EE-646

Lecture-8

Fuzzy Relations Contd

& Fuzzy Composition

Fuzzy Cartesian Product

• Fuzzy relations are in general fuzzy sets

• We can define Cartesian product as a relation between two or more fuzzy sets

• Let A & B be two fuzzy sets defined on the universes X & Y , then the Cartesian product between A & B will result in fuzzy relation which is contained in full Cartesian product space

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R

Fuzzy Cartesian Product

i. e.

Where, the fuzzy relation has membership function

The Cartesian product defined by is implemented in the same fashion as the cross product of two vectors

Again, the Cartesian product is not the same operation as the arithmetic product.

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A B R X Y

R

( , ) ( , ) min[ ( ), ( )]BR A B Ax y x y x y

A B R

2D Fuzzy Relation

• In the case of two-dimensional relations (r = 2), the Cartesian product employs the idea of pairing of elements among sets, whereas the arithmetic product uses actual arithmetic products between elements of sets.

• Each of the fuzzy sets could be thought of as a vector of membership values; each value is associated with a particular element in each set.

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2D Fuzzy Relation

• For example, for a fuzzy set (vector) that has four elements, hence column vector of size 4×1, and for a fuzzy set (vector) that has five elements, hence a row vector size of 1×5, the resulting fuzzy relation, , will be represented by a matrix of size 4 × 5

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A

R

B

Example

Suppose we have two fuzzy sets, A defined on a universe of three discrete temperatures, X = {x1, x2, x3} and B

defined on a universe of two discrete pressures, Y =

{y1, y2}, and we want to find the fuzzy Cartesian product between them. Fuzzy set A could represent the ‘‘ambient’’ temperature and fuzzy set B the ‘‘near optimum’’ pressure for a certain heat exchanger, and the Cartesian product might represent the conditions (temperature–pressure pairs) of the exchanger that are associated with ‘‘efficient’’ operations.

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Example

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Let

A can be represented as a column vector of size

3×1 and B can be represented by a row vector of

1×2. Then the fuzzy Cartesian product results in

a fuzzy relation (of size 3×2) representing

‘‘efficient’’ conditions

1 2 3 1 2

0.2 0.5 1 0.3 0.9&A B

x x x y y

Example

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1 2

1

2

3

0.2 0.2

0.3 0.5

0.3 0.9

A B R y y

x

x

x

Fuzzy Composition

• Fuzzy composition can be defined in a similar way as for crisp relations. Let be a fuzzy relation on the Cartesian space X × Y , be a fuzzy relation on Y × Z, and be a fuzzy relation on X × Z, then the fuzzy set max–min composition is defined as (in set-theoretic notation):

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R

S

T

T R S

Contd...

• Also in function theoretic notation it can be expressed as:

• Fuzzy max-product composition is defined as:

It should be noted that neither crisp nor fuzzy relation hold commutativity i.e.

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R S S R

, , ,T R Sy Y

x z x y y z

, , ,T R Sy Y

x z x y y z

Example

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For the fuzzy relations given below Find the relation using max–min and max–product composition

1 2

1

2

1 2 3

1

2

0.6 0.3 and

0.2 0.9

1.0 0.5 0.3

0.1 0.4 0.7

y y

xR

x

z z z

yS

y

T R S

Solution

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