Antisymmetric Relation

Relation and its types are an essential aspect of the set theory. You must know that sets, relations, and functions are interdependent topics. Sets indicate the collection of ordered elements, while functions and relations are there to denote the operations performed on sets. Relations, specifically, show the connection between two sets. You can find out relations in real life like mother-daughter, husband-wife, etc. Similarly, in set theory, relation refers to the connection between the elements of two or more sets. Antisymmetric relation is a concept based on symmetric and asymmetric relation in discrete math. To put it simply, you can consider an antisymmetric relation of a set as one with no ordered pair and its reverse in the relation.   

Basics of Antisymmetric Relation

A relation becomes an antisymmetric relation for a binary relation R on a set A. In that, there is no pair of distinct elements of A, each of which gets related by R to the other. Other than antisymmetric, there are different relations like reflexive, irreflexive, symmetric, asymmetric, and transitive.   

The relation R is antisymmetric, specifically for all a and b in A; if R(x, y) with x ≠ y, then R(y, x) must not hold. Or similarly, if R(x, y) and R(y, x), then x = y. Therefore, when (x,y) is in relation to R, then (y, x) is not. Here, x and y are nothing but the elements of set A.

Rules of Antisymmetric Relation

Typically, relations can follow any rules. Consider the relation ‘is divisible by,’ it’s a relation for ordered pairs in the set of integers. For a relation R, an ordered pair (x, y) can be found where x and y are whole numbers or integers, and x is divisible by y. However, it’s not necessary for antisymmetric relation to hold R(x, x) for any value of x. That’s a property of reflexive relation. 

Antisymmetric Relation: Definition

As per the set theory, the relation R gets considered as antisymmetric on set A, if x R y and y R x holds, given that x = y. You can also say that relation R is antisymmetric with (x, y) ∉ R or (y, x) ∉ R when x ≠ y.

(Note: The use of the graphic symbol ‘∈’ stands for ‘an element of,’ e.g., the letter A ∈ the set of letters in the English language.)

Relation R is not antisymmetric if x, y ∈ A holds, such that (x, y) ∈ R and (y, a) ∈ R but x ≠ y. You also need to keep in mind that if a relationship is not symmetric, it doesn’t imply that it’s antisymmetric. 

In antisymmetric relation, it’s like a thing in one set has a relation with a different thing in another set. And that different thing has relation back to the thing in the first set. To simplify it, a has a relation with b by some function and b has a relation with a by the same function. That can only become true when the two things are equal.

Symmetric, Asymmetric, and Antisymmetric Relations

Many students often get confused with symmetric, asymmetric, and antisymmetric relations. We are here to learn about the last type when you understand the first two types as well. It can indeed help you quickly solve any antisymmetric relation example.

• Symmetric: Relation R of a set X becomes symmetric if (b, a) ∈ R and (a, b) ∈ R. Keep in mind that the relation R ‘is equal to’ is a symmetric relation like, 5 = 3 + 2 and 3 + 2 = 5. The relation is like a two-way street. 

• Asymmetric: Relation R of a set X becomes asymmetric if (a, b) ∈ R, but (b, a) ∉ R. You should know that the relation R ‘is less than’ is an asymmetric relation such as 5 < 11 but 11 is not less than 5.

• Antisymmetric: Relation R of a set X becomes antisymmetric if (a, b) ∈ R and (b, a) ∈ R, which means a = b. But, if a ≠ b, then (b, a) ∉ R, it’s like a one-way street. 

Solved Examples

Below you can find solved antisymmetric relation examples that can help you understand the topic better. 

Question 1: Which of the following are antisymmetric? 

1. R = { (1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4) } 

2. R = { (1, 1), (1, 3), (3, 1) }

3. R = { (1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3),(4, 1), (4, 4) }

Solution: Rule of antisymmetric relation says that, if (a, b) ∈ R and (b, a) ∈ R, then it means a = b.

In case a ≠ b, then even if (a, b) ∈ R and (b, a) ∈ R holds, the relation cannot be antisymmetric.  

Keeping that in mind, below are the final answers.

1. Here, R is not antisymmetric as (1, 2) ∈ R and (2, 1) ∈ R, but 1 ≠ 2. 

2. R is not antisymmetric because of (1, 3) ∈ R and (3, 1) ∈ R, however, 1 ≠ 3. 

3. Here, R is not antisymmetric because of (1, 2) ∈ R and (2, 1) ∈ R, but 1 ≠ 2. Also, (1, 4) ∈ R, and (4, 1) ∈ R, but 1 ≠ 4.     

Question 2: R is the relation on set A and A = {1, 2, 3, 4}. Find the antisymmetric relation on set A. 

Solution: The antisymmetric relation on set A = {1, 2, 3, 4} is; 

R = { (1, 1), (2, 2), (3, 3), (4, 4) }.

In Maths, we can conclude that a binary relation on a set is called as antisymmetric if there is no pair of distinct elements.