Fuzzy relation – Definition, types and operations

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

fuzzy relation on youtube

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 !

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5 Responses

  1. Parekh Payal says:

    Very well example topic with video. It’s very helpful.

  2. Himani says:

    Reffered too much content on various sources on this topic, but couldn’t find better than this.

  3. Alachew Amaneh says:

    Wow thanks that was nice explanation about opretiom of fuzzy set.
    Can you explain about fuzzy subghroup

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