What is Antisymmetric Relation: Example, Formula & Questions

An antisymmetric Relation is a form of relation in which if the variables are shifted, it does not give the result of the actual relation.

Mathematically, relation R is antisymmetric, especially if:

R(x, y) with x ≠ y, then R(y, x) must not hold good.

If it holds good, then it must only be because x = y.

The antisymmetric function is true every time (x,y) is in relation to R, but (y, x) is not. 

Do note that this topic is important for both JEE Main and IIT JAM exam aspirants.

Example: If the relation R is “is divisible by”, and x and y are 6 and 2 respectively,

R (x, y) = R (6, 2) = 6 is divisible by 2 which holds good.
R (y, x) = R (2, 6) but 2 is not divisible by 6.

This forms an antisymmetric relation.

By extrapolating this to set theory, it can be written like this:

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,x) ∈ R but x ≠ y.

Note: If a relation is not symmetric, it doesn’t immediately mean it is antisymmetric. It could be Reflexive, Irreflexive, Asymmetric or Transitive.

There is a set S = {1,2,3}. Let us define the relation R={(1,1),(1,2),(2,2),(2,3),(3,3)}. Let us now check every ordered pair in R 

Since there is no counter example, R is an antisymmetric relation.

CBSE board students must understand these examples so that they can solve questions that require the student to identify an antisymmetric relation.

Follow the below-given steps to determine whether a relation is antisymmetric or not:

• Check whether (b, a) exists for every (a,b) in a given relation.
• In case (b, a) is present and b ≠ a, then the relation will not be antisymmetric.
• For every (a,b), there will be (b, a) and in every pair, a = b. In this case, the relation will be antisymmetric.
• If (b, a) is not present, then the relation will be antisymmetric.

The following points explain the properties of antisymmetric relations:

• Antisymmetric relations can be either reflexive or irreflexive. A relation will be reflexive if each element is related to itself. A reflexive type antisymmetric relation will have (a,a) in R for each a in A.
• An antisymmetric relation may have a possibility that neither (a,b) nor (b,a) is in R unless a=b.
• Subset (⊆) relation on power set of any set will be antisymmetric. Here, if  A⊆B and B⊆A, then A=B.
• Many of the important antisymmetric relations, especially those used in posets, are transitive. 

Let us take a look at some sample questions that may be asked in NEET and CUET exams:

1. Which of these is 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 = {(0,1), (1,2), (1,4), (2,3), (2,5), (3,1), (4,5), (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 antisymmetric here because all the (x, y) terms are present with no (y, x).

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)}

3.Is the relation R(x, y) = x + y antisymmetric?

Solution.

No. It is symmetric as R(x, y) = R(y, x) = x + y = y + x. Taking x and y as 2 and 3 will give you 5 whether 2 + 3 or 3 + 2.