EE-646

Lecture-7

Fuzzy Relations

Fuzzy Relations

• Fuzzy relations map elements of one universe, X, to those of another universe, Y, through the Cartesian product of the two universes

• This is also referred to as fuzzy sets defined on universal sets, which are Cartesian product

• A fuzzy relation is based on the concept that everything is related to some extent or unrelated

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Fuzzy Relations

• A fuzzy relation is a fuzzy set defined on the Cartesian product of classical sets {X1, X2,..., Xn} where n-tuples (x1, x2,..., xn) may have varying degree of membership µR(x1, x2,..., xn) within the relation i.e.

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1 2

1 2

1 2

1 2, ,...

, ,..., ,...

, ,...n

R n

n

nX X X

x x xR X X X

x x x

Fuzzy Relations

• A fuzzy relation between two sets X & Y is called binary fuzzy relation & is denoted by:

• A binary relation is referred to as bipartite graph when X ≠ Y

• The binary relation defined on a single set X is called directed graph or digraph. This occurs when X = Y & is denoted by

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,R X Y

2, or R X X R X

,R X Y

Fuzzy Relations

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Let X = {x1, x2,..., xn} , Y = {y1, y2,..., ym}. The fuzzy relation can be expressed by n × m matrix called Fuzzy Matrix denoted as:

1 1 1 2 1

2 1 2 2 2

1 2

, , ... ,

, , ... ,,

... ... ... ...

, , ... ,

R R R m

R R R m

R n R n R n m

x y x y x y

x y x y x yR X Y

x y x y x y

,R X Y

Fuzzy Relations

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A fuzzy relation is mapping from Cartesian space (X, Y) to the interval [0, 1] where the mapping strength is expressed by the membership function of the relation for ordered pairs from the two universes

A fuzzy graph is the graphical representation of a binary fuzzy relation

,R x y

Domain & Range

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The domain of the fuzzy relation is the fuzzy set, having the membership function as:

The range of the fuzzy relation is the fuzzy set, having the membership function as:

dom max , ,R Ry Y

x x y x X

,R X Y

dom ,R X Y

Range max , ,R Rx X

y x y y Y

,R X Y

Range ,R X Y

Example

Consider a universe X = {x1, x2, x3, x4} & a binary fuzzy relation R as:

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0.2 0 0.5 0

0 0.3 0.7 0.8,

0.1 0 0.4 0

0 0.6 0 0.1

R X X

x1

x2

x3

x4

x1 x2 x3 x4

Example

Domain = {0.5, 0.8, 0.4, 0.6} (Take max on rows) &

Range = {0.2, 0.6, 0.7, 0.8} (Take max on columns)

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Digraph

See Board

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x1

x2

x3

x4

Sagittal Diagram

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x1

x2

x3

x4

X X

0.2

0.5 0.3

0.7

0.8

0.4

0.1

0.1

0.6

Operations on Fuzzy Relations

Let be fuzzy relations on the Cartesian space X × Y. Then the following operations apply for the membership values for various set operations:

1. Union:

2. Intersection:

3. Complement:

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~ ~~ ~

( , ) max[ ( , ), ( , )]R S R S

x y x y x y

~ ~

( , ) 1 ( , )R R

x y x y

and R S

~ ~~ ~

( , ) min[ ( , ), ( , )]R S R S

x y x y x y

Operations on Fuzzy Relations

4. Containment:

• All properties like commutativity, associativity, distributivity, involution, idempotency, De’Morgan’s laws also hold for fuzzy relation as they do for crisp relations.

• However, the law of excluded middle and law of contradiction does not hold good for fuzzy relations (as for fuzzy sets)

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~~ ~ ~

( , ) ( , )R S

R S x y x y

~ ~ ~ ~ ~ ~ and R R E R R O

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