# 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 â (X x Y)

where the relation R has a membership function,

Îŧ_{R}(x, y) = Îŧ_{A x B}(x, y) = min( Îŧ_{A}(x), Îŧ_{B}(y) )

A binary fuzzy relation R(X, Y) is called a **bipartite graph **if X â 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(X^{2})

Let A = {a_{1}, a_{2}, …, a_{n}} and B = {b_{1}, b_{2}, .., 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 Îŧ_{R}_{(x, y)}

**Example:**

Given A = { (a_{1}, 0.2), (a_{2}, 0.7), (a_{3}, 0.4) } and B = { (b_{1}, 0.5), (b_{2}, 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 âapproximately equalâ 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 âĒ S = { (a, b), Îŧ_{A âĒ B}(a, b) }

Îŧ_{R âĒ S}(a, b) = max( Îŧ_{R}(a, b), Îŧ_{S}(a, b))

Îŧ_{R âĒ S}(x_{1}, y_{1}) = max( Îŧ_{R}( x_{1}, y_{1} ), Îŧ_{S}( x_{1}, y_{1} ))

= max(0.8, 0.4) = 0.8

Îŧ_{R âĒ S}(x_{1}, y_{2}) = max( Îŧ_{R}( x_{1}, y_{2} ), Îŧ_{S}( x_{1}, y_{2} ))

= max(0.1, 0.0) = 0.1

Îŧ_{R âĒ S}(x_{1}, y_{3}) = max( Îŧ_{R}( x_{1}, y_{3} ), Îŧ_{S}( x_{1}, y_{3} ))

= max(0.1, 0.9) = 0.9

Îŧ_{R âĒ S}(x_{1}, y_{4}) = max( Îŧ_{R}( x_{1}, y_{4} ), Îŧ_{S}( x_{1}, y_{4} ))

max(0.7, 0.6) = 0.7

.

.

.

Îŧ_{R âĒ S}(x_{3}, y_{4}) = max( Îŧ_{R}( x_{3}, y_{4} ), Îŧ_{S}( x_{3}, y_{4} ) )

= max(0.8, 0.5) = 0.8

Thus, the final matrix for union operation would be,

### Intersection:

R âŠ S = { (a, b), Îŧ_{A âŠ B}(a, b) }

Îŧ_{R âŠ S}(a, b) = min( Îŧ_{R}(a, b), Îŧ_{S}(a, b) )

Îŧ_{R âŠ S}(x_{1}, y_{1}) = min( Îŧ_{R}( x_{1}, y_{1} ), Îŧ_{S}( x_{1}, y_{1} ) )

= min(0.8, 0.4) = 0.4

Îŧ_{R âŠ S}(x_{1}, y_{2}) = min( Îŧ_{R}( x_{1}, y_{2} ), Îŧ_{S}( x_{1}, y_{2} ) )

= max(0.1, 0.0) = 0.0

Îŧ_{R âŠ S}(x_{1}, y_{3}) = min( Îŧ_{R}( x_{1}, y_{3} ), Îŧ_{S}( x_{1}, y_{3} ) )

= max(0.1, 0.9) = 0.1

Îŧ_{R âŠ S}(x_{1}, y_{4}) = min( Îŧ_{R}( x_{1}, y_{4} ), Îŧ_{S}( x_{1}, y_{4} ) )

max(0.7, 0.6) = 0.6

.

.

.

Îŧ_{R âŠ S}(x_{3}, y_{4}) = min( Îŧ_{R}( x_{3}, y_{4} ), Îŧ_{S}( x_{3}, y_{4} ) )

= max(0.8, 0.5) = 0.5

### Complement:

R^{c} = { (a, b), Îŧ_{R}^{c}(a, b) }

Îŧ_{R}^{c}(a, b) = 1 – Îŧ_{R}(a, b)

Îŧ_{R}^{c}(x_{1}, y_{1}) = 1 – Îŧ_{R}(x_{1}, y_{1}) = 1 – 0.8 = 0.2

Îŧ_{R}^{c}(x_{1}, y_{2}) = 1 – Îŧ_{R}(x_{1}, y_{2}) = 1 – 0.1 = 0.9

Îŧ_{R}^{c}(x_{1}, y_{3}) = 1 – Îŧ_{R}(x_{1}, y_{3}) = 1 – 0.1 = 0.9

.

.

Îŧ_{R}^{c}(x_{3}, y_{4}) = 1 – Îŧ_{R}(x_{3}, y_{4}) = 1 – 0.8 = 0.2

The complement of relation R would be,

### Projection:

**The projection of R on X :**

â_{X}(x) = sup( R(x, y) | y â Y)

**The projection of R on Y : **

â_{Y}(y) = sup( R(x, y) | x â X)

sup: Supremum of the set

**The projection of R on X :**

â_{X}(x_{1}) = 0.8

â_{X}(x_{2}) = 0.8

â_{X}(x_{3}) = 1.0

**The projection of R on Y :**

â_{Y}(y_{1}) = 0.9

â_{Y}(y_{2}) = 1.0

â_{Y}(y_{3}) = 0.7

â_{Y}(y_{4}) = 0.8

## Watch on YouTube: Fuzzy relation

### Fuzzy composition:

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

**Fuzzy MaxâMin composition **is defined as:

**Fuzzy MaxâProduct 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 !**

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

Thank you Payal

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Thanks for your words Himani

Wow thanks that was nice explanation about opretiom of fuzzy set.

Can you explain about fuzzy subghroup