Ming-Feng Yeh 2-1

2. Fuzzy Relations2. Fuzzy Relations

Objectives:

Crisp and Fuzzy Relations

Projections and Cylindrical Extensions*

Extension Principle

Compositions of Fuzzy Relations

Ming-Feng Yeh 2-2

Cartesian Product: crispCartesian Product: crisp

Let A and B be two crisp subsets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is defined by

Let X={0, 1}, Y={a,b,c}. If A=X and B=Y, then

AB={(0,a), (0,b), (0,c), (1,a), (1,b), (1,c)}

BA={(a,0), (b,0), (c,0), (a,1), (b,1), (c,1)}

}.,),({ ByAxyxBA

Ming-Feng Yeh 2-3

Fuzzy RelationshipsFuzzy Relationships

A fuzzy relationship over the pair X, Y is defined as a fuzzy subset of the Cartesian product XY.If X={0, 1}, Y={a,b,c}, then

A = {0.1/(0,a), 0.6/(0,b), 0.8/(0,c),

0.3/(1,a), 0.5/(1,b), 0.7/(1,c)}

is a fuzzy relationship over the space XY.

7.05.03.0

8.06.01.0

1

0 cba

Ming-Feng Yeh 2-4

Cartesian Product: fuzzyCartesian Product: fuzzy

Let A and B be fuzzy sets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is a fuzzy set in the product space XY with the membership function:

Assume X={0,1} and Y={a,b,c}

Let A=1.0/0 + 0.6/1, B=0.2/a + 0.5/b+ 0.8/c.

Then AB is a fuzzy relationship over XY.

)].(),(min[),(, yBxAyxTBAT

)},1(6.0,),1(5.0,),1(2.0,),0(8.0,),0(5.0,),0(2.0{ cbccbaBA

Ming-Feng Yeh 2-5

Cylindrical Extension*Cylindrical Extension*

Assume X and Y are two crisp sets and let A be a fuzzy subset of X. The cylindrical extension of A to XY, denoted by , is a fuzzy relationship on XY.

Assume X={a,b,c} and Y={1,2}. Let A={1/a, 0.6/b, 0.3/c}. Then the cylindrical

extension of A to XY is {1/(a,1), 1/(a,2), 0.6/(b,1), 0.6/(b,2), 0.3/(c,1), 0.3/(c,2)}

YAA ˆ

.),(),(1)()()(),(ˆ YXyxxAxAyYxAyxA

A

Ming-Feng Yeh 2-6

Cylindrical Extension*Cylindrical Extension*

x

A(x)

x y

),(ˆ yxA

Ming-Feng Yeh 2-7

Projection*Projection*

Assume A is a fuzzy relationship on XY. The projection of A onto X is a fuzzy subset A of X, denoted by A=Projx A,

Assume X = {a,b,c} and Y = {1,2}. Let A={1/(a,1), 0.6/(a,2), 0.8/(b,1), 0.6/(b,2), 0.3/(c,1), 0.5/(c,2)}. Then Projx A = {1/a, 0.6/b, 0.5/c}.

Projy A = {0.8/1, 0.6/2}.

)].,([max)( yxAxA y

Ming-Feng Yeh 2-8

Projection*Projection*

Ming-Feng Yeh 2-9

Extension PrincipleExtension Principle

Assume X and Y are two crisp sets and let f be a mapping form X into Y, f: XY, such that xX, f(x) = y Y. Assume A is a fuzzy subset of X, using the extension principle, we can define f(A) as a fuzzy subset of Y such that

Denote B = f(A), then B is a fuzzy subset of Y such that for each y Y

})()({)( xfxAAf x

)(max)()(1

xAyByfx

Ming-Feng Yeh 2-10

Example 2-3Example 2-3

Assume X = {1, 2, 3} and Y = {a, b, c, d, e}.

Let f be defined by f(1) = a, f(2) = e, f(3) = b.

Let A = {1.0/1, 0.3/2, 0.7/3} be a fuzzy subset,

then B = f(A) = {1.0/a, 0.3/e, 0.7/b}.

Let A = 0.1/2 + 0.4/1 + 0.8/0 + 0.9/1 + 0.3/2

and f(x) = x2 3.

Then B = 0.1/1 + 0.4/2 + 0.8/3 + 0.9/2 + 0.3/1

= 0.8/3 + (0.40.9)/2 + (0.10.3)/1

= 0.8/3 + 0.9/2 + 0.3/1

Ming-Feng Yeh 2-11

Binary Fuzzy RelationsBinary Fuzzy Relations

Let X and Y be two universes of discourse. Then

is a binary fuzzy relation in XY.

Examples of binary fuzzy relation: y is much greater than x. (x and y are numbers) x is close to y. (x and y are numbers) x depends on y. (x and y are events) x and y look alike. (x and y are persons, objects, etc.) If x is large, then y is small. (x is an observed reading and y is a

corresponding action)

YXyxyxyxR R ),(),(),,(

Ming-Feng Yeh 2-12

Max-min CompositionMax-min Composition

Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-min composition of R1 and R2 is a fuzzy set defined by

Max-min product: the calculation of is almost the same as matrix multiplication, except that and are replaced by and , respectively.

ZzYyXxzyyxzxRR RR

y,,]),(),,(minmax),,[(

2121

)],(),([

)],(),,(min[max),(

21

2121

zyyx

zyyxzx

RRy

RRy

RR

21 RR

Ming-Feng Yeh 2-13

Max-product CompositionMax-product Composition

Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-product composition of R1 and R2 is a fuzzy set defined by

),(),(max),(2121

zyyxzx RRy

RR

Ming-Feng Yeh 2-14

Example 2-3Example 2-3

R1 = “x is relevant to y”, R2 = “y is relevant to z”,

X = {1,2,3}, Y={,,,} and Z={a,b}.

Max-min composition:

Max-product composition:

2.07.06.05.03.02.01.09.0

,2.03.08.06.09.08.02.04.07.05.03.01.0

21 RR

7.0)7.09.0,5.08.0,2.02.0,9.04.0max(

21

RR

63.0)7.09.0,5.08.0,2.02.0,9.04.0max(

21

RR