Antisymmetric Relation (original) (raw)

Last Updated : 23 Jul, 2025

Antisymmetric Relation is a type of binary relation on a set where any two distinct elements related to each other in one direction cannot be related in the opposite direction. For example, consider the relation "less than or equal to" (≤) on the set of integers. This relation is antisymmetric because if a ≤ _b and _b ≤ _a, then _a must be equal to _b. This article deals with Antisymmetric Relation including their definition, examples, as well as properties.

Table of Content

• What is Antisymmetric Relation?
• Examples of Antisymmetric Relations
• How to Check Relation is Antisymmetric or not?
• Number of Antisymmetric Relations
• Properties of Antisymmetric Relations
• Symmetric and Antisymmetric Relations

What is Relation in Maths?

A relation refers to a set of ordered pairs, where each pair consists of elements from two sets. These sets can be the same or different. A relation R between two sets A and B is defined as a subset of the Cartesian product A × B. In other words, if (a, b) is an ordered pair in the relation R, it means that there is some kind of relationship between a and b.

For example, let's consider two sets:

• A = {1, 2, 3}
• B = {4, 5, 6}

A relation between A and B could be R={(1,4), (2,5), (3,6)}.

Types of Relation

There can be various types of relations in mathematics, i.e.,

• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Antisymmetric Relation
• Asymmetric Relation
• Equivalence Relation

In this article, we will learn about antisymmetric relation in detail.

What is Antisymmetric Relation?

An antisymmetric relation is a relation in which is two elements of set are related with relation R i.e., first element R second element and second element R first element then, first element is equal to second element.

In other words, antisymmetric relation is defined as if aRb and bRa then, a = b. A relation R = {(a, b) → R | a ≤ b} is an asymmetric relation since a ≤ b and b ≤ a implies a = b.

Antisymmetric Relation Definition

The relation is said to be an antisymmetric relation if in a set S the two elements p and q are related with relation R then, p = q. Also, if for every (p, q) ∈ R, (q, p) ∉ R then, R is antisymmetric.Mathematically, the antisymmetric relation is defined as:

If x and y are two elements in set X and R is a relation then, conditions for relation to be antisymmetric:

****(xRy and yRx) ⇒ (x = y) ∀ x, y ∈ X**

**or

****(x, y) ∈ R then, (y, x) ∉ R**

Examples of Antisymmetric Relations

There are multiple examples of antisymmetric relation. Some of these examples are listed below.

• Less than operation on two elements.
• Equality relation on any set.
• Divisibility relation
• Subset

Let's consider an example to check for antisymmetry:

**Example: If relation R = {(1, 1), (4, 7), (7, 4)} then, find the given relation is an antisymmetric relation or not?

**Solution:

R = {(1, 1), (4, 7), (7, 4)}

The above relation is antisymmetric as

• (1, 1) ∈ R and (1, 1) ∈ R and 1 = 1.
• (4, 7) ∈ R and (7, 4) ∈ R and 4 ≠ 7.

R is not an antisymmetric relation.

Properties of Antisymmetric Relations

The properties of antisymmetric relations are listed below:

• Empty relation on any set is always antisymmetric.
• A relation can be symmetric and antisymmetric at same time.
• If R is an antisymmetric relation, then R-1 is also antisymmetric.
• It R1 and R2 are two antisymmetric relations, then R1 ∩ R2 is also antisymmetric.
• In the matrix representation of antisymmetric relation, either Mij = 0 or Mij ≠ 0 when i ≠ j.

Number of Antisymmetric Relations

The formula for number of antisymmetric relations with n-elements is given by:

**Total number of antisymmetric relation = 2 n × 3 **[n(n-1)]/2

How to Check Relation is Antisymmetric or not?

To check whether the given relation is antisymmetric or not follow the below steps.

• First, check for every (a, b) in the given relation the existence of (b, a).
• If (b, a) is present and b ≠ a then, relation is not antisymmetric.
• If for every (a, b) there is (b, a) and in all the pairs a = b then, relation is antisymmetric.
• If (b, a) is absent then, relation is antisymmetric.

Symmetric and Antisymmetric Relations

Below table represents the difference between the symmetric and antisymmetric relation.

Conclusion

From the above discussion we can conclude that a relation R is said to be an antisymmetric relation when if x and y holds the relation R i.e., if xRy and yRx then, x = y. The formula for calculating the total number of antisymmetric relations from a set of n elements is **2 n × 3 **[n(n-1)]/2 . Also, we have learnt that a relation can be symmetric or antisymmetric at a same point of time.

**Read More About

• Relation
• Domain and Range of Relation

Sample Problems on Antisymmetric Relations

**Example 1: Check whether the relation R = {(1,4), (2,5)} is antisymmetric or not?

**Solution:

R = {(1,4), (2,5)}

The above relation is antisymmetric as

• (1, 4) ∈ R and (4, 1) ∉ R.
• (2, 5) ∈ R and (5, 2) ∉ R.

R is antisymmetric.

**Example 2: Prove the given relation R = {(2,2), (3,7)} is an antisymmetric relation?

**Solution:

R = {(2, 2), (3, 7)}

The above relation is antisymmetric as

• (2, 2) ∈ R and (2, 2) ∈ R and 2 = 2.
• (3, 7) ∈ R and (7, 3) ∉ R.

R is antisymmetric.

**Example 3: Find the number of antisymmetric relations on set V with 2 elements.

**Solution:

The total number of antisymmetric relation = 2n × 3 [n(n-1)]/2

The total number of antisymmetric relation on given set V = 22 × 3 [2(2-1)]/2

The total number of antisymmetric relations on given set V = 22 × 3

The total number of antisymmetric relations on given set V = 12