Antisymmetric Relation: Overview, Questions, Preparation

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 every time (x,y) is in relation to R, but (y, x) is not. 

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 the 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.

This topic is introduced in Class 12 for the students to understand how certain relations do not apply in the reverse direction. This topic is crucial for them to grasp certain equations and concepts. This broad range topic has wide applications in set theory, basic mathematics, and geometry. 

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.

Q: What is a relation?

Q: Give the mathematical representation of an antisymmetric relation.

Q: Is R (x, x) an antisymmetric relation?

Q: What is the difference between antisymmetric and asymmetric relation?

Q: Name a few types of relations.