CHAPTER 3 FUZZY 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 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) 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

(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.

Example Fuzzy relation

Projection For all x ∈ A, y ∈ B,

µ R (x) = Max µR (x, y ) : projection to A A y : projection to B µR (y)= MaxµR (x, y) B x

n Example Fuzzy relation

n Projection in n dimension

µR (xi1, xi2 ,…, xik ) = µR (x1, x2 ,…, xn ) Xi1×Xi 2×…×Xik Max X j1 ,X j 2 ,…,X jm n Cylindrical extension

µC(R) (a, b, c) = µR (a, b) a ∈ A, b ∈ B, c ∈ C 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)

f -1(y) for y ∈ B µ B ʹ ( y ) = Max [µ A (x)] if ≠∅ x∈f −1(y)

Example A = {(a1, 0.4), (a2, 0.5), (a3, 0.9), (a4, 0.6)}, B = {b1, b2, b3}

-1 f (b1) = {(a1, 0.4), (a3, 0.9)}, Max [0.4, 0.9] = 0.9

⇒ µB' (b1) = 0.9 -1 f (b2) = {(a2, 0.5), (a4, 0.6)}, Max [0.5, 0.6] = 0.6

⇒ µB' (b2) = 0.6 -1 f (b3) = {(a4, 0.6)}

⇒ µB' (b3) = 0.6

B' = {(b1, 0.9), (b2, 0.6), (b3, 0.6)} Extension of fuzzy set

Extension by fuzzy relation For x ∈ A, y ∈ B, and Bʹ ⊆ B

µB' (y) = Max [Min (µA (x), µR (x, y))] x ∈ f -1(y) n Example For b1 Min [µA (a1), µR (a1, b1)] = Min [0.4, 0.8] = 0.4 Min [µA (a3), µR (a3, b1)] = Min [0.9, 0.3] = 0.3 Max [0.4, 0.3] = 0.4 è µB' (b1) = 0.4

For b2, Min [µA (a2), µR (a2, b2)] = Min [0.5, 0.2] = 0.2 Min [µA (a4), µR (a4, b2)] = Min [0.6, 0.7] = 0.6 Max [0.2, 0.6] = 0.6 èµB' (b2) = 0.6

For b3, Max Min [µA (a4), µR (a4, b3)] = Max Min [0.6, 0.4] = 0.4

è µB' (b3) = 0.4

B' = {(b1, 0.4), (b2, 0.6), (b3, 0.4)} Extension of fuzzy set

n Example

A = {(a1, 0.8), (a2, 0.3)}

B = {b1, b2, b3}

C = {c1, c2, c3}

B' = {(b1, 0.3), (b2, 0.8), (b3, 0)}

C' = {(c1, 0.3), (c2, 0.3), (c3, 0.8)} Extension of fuzzy set

Fuzzy distance between fuzzy sets

n Pseudo-metric distance (1) d(x, x) = 0, ∀ x ∈ X

(2) d(x1, x2) = d(x2, x1), ∀ x1, x2 ∈ X

(3) d(x1, x3) ≤ d(x1, x2) + d(x2, x3), ∀ x1, x2, x3 ∈ X

+ (4) if d(x1, x2)=0, then x1 = x2 è metric distance

n Distance between fuzzy sets + n ∀ δ ∈ ℜ , µd(A, B)(δ) =Max [Min (µA(a), µB(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)}.