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 = {a1, a2, a3}, B = {b1, b2} A × B = {(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2)} b2 (a1, b2) (a2, b2) (a3, b2) b1 (a1, b1) (a2, b1) (a3, b1) a1 a2 a3 Crisp relation 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" n-ary relation For sets A1, A2, A3, ..., An, the relation among elements x1 ∈ A1, x2 ∈ A2, x3 ∈ A3, ..., xn ∈ An can be described by n-tuple (x1, x2, ..., xn). A collection o f such n-tuples (x1, x2, x3, ..., xn) is a relation R among A1, A2, A3, ..., An. (x1, x2, x3, … , xn) ∈ R , R ⊆ A1 × A2 × A3 × … × An 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 dom (R ) 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 x1 f y x2 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 A B f x f y x1 y1 1 1 x2 y2 x2 y2 x3 y3 x3 y3 y 4 x4 y4 Injection 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 4 a1 b1 a2 b2 x a3 -4 4 a4 b3 -4 Fig 3.7 Binary relation from A to B Fig 3.8 Relation of x2 + y2 = 4 Crisp relation (3) Matrix manipulating relation matrix (4) Digraph the directed graph or digraph method MR = (mij) R b1 b2 b3 ⎪⎧1, (ai,bj )∈ R a 1 0 0 1 mij = ⎨ 1 0, (a ,b ) R ⎩⎪ i j ∉ a 0 1 0 2 2 3 i = 1, 2, 3, …, m a3 0 1 0 j = 1, 2, 3, …, n a4 0 0 1 4 Matrix 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) ∈ R (4) Inverse relation R-1 = {(y, x) ∈ B × A | (x, y) ∈ R, x ∈ A, y ∈ B} (5) Composition T 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, x1, x2, ..., xn-1, b where each element should be a R x1, x1 R x2, ..., xn-1 R b. Besides, when n refers to a positive integer (1) Relation Rn on A is defined, x Rn 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 R2 y or x R3 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 µR(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 µR(x, y) = µR(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 ∪ R2 ∪ R3 ∪ … ={(1, 1), (1, 2), (1, 3), (1,4), (2,1), (2,2), (2, 3), (2, 4), (3, 4)}. 2 2 1 3 1 3 4 4 (a) R (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 A1, A2, ... , An A A1 A2 b b d e a d e a c c (a) Expression by set (b) Expression by undirected graph Fig 3.11 Partition by equivalence relation π(A/R) = {A1, A2} = {{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 (x, y) ∈ R → (y, x) ∈ R A A1 A2 b b d e a d e a 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 e e b a f a d b, d f, h c g h g c (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 Reflexive Antireflexive Symmetric Antisymmetric Transitive Relation 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 µR(x, y) 1 iff (x, y) ∈ R µ (x, y) = R 0 iff (x, y) ∉ R n" µR : A × B → {0, 1} n" Fuzzy relation n" µR : A × B → [0, 1] n" R = {((x, y), µR(x, y))| µR(x, y) ≥ 0 , x ∈ A, y ∈ B} Fuzzy relation Example Crisp relation R µR(a, c) = 1, µR(b, a) = 1, µR(c, b) = 1 and µR(c, d) = 1. Fuzzy relation R µR(a, c) = 0.8, µR(b, a) = 1.0, µR(c, b) = 0.9, µR(c, d) = 1.0 a a A a b c d 0.8 A c a 0.0 0.0 0.8 0.0 1.0 c 0.9 1.0 b 1.0 0.0 0.0 0.0 b b c 0.0 0.9 0.0 1.0 d d d 0.0 0.0 0.0 0.0 (a) Crisp relation (b) Fuzzy relation corresponding fuzzy matrix Fig 3.16 crisp and fuzzy relations Fuzzy relation ! " Fuzzy matrix 1) Sum A + B = Max [aij, bij] 2) Max product A • B = AB = Max [ Min (aik, bkj) ] k 3) Scalar product λA where 0 ≤ λ ≤ 1 Fuzzy relation ! " Example Fuzzy relation ! " Operation of fuzzy relation 1) Union relation ∀ (x, y) ∈ A × B µR ∪ S (x, y) = Max [µR (x, y), µS (x, y)] = µR (x, y) ∨ µS (x, y) 2) Intersection relation µR ∩ S (x) = Min [µR (x, y), µS (x, y)] = µR (x, y) ∧ µS (x, y) 3) Complement relation ∀ (x, y) ∈ A × B µR (x, y) = 1 - µR (x, y) 4) Inverse relation -1 For all (x, y) ⊆ A × B, µR (y, x) = µR (x, y) Fuzzy relation ! " Fuzzy relation matrix Fuzzy relation ! " Composition of fuzzy relation For (x, y) ∈ A × B, (y, z) ∈ B × C, µS•R (x, z) = Max [Min (µR (x, y), µS (y, z))] y = ∨ [µR (x, y) ∧ µS (y, z)] y MS • R = MR • MS Fuzzy relation ! " Example => => Composition of fuzzy relation Fuzzy relation ! " α-cut of fuzzy relation Rα = {(x, y) | µR(x, y) ≥ α, x ∈ A, y ∈ B} : a crisp relation.