Cartesian product

Fuzzy relation defines the mapping of variables from one fuzzy set to another. Like crisp relation, we can also define the relation over fuzzy sets.

Let A be a fuzzy set on universe X and B be a fuzzy set on universe Y, then the Cartesian product between fuzzy sets A and B will result in a fuzzy relation R which is contained with the full Cartesian product space or it is subset of cartesian product of fuzzy subsets. Formally, we can define fuzzy relation as,

R = A x B

and

R ⊂ (X x Y)

where the relation R has membership function,

μR(x, y) = μA x B(x, y) = min( μA(x), μB(y) )

A binary fuzzy relation R(X, Y) is called bipartite graph if X ≠ Y.

A binary fuzzy relation R(X, Y) is called directed graph or digraph if X = Y. , which is denoted as R(X, X) = R(X2)

Let A = {a1, a2, …, an} and B = {b1, b2, .., bm}, then fuzzy relation between A and B is described by the fuzzy relation matrix as,

fuzzy relation matrix
Fuzzy relation matrix

We can also consider fuzzy relation as a mapping from the cartesian space (X, Y) to the interval [0, 1]. The strength of this mapping is represented by the membership function of the relation for every tuple μR(x, y)

Example:

Given A = { (a1, 0.2), (a2, 0.7), (a3, 0.4) } and B = { (b1, 0.5), (b2, 0.6)}, find the relation over A x B

cartesian product
Cartesian product

Fuzzy relation

Fuzzy relations are very important because they can describe interactions between variables.

Example: A simple example of a binary fuzzy relation on X = {1, 2, 3}, called ”approximately equal” can be defined as

R(1, 1) = R(2, 2) = R(3, 3) = 1

R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2) = 0.8

R(1, 3) = R(3, 1) = 0.3

The membership function and relation matrix of R is given by

\bar{R}(x, y) = \begin{cases} 1, & if x = y \\ 0.7, & if |x-y| = 1 \\ 0.3, & if |x-y| = 2 \end{cases}

fuzzy relation R

Operations on fuzzy relation:

For our discussion, we will be using following two relation matrices:

relation R
relation S

Union:

RS = { (a, b), μAB(a, b) }

μRS(a, b) = max( μR(a, b), μS(a, b))

μRS(x1, y1) = max( μR( x1, y1 ), μS( x1, y1 ))

= max(0.8, 0.4) = 0.8

μRS(x1, y2) = max( μR( x1, y2 ), μS( x1, y2 ))

= max(0.1, 0.0) = 0.1

μRS(x1, y3) = max( μR( x1, y3 ), μS( x1, y3 ))

= max(0.1, 0.9) = 0.9

μRS(x1, y4) = max( μR( x1, y4 ), μS( x1, y4 ))

max(0.7, 0.6) = 0.7

.

.

.

μRS(x3, y4) = max( μR( x3, y4 ), μS( x3, y4 ) )

= max(0.8, 0.5) = 0.8

Thus, the final matrix for union operation would be,

union
Union of fuzzy relations

Intersection:

RS = { (a, b), μAB(a, b) }

μRS(a, b) = min( μR(a, b), μS(a, b) )

μRS(x1, y1) = min( μR( x1, y1 ), μS( x1, y1 ) )

= min(0.8, 0.4) = 0.4

μRS(x1, y2) = min( μR( x1, y2 ), μS( x1, y2 ) )

= max(0.1, 0.0) = 0.0

μRS(x1, y3) = min( μR( x1, y3 ), μS( x1, y3 ) )

= max(0.1, 0.9) = 0.1

μRS(x1, y4) = min( μR( x1, y4 ), μS( x1, y4 ) )

max(0.7, 0.6) = 0.6

.

.

.

μRS(x3, y4) = min( μR( x3, y4 ), μS( x3, y4 ) )

= max(0.8, 0.5) = 0.5

intersection
Intersection of relation

Complement:

Rc = { (a, b), μRc(a, b) }

μRc(a, b) = 1 – μR(a, b)

μRc(x1, y1) = 1 – μR(x1, y1) = 1 – 0.8 = 0.2

μRc(x1, y2) = 1 – μR(x1, y2) = 1 – 0.1 = 0.9

μRc(x1, y3) = 1 – μR(x1, y3) = 1 – 0.1 = 0.9

.

.

μRc(x3, y4) = 1 – μR(x3, y4) = 1 – 0.8 = 0.2

The complement of relation R would be,

complement
Complement of fuzzy relation

Projection:

The projection of \bar{R}  on X :

X(x) = sup( R(x, y) | y ∈ Y)

The projection of \bar{R}  on Y :

Y(y) = sup( R(x, y) | x ∈ X)

sup: Supremum of the set

projection

The projection of R on X :

X(x1) = 0.8

X(x2) = 0.8

X(x3) = 1.0

The projection of R on Y :

Y(y1) = 0.9

Y(y2) = 1.0

Y(y3) = 0.7

Y(y4) = 0.8

Watch on YouTube: Fuzzy relation

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Fuzzy composition:

Fuzzy composition can be defined just as it is for crisp (binary) relations. Suppose R is a fuzzy relation on X × Y, S is a fuzzy relation on Y × Z, and T is a fuzzy relation on X × Z; then,

Fuzzy Max–Min composition is defined as:

T = RS = μT(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μR(x, y) ∨ μS(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( min ( μR(x, y) ∨ μS(y, z) ) )

Fuzzy Max–Product composition is defined as:

T = RS = μT(x, z) = \begin{matrix} \vee\\ y \in Y \end{matrix} ( μR(x, y) ∙ μS(y, z) )

= \begin{matrix} max \\ y \in Y \end{matrix} ( ( μR(x, y) ∙ μS(y, z) ) )


Test Your Knowledge:

fuzzy relation r
fuzzy relation s

Find max-Min composition and Max-Product composition for above given fuzzy relations

Please post your answer / query / feedback in comment section below !