CHAPTER 3 FUZZY RELATION and COMPOSITION

3.1 Crisp relation Product set Definition (Product set) Let A and B be two non-empty sets, the product set or Cartesian product A  B is defined as follows, A  B  {(a, b) | a  A, b  B } extension to n sets A 1  A 2 ...  A n (= )={(a 1,..., a n ) | a 1  A 1, a 2  A 2,..., a n  A n }

3.1 Crisp relation Example 3.1 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 )} Fig 3.1 Product set A  B

3.1 Crisp relation A  A  {(a 1, a 1 ), (a 1, a 2 ), (a 1, a 3 ), (a 2, a 1 ), (a 2, a 2 ), (a 2, a 3 ), (a 3, a 1 ), (a 3, a 2 ), (a 3, a 3 )} Fig 3.2 Cartesian product A  A

3.1 Crisp relation Definition of relation 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 of 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

3.1 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 } R dom (R ) ran(R ) AB f x 1 x 2 x 3 AB y 1 y 2 y 3 Mapping y  f(x) dom(R), ran(R)

3.1 Crisp relation Characteristics of relation (1) One-to-many  x  A, y1, y2  B (x, y1)  R, (x, y2)  R (2) Surjection (many-to-one) f(A)  B or ran(R)  B.  y  B,  x  A, y  f(x) Thus, even if x1  x2, f(x1)  f(x2) can hold. AB y1y1 x y2y2 One-to-many relation (not a function) f x 1 x 2 AB y Surjection

3.1 Crisp relation (3) Injection (into, one-to-one) for all x1, x2  A, x1  x2, f(x1)  f(x2). if R is an injection, (x1, y)  R and (x2, y)  R then x1  x2. (4) Bijection (one-to-one correspondence) both a surjection and an injection. f x 1 AB y 1 x 2 y 2 x 3 y 3 y 4 Injection f x 1 x 2 AB y 1 y 2 x 3 y 3 x 4 y 4 Bijection

3.1 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 Fig 3.8 Relation of x 2 + y 2  4 Fig 3.7 Binary relation from A to B y x

3.1 Crisp relation (3) Matrix manipulating relation matrix (4) Digraph(Fig 3.9) the directed graph or digraph method Rb1b1 b2b2 b3b3 a1a1 100 a2a2 010 a3a3 010 a4a4 001 M R  (m ij ) i  1, 2, 3, …, m j  1, 2, 3, …, n Matrix Fig 3.9 Directed graph

3.1 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  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}

3.1 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 whose 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.

3.2 Properties of relation on a single set Fundamental properties 1) Reflexive relation x  A  (x, x)  R or  R (x, x) = 1,  x  A irreflexive if it is not satisfied for some x  A antireflexive if it is not satisfied for all x  A 2) Symmetric relation (x, y)  R  (y, x)  R or  R (x, y) =  R (y, x),  x, y  A asymmetric or nonsymmetric when for some x, y  A, (x, y)  R and (y, x)  R. antisymmetric if for all x, y  A, (x, y)  R and (y, x)  R

3.2 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 The requisites for closure (1) Set A should satisfy a certain specific property. (2) Intersection between A's subsets should satisfy the relation R. Closure of R the smallest relation R' containing the specific property

3.2 Properties of relation on a single set Example 3.3 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)} (a) R (b) R  Fig 3.10 Transitive closure

3.2 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

3.2 Properties of relation on a single set Equivalence classes a partition of A into n disjoint subsets A 1, A 2,..., A n (a) Expression by set(b) Expression by undirected graph Fig 3.11 Partition by equivalence relation  (A/R)  {A 1, A 2 }  {{a, b, c}, {d, e}}

3.2 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) Expression by set (b) Expression by undirected graph Fig 3.12 Partition by compatibility relation

3.2 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 hg f d b c e a A (a) Pre-order relation a c b, d e f, h g (b) Pre-order Fig 3.13 Pre-order relation

3.2 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 strict order relation (1’) Antireflexive relation x  A  (x, x)  R total order or linear order relation (4)  x, y  A, (x, y)  R or (y, x)  R

3.2 Properties of relation on a single set Ordinal function For all x, y  A (x  y), 1) If (x, y)  R, xRy or x > y,f(x)  f(y) + 1 2) If reachability relation exists in x and y, i.e. if xR  y, f(x) > f(y) b a d c e f g (a) Order relation a d e b cfg (b) Ordinal function Fig 3.14 Order relation and ordinal function

3.2 Properties of relation on a single set Table 3.1 Comparison of relations Property Relation ReflexiveAntireflexiveSymmetricAntisymmetricTransitive Equivalence Compatibility Pre-order Order Strict order

3.3 Fuzzy relation Definition of fuzzy relation Crisp relation membership function  R (x, y)  R : A  B  {0, 1} Fuzzy relation  R : A  B  [0, 1] R = {((x, y),  R (x, y))|  R (x, y)  0, x  A, y  B}  R (x, y) = 1 iff (x, y)  R 0 iff (x, y)  R

3.3 Fuzzy relation (a 1, b 1 )...(a 1, b 2 ) ( a 2, b 1 ) R Fig 3.15 Fuzzy relation as a fuzzy set

3.3 Fuzzy relation Examples of fuzzy relation Example 3.3 Crisp relation R  R (a, c)  1,  R (b, a)  1,  R (c, b)  1 and  R (c, d)  1. Fuzzy relation R  R (a, c)  0.8,  R (b, a)  1.0,  R (c, b)  0.9,  R (c, d)  1.0 a b c d a b c d (a) Crisp relation (b) Fuzzy relation Fig 3.16 crisp and fuzzy relations corresponding fuzzy matrix

3.3 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 ) ] k 3) Scalar product A where 0   1

3.3 Fuzzy relation Example 3.4

3.3 Fuzzy relation Operation of fuzzy relation 1) Union relation  (x, y)  A  B  R  S (x, y)  Max [  R (x, y),  S (x, y)]   R (x, y)   S (x, y) 2) Intersection relation  R  S (x) = Min [  R (x, y),  S (x, y)] =  R (x, y)   S (x, y) 3) Complement relation  (x, y)  A  B  R (x, y)  1 -  R (x, y) 4) Inverse relation For all (x, y)  A  B,  R -1 (y, x)   R (x, y)

3.3 Fuzzy relation Fuzzy relation matrix

3.3 Fuzzy relation Composition of fuzzy relation For (x, y)  A  B, (y, z)  B  C,  S  R (x, z) = Max [Min (  R (x, y),  S (y, z))] y =  [  R (x, y)   S (y, z)] y M S  R  M R  M S

3.3 Fuzzy relation Example 3.6 => Composition of fuzzy relation

3.3 Fuzzy relation  -cut of fuzzy relation R  = {(x, y) |  R (x, y)  , x  A, y  B} : a crisp relation. Example 3.7

3.3 Fuzzy relation Projection and cylindrical extension for (x, y)  A  B  is a value in the level set; R  is a  -cut relation;  R  is a fuzzy relation Example 3.8

3.3 Fuzzy relation Projection For all x  A, y  B, : projection to A : projection to B Example 3.9

3.3 Fuzzy relation Projection in n dimension Cylindrical extension  C(R) (a, b, c)   R (a, b) a  A, b  B, c  C Example 3.10

3.4 Extension of fuzzy set Extension by relation Extension of fuzzy set x  A, y  B y  f(x) or x  f -1 (y) for y  B if f -1 (y)  Example 3.11 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   B' (b 1 )  0.9 f -1 (b 2 )  {(a 2, 0.5), (a 4, 0.6)}, Max [0.5, 0.6]  0.6   B' (b 2 )  0.6 f -1 (b 3 )  {(a 4, 0.6)}   B' (b 3 )  0.6 B'  {(b 1, 0.9), (b 2, 0.6), (b 3, 0.6)}

3.4.2 Extension principle Extension principle  A­­1  A2 ...  Ar ( x 1  x 2 ...  x r )  Min [  A1 (x 1 ),...,  Ar (x r ) ] f(x 1, x 2,..., x r ) : X  Y 3.4 Extension of fuzzy set

3.4.3 Extension by fuzzy relation For x  A, y  B, and B  B  B' (y)  Max [Min (  A (x),  R (x, y))] x  f -1(y) Example 3.12 For b 1 Min [  A (a 1 ),  R (a 1, b 1 )]  Min [0.4, 0.8]  0.4 Min [  A (a 3 ),  R (a 3, b 1 )]  Min [0.9, 0.3]  0.3 Max [0.4, 0.3]  0.4   B' (b 1 )  0.4 For b 2, Min [  A (a 2 ),  R (a 2, b 2 )]  Min [0.5, 0.2]  0.2 Min [  A (a 4 ),  R (a 4, b 2 )]  Min [0.6, 0.7]  0.6 Max [0.2, 0.6]  0.6   B' (b 2 )  0.6 For b 3, Max Min [  A (a 4 ),  R (a 4, b 3 )]  Max Min [0.6, 0.4]  0.4   B' (b 3 )  0.4 B'  {(b 1, 0.4), (b 2, 0.6), (b 3, 0.4)} 3.4 Extension of fuzzy set

Example 3.13 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)} 3.4 Extension of fuzzy set

3.4.4 Fuzzy distance between fuzzy sets 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  metric distance Distance between fuzzy sets     ,  d(A, B) (  )  Max[Min (  A (a),  B (b))]   d(a, b) 3.4 Extension of fuzzy set

Example 3.14 A  {(1, 0.5), (2, 1), (3, 0.3)} B  {(2, 0.4), (3, 0.4), (4, 1)}. 3.4 Extension of fuzzy set