Antisymmetric Relation

In discrete Maths, a relation is said to be antisymmetric relation for a binary relation R on a set A, if there is no pair of distinct or dissimilar elements of A, each of which is related by R to the other. In a formal way, relation R is antisymmetric, specifically if for all a and b in A, if R(x, y) with x ≠ y, then R(y, x) must not hold, or, equivalently, if R(x, y) and R(y, x), then x = y. Hence, as per it, whenever (x,y) is in relation R, then (y, x) is not. Here x and y are the elements of set A. Apart from antisymmetric, there are different types of relations, such as:

  • Reflexive
  • Irreflexive
  • Symmetric
  • Asymmetric
  • Transitive

An example of antisymmetric is: for a relation “is divisible by” which is the relation for ordered pairs in the set of integers. For relation, R, an ordered pair (x,y) can be found where x and y are whole numbers and x is divisible by y. It is not necessary that if a relation is antisymmetric then it holds R(x,x) for any value of x, which is the property of reflexive relation.

Antisymmetric Relation Definition

In set theory, the relation R is said to be antisymmetric on a set A, if xRy and yRx hold when x = y. Or it can be defined as, relation R is antisymmetric if either (x,y)∉R or (y,x)∉R whenever x ≠ y.

A relation R is not antisymmetric if there exist x,y∈A such that (x,y) ∈ R and (y,a) ∈ R but x ≠ y.

Note: If a relation is not symmetric that does not mean it is antisymmetric.

Also, read:

Antisymmetric Relation Examples

Q.1: Which of these are antisymmetric?

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

(ii) R = {(1,1),(1,3),(3,1)}

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

Solution:

(i) R is not antisymmetric here because of (1,2) ∈ R and (2,1) ∈ R, but 1 ≠ 2.

(ii) R is not antisymmetric here because of (1,3) ∈ R and (3,1) ∈ R, but 1 ≠ 3.

(iii) R is not antisymmetric here because of (1,2) ∈ R and (2,1) ∈ R, but 1 ≠ 2 and also (1,4) ∈ R and (4,1) ∈ R but 1 ≠ 4.

Q.2: If A = {1,2,3,4} and R is the relation on set A, then find the antisymmetric relation on set A.

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

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

Symmetric, Asymmetric and Antisymmetric Relation

Symmetric

Asymmetric

Antisymmetric

Relation R on set A is symmetric if (b, a)∈R and (a,b)∈R

Relation R on a set A is asymmetric (a,b)∈R but (b,a)∈ R

Relation R of a set A is antisymmetric if (a,b) ∈ R and (b,a) ∈ R, then a=b

“Is equal to” is a symmetric relation, such as 3 = 2+1 and 1+2=3.

“Is less than” is an asymmetric, such as 7<15 but 15 is not less than 7

If a ≠ b, then (b,a)∈R

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