Properties of relation: Reflexivity, Symmetricity and more

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

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

Properties of relation:

Reflexivity:

Relation R is reflexive if every element in the set is associated with itself, i.e. μ_R(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 μ_R(1, 1) = μ_R(2, 2) = μ_R(3, 3) = 1.

This relation holds the condition μ_R(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. μ_R(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 μ_R(1, 1) = μ_R(2, 2) = μ_R(3, 3) = 0.

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

Symmetricity:

A relation R is called symmetric if element x is related to element y then element y must be related to x, i.e. μ_R(x, y) = μ_R(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 itself. So the given relation is symmetric relation.

Anti-symmetricity:

Relation R is called anti-symmetric if, μ_R(x, y) > 0, then μ_R(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= μ_R(x_i, x_j ), λ₂= μ_R(x_j, x_k) and λ = μ_R(x_i, x_k)

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

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

How to test transitivity?

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

We can compute R² by taking the composition of relation R with itself, i.e. R² = R ∘ R

μ_R²(x, z) = max( μ_R(x, y), μ_R(y, z) ) where y ∈ X

R is transitive if R² ⊂ R, i.e. μ_R²(x, y) ≤ μ_R(x, y)

Example:

Check if the 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,

μ_R²(1, 1) ≤ μ_R(1, 1)

μ_R²(1, 2) ≤ μ_R(1, 2)

μ_R²(1, 3) ≤/ μ_R(1, 3) ( ≤/ indicates not less than or equal to)

As μ_R²(x, y) is not always less than or equal to μ_R(x, y), for all (x, y), hence R is not transitive

Similarity / Equivalence Relation:

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

Anti-Similarity / Partial Order Relation:

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

i.e. μ_R‘(x, y)=1 – μ_R(x, y)

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

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

Example:

Check if the given relation is anti-similarity relation or not.

μ_R‘(x, y) = 1 – μ_R(x, y)

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

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Weak Similarity / Tolerance Relation:

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

Order Relations:

Relation R is an order relation if it is a transitive relation

Relation R is pre-order relation if it is a 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 the composition of relation with itself

R′ → R ° R

R′′ → R′ ° R′

Example – 1:

Check if the given crisp relation is equivalency or not:

Solution:

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

R = R^T, 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 the following relation into equivalence if it is not already

Solution:

μ_R(x_i, x_i) = 1, ∀i, so relation is reflexive

μ_R(x_i, x_j) = μ_R(x_j, x_i) , so relation is symmetric

λ₁ = μ_R(x₁, x₂) = 0.8

λ₂ = μ_R(x₂, x₅) = 0.9

λ = μ_R(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 the composition of relation with itself. This process is to be repeated until it becomes equivalence.

λ₁ = μ_R(x₁, x₂) = 0.8

λ₂ = μ_R(x₂, x₄) = 0.5

λ = μ_R(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 = μ_R(x₁, x₃) = 0.4

λ₂ = μ_R(x₃, x₂) = 0.4

λ = μ_R(x₁, x₂) = 0.8

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

Thus, this article concludes all the properties of relation.

Test Your Knowledge!

Given the relation matrix,

Answer the following questions:

1. Is the relation reflexive? Why?
2. Is the relation symmetric? Why?
3. Is relation transitive? Why?
4. Is relation equivalence? Why?
5. Is the relation partial order / anti-similar? Why?
6. Is the relation weak similar / tolerance? Why?

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