Properties of relations is important to understand the characteristics or the behavior of relation. Many set properties are similar for crisp relations and fuzzy relations. We have already discussed properties of crisp sets and properties of fuzzy sets. In this article, we will learn about the properties of relation

Note: The Greek letter \chi (chi) is represented as χ in the article.

## Properties of relation:

### Reflexivity:

Relation R is reflexive if every element in set is associated with it self, i.e. μR(x, x) = 1, ∀x ∈ X

Example:

Consider the universe X = {1, 2, 3}. The fuzzy relation R defined over X is,

As we can see that μR(1, 1) = μR(2, 2) = μR(3, 3) = 1.

This relation holds the condition μR(x, x) = 1 for all x in X. So this is reflexive relation.

### Anti-reflexivity:

Relation R is an anti-reflexive if ∀x ∈ X, (x, x) ∉ R, i.e. μR(x, x) = 0, ∀x ∈ X

Example:

Consider the universe X = {1, 2, 3}. The fuzzy relation R defined over X is,

As we can see that μR(1, 1) = μR(2, 2) = μR(3, 3) = 0.

This relation holds the condition μR(x, x) = 0 for all x in X. So this is anti-reflexive relation.

### Symmetricity:

Relation R is called symmetric if element x is related to element y then element y must be related to x, i.e. μR(x, y) = μR(y, x), ∀x, y ∈ X

Example:

Consider the universe X = {1, 2, 3}. The fuzzy relation R defined over X is,

As we can see that the transpose of relation matrix R is the matrix it self. So the given relation is symmetric realtion.

### Anti-symmetricity:

Relation R is called anti-symmetric if, μR(x, y) > 0, then μR(y, x) = 0,    x, y ∈ X, x ≠ y

Example:

Consider the universe X = {1, 2, 3}. The fuzzy relation R defined over X is,

As we can observe in above relation matrix, R(1, 3), R(2, 1) and R(3, 2) are non zero, and their corresponding R(y, x), i.e. R(3, 1), R(1, 2) and R(2, 3) are zero and hence the relation is anti-symmetric.

### Transitivity:

For crisp Relation:

Crisp Relation R is called transitive if x is related to y and y is related to z then x must be related to z.

That states if, (x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R

For fuzzy relation:

Let λ1= μR(xi, xj ), λ2= μR(xj, xk) and λ = μR(xi, xk)

For fuzzy relation to be transitive, λ ≥ min⁡(λ1, λ2 ) for all λ

Note: It is clear that ≤ is reflexive, anti-symmetric and transitive, < is anti-reflexive, anti-symmetric and transitive.

How to test transitivity?

For any given matrix, it is hard to manually test all the pair for different λ values. It takes tremendous amount of time. Can we do it better way? Of course, there is a smarter and easier way to test if given relation is transitive or not.

We can compute R2 by taking composition of relation R with it self, i.e. R2= RR

μR2(x, z) = max( μR(x, y), μR(y, z) ) where y ∈ X

R is transitive if R2R, i.e. μR2(x, y) ≤ μR(x, y)

Example:

Check if given relation is transitive or not?

Solution:

As discussed above, to check the transitivity of relation R, we shall compute R2

For the above matrices,

μR2(1, 1) ≤ μR(1, 1)

μR2(1, 2) ≤ μR(1, 2)

μR2(1, 3) ≤/ μR(1, 3) ( ≤/ indicates not less than or equal to)

As μR2(x, y) is not always less than or equal to μR(x, y), for all (x, y), hence R is not transitive

### Similarity / Equivalence Relation:

If relation R ̅ is reflexive, symmetric and transitive than it is called similarity (equivalence) relation

### Anti-Similarity / Partial Order Relation:

If R is similarity relation then its complement is anti-similarity (partial order) relation

i.e. μR‘(x, y)=1 – μR(x, y)

Anti-similarity relation is anti-reflexive, symmetric and transitive, in the sense of max-min

μR‘(x, y) ≥ min( max( μR‘(x, y), μR‘(y, z)) ) where y ∈ X

Example:

Check if given relation is anti-similarity relation or not?

μR‘(x, y)=1 – μR(x, y)

As it is anti-reflexive, symmetric and transitive, so R is anti-similarity relation

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### Weak Similarity / Tolerance Relation:

If relation R is reflexive and symmetric but not transitive than it is called weak similarity (tolerance) relation

### Order Relations:

Relation R is order relation if it is transitive relation

Relation R is pre-order relation if it is reflexive and transitive relation

Relation R is half-order relation if it is reflexive and weak anti-symmetric relation

### Tolerance to Equivalence relation conversion:

Tolerance (weak similarity) relation can be converted to equivalence (similarity) by taking composition of relation with itself

R′→ R ° R

R′′→ R′ ° R′

Example – 1:

Check if given crisp relation is equivalence or not:

Solution:

Here, (xi, xi ) ∈ R, ∀i, so R is reflexive

R = RT, so R is symmetric

χ(x1, x2) = 1, χ(x2, x5) = 1 but χ(x1, x5) = 0, so R is not transitive

Hence R is Not equivalence but it is Tolerance

Example 2: χ

Convert following relation into equivalence if it is not already

Solution:

μR(xi, xi) = 1, ∀i, so relation is reflexive

μR(xi, xj) = μR(xj, xi) , so relation is symmetric

λ1 = μR(x1, x2) = 0.8

λ2 = μR(x2, x5) = 0.9

λ = μR(x1, x5) = 0.2

For fuzzy relation to be transitive, λ ≥ min⁡(λ1, λ2)

For given relation matrix λ < min⁡( λ1, λ2 ), so given relation is tolerance but not equivalence

We can convert tolerance relation into equivalence relation by taking composition of relation with it self. This process is to be repeated until it becomes equivalence.

λ1 = μR(x1, x2) = 0.8

λ2 = μR(x2, x4) = 0.5

λ = μR(x1, x4) = 0.2

For given relation matrix λ < min⁡( λ1, λ2 ), so given relation is yet not equivalence

Let us repeat the step again and take composition once again.

λ1 = μR(x1, x3) = 0.4

λ2 = μR(x3, x2) = 0.4

λ = μR(x1, x2) = 0.8

For given relation matrix λ > min⁡( λ1, λ2 ), and this is true for any pair in the above matrix. So given relation is now equivalence