# 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 λ_{1}= μ_{R}(x_{i}, x_{j} )*, *λ_{2}= μ_{R}(x_{j}, x_{k})* *and λ = μ_{R}(x_{i}, x_{k})

For fuzzy relation to be transitive, λ ≥ min(λ_{1}, λ_{2} ) 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 R^{2} by taking the composition of relation R with itself, i.e. R^{2} = R ∘ R

μ_{R}^{2}(x, z) = max( μ_{R}(x, y), μ_{R}(y, z) ) where y ∈ X

R is transitive if R^{2} ⊂ R, i.e. μ_{R}^{2}(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 R^{2}

For the above matrices,

μ_{R}^{2}(1, 1) ≤ μ_{R}(1, 1)

μ_{R}^{2}(1, 2) ≤ μ_{R}(1, 2)

μ_{R}^{2}(1, 3) ≤/ μ_{R}(1, 3) ( ≤/ indicates not less than or equal to)

As μ_{R}^{2}(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

## Watch on YouTube: Properties of relation

### 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(x_{1}, x_{2}) = 1, χ(x_{2}, x_{5}) = 1 but χ(x_{1}, x_{5}) = 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

λ_{1} = μ_{R}(x_{1}, x_{2}) = 0.8

λ_{2} = μ_{R}(x_{2}, x_{5}) = 0.9

λ = μ_{R}(x_{1}, x_{5}) = 0.2

For fuzzy relation to be transitive, λ ≥ min(λ_{1}, λ_{2})

For given relation matrix λ < min( λ_{1}, λ_{2} ), 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.

λ_{1} = μ_{R}(x_{1}, x_{2}) = 0.8

λ_{2} = μ_{R}(x_{2}, x_{4}) = 0.5

λ = μ_{R}(x_{1}, x_{4}) = 0.2

For given relation matrix λ < min( λ_{1}, λ_{2} ), so given relation is yet **not equivalence**

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

λ_{1} = μ_{R}(x_{1}, x_{3}) = 0.4

λ_{2} = μ_{R}(x_{3}, x_{2}) = 0.4

λ = μ_{R}(x_{1}, x_{2}) = 0.8

For the given relation matrix λ > min( λ_{1}, λ_{2} ), 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:

- Is the relation reflexive? Why?
- Is the relation symmetric? Why?
- Is relation transitive? Why?
- Is relation equivalence? Why?
- Is the relation partial order / anti-similar? Why?
- Is the relation weak similar / tolerance? Why?

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