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 a subset of the 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 a membership function,

μ_R(x, y) = μ_{A x B}(x, y) = min( μ_A(x), μ_B(y) )

A binary fuzzy relation R(X, Y) is called a 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, …, a_n} and B = {b₁, b₂, .., b_m}, then the fuzzy relation between A and B is described by the fuzzy relation matrix as,

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 μ_{R(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

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 are given by

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

Operations on fuzzy relation:

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

Union:

R ∪ S = { (a, b), μ_{A ∪ B}(a, b) }

μ_{R ∪ S}(a, b) = max( μ_R(a, b), μ_S(a, b))

μ_{R ∪ S}(x₁, y₁) = max( μ_R( x₁, y₁ ), μ_S( x₁, y₁ ))

= max(0.8, 0.4) = 0.8

μ_{R ∪ S}(x₁, y₂) = max( μ_R( x₁, y₂ ), μ_S( x₁, y₂ ))

= max(0.1, 0.0) = 0.1

μ_{R ∪ S}(x₁, y₃) = max( μ_R( x₁, y₃ ), μ_S( x₁, y₃ ))

= max(0.1, 0.9) = 0.9

μ_{R ∪ S}(x₁, y₄) = max( μ_R( x₁, y₄ ), μ_S( x₁, y₄ ))

max(0.7, 0.6) = 0.7

.

.

.

μ_{R ∪ S}(x₃, y₄) = max( μ_R( x₃, y₄ ), μ_S( x₃, y₄ ) )

= max(0.8, 0.5) = 0.8

Thus, the final matrix for union operation would be,

Intersection:

R ∩ S = { (a, b), μ_{A ∩ B}(a, b) }

μ_{R ∩ S}(a, b) = min( μ_R(a, b), μ_S(a, b) )

μ_{R ∩ S}(x₁, y₁) = min( μ_R( x₁, y₁ ), μ_S( x₁, y₁ ) )

= min(0.8, 0.4) = 0.4

μ_{R ∩ S}(x₁, y₂) = min( μ_R( x₁, y₂ ), μ_S( x₁, y₂ ) )

= max(0.1, 0.0) = 0.0

μ_{R ∩ S}(x₁, y₃) = min( μ_R( x₁, y₃ ), μ_S( x₁, y₃ ) )

= max(0.1, 0.9) = 0.1

μ_{R ∩ S}(x₁, y₄) = min( μ_R( x₁, y₄ ), μ_S( x₁, y₄ ) )

max(0.7, 0.6) = 0.6

.

.

.

μ_{R ∩ S}(x₃, y₄) = min( μ_R( x₃, y₄ ), μ_S( x₃, y₄ ) )

= max(0.8, 0.5) = 0.5

Complement:

R^c = { (a, b), μ_R^c(a, b) }

μ_R^c(a, b) = 1 – μ_R(a, b)

μ_R^c(x₁, y₁) = 1 – μ_R(x₁, y₁) = 1 – 0.8 = 0.2

μ_R^c(x₁, y₂) = 1 – μ_R(x₁, y₂) = 1 – 0.1 = 0.9

μ_R^c(x₁, y₃) = 1 – μ_R(x₁, y₃) = 1 – 0.1 = 0.9

.

.

μ_R^c(x₃, y₄) = 1 – μ_R(x₃, y₄) = 1 – 0.8 = 0.2

The complement of relation R would be,

Projection:

The projection of R on X :

∏_X(x) = sup( R(x, y) | y ∈ Y)

The projection of R on Y :

∏_Y(y) = sup( R(x, y) | x ∈ X)

sup: Supremum of the set

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Fuzzy composition:

The 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:

Fuzzy Max–Product composition is defined as:

Test Your Knowledge:

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

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