = { (x,y) x A, y B } n-ary relation For sets A 1, A 2, A 3,..., A n, the relation among elements x 1 A 1, x 2 A 2, x 3 A 3,..., x n A n can be described by n-tuple (x 1, x 2,..., x n ). A collection o f such n-tuples (x 1, x 2, x 3,..., x n ) is a relation R among A 1, A 2, A 3,..., A n. (x 1, x 2, x 3,, x n ) R, R A 1 A 2 A 3 A n
Crisp relation! Domain and Range dom(r) = { x x A, (x, y) R for some y B } ran(r) = { y y B, (x, y) R for some x A } A B R dom ( R ) ran( R ) dom(r), ran(r)
Crisp relation! Characteristics of relation (1) Surjection (many-to-one) n f(a) = B or ran(r) = B. y B, x A, y = f(x) n Thus, even if x1 x2, f(x1) = f(x2) can hold. A B x 1 x 2 f y Surjection
Crisp relation (3) Injection (into, one-to-one) n for all x1, x2 A, x1 x2, f(x1) f(x2). n if R is an injection, (x1, y) R and (x2, y) R then x1 = x2. (4) Bijection (one-to-one correspondence) n both a surjection and an injection. A B x 1 f y 1 x 2 y 2 x 3 y 3 y 4 Injection A B x 1 x 2 f y 1 y 2 x 3 y 3 x 4 y 4 Bijection
Crisp relation! Representation methods of relations (1)Bipartigraph(Fig 3.7) representing the relation by drawing arcs or edges (2)Coordinate diagram(fig 3.8) plotting members of A on x axis and that of B on y axis A B y a 1 b 1 4 a 2 b 2 a 3 a 4 b 3-4 4-4 x Fig 3.7 Binary relation from A to B Fig 3.8 Relation of x 2 + y 2 = 4
Crisp relation (3) Matrix manipulating relation matrix (4) Digraph the directed graph or digraph method M R = (m ij ) m ij 1, = 0, ( a, b i ( a, b i j ) R i = 1, 2, 3,, m j = 1, 2, 3,, n j ) R Matrix R b 1 b 2 b 3 a 1 1 0 0 a 2 0 1 0 a 3 0 1 0 a 4 0 0 1 1 2 3 4 Directed graph
Crisp relation! Operations on relations R, S A B (1) Union of relation T = R S If (x, y) R or (x, y) S, then (x, y) T (2) Intersection of relation T = R S If (x, y) R and (x, y) S, then (x, y) T. (3) Complement of relation If (x, y) R, then (x, y) (4) Inverse relation R -1 = {(y, x) B A (x, y) R, x A, y B} (5) Composition T R R A B, S B C, T = S R A C T = {(x, z) x A, y B, z C, (x, y) R, (y, z) S}
Crisp relation! Path and connectivity in graph Path of length n in the graph defined by a relation R A A is a finite series of p = a, x 1, x 2,..., x n-1, b where each element should be a R x 1, x 1 R x 2,..., x n-1 R b. Besides, when n refers to a positive integer (1) Relation R n on A is defined, x R n y means there exists a path from x to y w hose length is n. (2) Relation R on A is defined, x R y means there exists a path from x to y. That is, there exists x R y or x R 2 y or x R 3 y... and. This relation R is the reachability relation, and denoted as xr y (3) The reachability relation R can be interpreted as connectivity relation of A.
Properties of relation on a single set! Fundamental properties 1) Reflexive relation x A (x, x) R or µ R (x, x) = 1, x A n irreflexive if it is not satisfied for some x A n antireflexive if it is not satisfied for all x A 2) Symmetric relation (x, y) R (y, x) R or µ R (x, y) = µ R (y, x), x, y A n asymmetric or nonsymmetric when for some x, y A, (x, y) R and (y, x) R. n antisymmetric if for all x, y A, (x, y) R and (y, x) R
Properties of relation on a single set 3) Transitive relation For all x, y, z A (x, y) R, (y, z) R (x, z) R 4) Closure n Closure of R with the respect to a specific property is the smallest rela tion R' containing R and satisfying the specific property
Properties of relation on a single set! Example The transitive closure (or reachability relation) R of R for A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (3, 4), (2, 1)} is R = R R 2 R 3 ={(1, 1), (1, 2), (1, 3), (1,4), (2,1), (2,2), (2, 3), (2, 4), (3, 4)}. 2 1 3 2 1 3 4 (a) R 4 (b) R Fig 3.10 Transitive closure
Properties of relation on a single set! Equivalence relation (1) Reflexive relation x A (x, x) R (2) Symmetric relation (x, y) R (y, x) R (3) Transitive relation (x, y) R, (y, z) R (x, z) R
Properties of relation on a single set! Equivalence classes a partition of A into n disjoint subsets A 1, A 2,..., A n A A 1 A 2 b b a d e a d e c c (a) Expression by set (b) Expression by undirected graph Fig 3.11 Partition by equivalence relation π(a/r) = {A 1, A 2 } = {{a, b, c}, {d, e}}
Properties of relation on a single set! Compatibility relation (tolerance relation) (1) Reflexive relation (2) Symmetric relation x A (x, x) R A (x, y) R (y, x) R A 1 A 2 b b a d e a d e c c (a) Expression by set (b) Expression by undirected graph Fig 3.12 Partition by compatibility relation
Properties of relation on a single set! Pre-order relation (1) Reflexive relation x A (x, x) R (2) Transitive relation (x, y) R, (y, z) R (x, z) R A a c b d e h f g a c b, d e f, h g (a) Pre-order relation (b) Pre-order Fig 3.13 Pre-order relation
Properties of relation on a single set! Order relation (1) Reflexive relation x A (x, x) R (2) Antisymmetric relation (x, y) R (y, x) R (3) Transitive relation (x, y) R, (y, z) R (x, z) R n strict order relation (1 ) Antireflexive relation x A (x, x) R n total order or linear order relation (4) x, y A, (x, y) R or (y, x) R
Properties of relation on a single set! Comparison of relations Property Relation Reflexive Antireflexive Symmetric Antisymmetric Transitive Equivalence ü ü ü Compatibility ü ü Pre-order ü ü Order ü ü ü Strict order ü ü ü
Fuzzy relation n Crisp relation n membership function µ R (x, y) 1 iff (x, y) R µ R (x, y) = n µ R : A B {0, 1} 0 iff (x, y) R n Fuzzy relation n µ R : A B [0, 1] n R = {((x, y), µ R (x, y)) µ R (x, y) 0, x A, y B}
Fuzzy relation Example Crisp relation R µ R (a, c) = 1, µ R (b, a) = 1, µ R (c, b) = 1 and µ R (c, d) = 1. Fuzzy relation R µ R (a, c) = 0.8, µ R (b, a) = 1.0, µ R (c, b) = 0.9, µ R (c, d) = 1.0 a a 0.8 c 1.0 c b b 0.9 1.0 d d (a) Crisp relation (b) Fuzzy relation Fig 3.16 crisp and fuzzy relations A A a b c d a 0.0 0.0 0.8 0.0 b 1.0 0.0 0.0 0.0 c 0.0 0.9 0.0 1.0 d 0.0 0.0 0.0 0.0 corresponding fuzzy matrix
Fuzzy relation! Fuzzy matrix 1) Sum A + B = Max [a ij, b ij ] 2) Max product A B = AB = Max [ Min (a ik, b kj ) ] 3) Scalar product λa where 0 λ 1 k
Fuzzy relation! Example
Fuzzy relation! Operation of fuzzy relation 1) Union relation (x, y) A B µ R S (x, y) = Max [µ R (x, y), µ S (x, y)] = µ R (x, y) µ S (x, y) 2) Intersection relation µ R S (x) = Min [µ R (x, y), µ S (x, y)] = µ R (x, y) µ S (x, y) 3) Complement relation (x, y) A B µ R (x, y) = 1 - µ R (x, y) 4) Inverse relation For all (x, y) A B, µ -1 R (y, x) = µ R (x, y)
! Fuzzy relation matrix Fuzzy relation
Fuzzy relation! Composition of fuzzy relation For (x, y) A B, (y, z) B C, µ S R (x, z) = Max [Min (µ R (x, y), µ S (y, z))] y = [µ R (x, y) µ S (y, z)] y M S R = M R M S
Fuzzy relation! Example => => Composition of fuzzy relation
Fuzzy relation! α-cut of fuzzy relation R α = {(x, y) µ R (x, y) α, x A, y B} : a crisp relation. Example
Fuzzy relation! Projection For all x A, y B, µ x = Max µ x ( ) R( y) y ( y) Maxµ ( x y) R A, µ = R B R, x : projection to A : projection to B n Example
Fuzzy relation n Projection in n dimension µ x x,, x = R n Cylindrical extension µ C(R) (a, b, c) = µ R (a, b) a A, b B, c C ( ) ( x, x,, x ) Max i1, i2 ik R 1 2 X, X,, X µ Xi1 Xi 2 Xik j1 j 2 jm n n Example
Extension of fuzzy set! Extension by relation n Extension of fuzzy set x A, y B y = f(x) or x = f -1 (y) for y B ( y) µ B Max µ ʹ = 1 x f ( y) [ ( x) ] A if f -1 (y) Example A = {(a 1, 0.4), (a 2, 0.5), (a 3, 0.9), (a 4, 0.6)}, B = {b 1, b 2, b 3 } f -1 (b 1 ) = {(a 1, 0.4), (a 3, 0.9)}, Max [0.4, 0.9] = 0.9 µ B' (b 1 ) = 0.9 f -1 (b 2 ) = {(a 2, 0.5), (a 4, 0.6)}, Max [0.5, 0.6] = 0.6 µ B' (b 2 ) = 0.6 f -1 (b 3 ) = {(a 4, 0.6)} µ B' (b 3 ) = 0.6 B' = {(b 1, 0.9), (b 2, 0.6), (b 3, 0.6)}
Extension of fuzzy set! Extension by fuzzy relation For x A, y B, and Bʹ B µ B' (y) = Max [Min (µ A (x), µ R (x, y))] x f -1(y) n Example For b 1 Min [µ A (a 1 ), µ R (a 1, b 1 )] = Min [0.4, 0.8] = 0.4 Min [µ A (a 3 ), µ R (a 3, b 1 )] = Min [0.9, 0.3] = 0.3 Max [0.4, 0.3] = 0.4 è µ B' (b 1 ) = 0.4 For b 2, Min [µ A (a 2 ), µ R (a 2, b 2 )] = Min [0.5, 0.2] = 0.2 Min [µ A (a 4 ), µ R (a 4, b 2 )] = Min [0.6, 0.7] = 0.6 Max [0.2, 0.6] = 0.6 èµ B' (b 2 ) = 0.6 For b 3, Max Min [µ A (a 4 ), µ R (a 4, b 3 )] = Max Min [0.6, 0.4] = 0.4 è µ B' (b 3 ) = 0.4 B' = {(b 1, 0.4), (b 2, 0.6), (b 3, 0.4)}
Extension of fuzzy set n Example A = {(a 1, 0.8), (a 2, 0.3)} B = {b 1, b 2, b 3 } C = {c 1, c 2, c 3 } B' = {(b 1, 0.3), (b 2, 0.8), (b 3, 0)} C' = {(c 1, 0.3), (c 2, 0.3), (c 3, 0.8)}
Extension of fuzzy set! Fuzzy distance between fuzzy sets n Pseudo-metric distance (1) d(x, x) = 0, x X (2) d(x 1, x 2 ) = d(x 2, x 1 ), x 1, x 2 X (3) d(x 1, x 3 ) d(x 1, x 2 ) + d(x 2, x 3 ), x 1, x 2, x 3 X + (4) if d(x 1, x 2 )=0, then x 1 = x 2 è metric distance n Distance between fuzzy sets n δ R +, µ d(a, B) (δ) =Max [Min (µ A (a), µ B (b))] δ = d(a, b)
Extension of fuzzy set! Example A = {(1, 0.5), (2, 1), (3, 0.3)} B = {(2, 0.4), (3, 0.4), (4, 1)}.