Antisymmetric Relation

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.

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 M_ij = 0 or M_ij ≠ 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 = 2ⁿ × 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.

Characteristics	Symmetric Relation	Antisymmetric relation
Definition	A relation R is symmetric when two elements p and q of set A if p is related to q then, q is also related to p.	A relation R is antisymmetric when two elements p and q of set A if p is related to q and q is related to p then p is equal to q or if (p, q) belongs to R then, (q, p) does not belongs to R.
Mathematical Representation	pRq ⇔ qRp	(pRq and qRp) ⇒ (p = q) ∀ x, y ∈ X or (p, q) ∈ R then, (q, p) ∉ R
Example	Parallel lines if a || b then b || a	Greater than if a > b then, b≯ a.

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 2ⁿ × 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

FAQs on Antisymmetric Relations

What is an Antisymmetric Relation?

A relation R on a set A is antisymmetric if for every distinct pair (a,b) and (b,a) in R, a must be equal to b.

What is an Example of Antisymmetric Relation?

An example of antisymmetric relation is divisibility or subset.

Can a Relation be Symmetric and Antisymmetric at Same Time?

No, a relation cannot be both symmetric and antisymmetric at the same time.

Are all the Reflexive Relations Antisymmetric?

No, all the reflexive relations are not antisymmetric.

What is the difference between Reflexive and Anti Symmetric Relations?

Differenec between reflexive and anti symmetric relation is:

• Reflexivity focuses on ensuring that each element is related to itself.
• Antisymmetry focuses on ensuring that if two distinct elements are related to each other in both directions, they must be equal.