A Short Note On Anti-Symmetric Relation

Anti-symmetric relation can be understood by analyzing symmetric relation. A symmetric relation is a relation between two or more elements of any set, then that relation is known as symmetric relation. A symmetric relation is a binary relation. So, suppose there is a set S with a binary relation L for the elements available in the set S shares a relation (pLq). For the same elements, there should be a relation (qLp) also coexisting at the same time for the whole relation to be called symmetric relation. Therefore, A relation L on a set S is said to be antisymmetric if there is no pair of distinct elements of S that are related to each other by any relation L. This article will take you through anti-symmetric relation and antisymmetric relation examples at the end.

What Is Anti-Symmetric Relation?

Any binary relation L formed by the elements present in the set S is said to antisymmetric relation if no elements from the set S get involved in any relation L. If a relation L exists between two elements of set S and that relation L is formed with other elements of the same set S, then that relation L is said to be antisymmetric relation. Anti-symmetric and symmetric relations are two of seven relations generally studied in discrete mathematics. Some other relations are reflexive, irreflexive, trasitive, etc. 

For example, suppose there is a set S whose two elements are p and q. If a relation L exists, L(p, q) where p should not be equal to q, then L(p, q) does not have to hold. If L(p, q) and L(q, p) exist, only when p=q. Hence, whenever the pair of (p q) is in a relation L, then there should not be any pair of (q, p) present in the set S. Here, p and q are the elements of set S, that can be a number or alphabet.

For relation L, a pair (p, q) can be found where p and q are natural numbers; p is divisible by q. A relation L doesn’t need to be antisymmetric only if it holds L(p, p) or L(q, q) should exist for any value of p or q.

In the set theory, the relation L is said to be antisymmetric on a set S if pLq and qLp hold when p is equal to q. Or it can be described as relation L is claimed to be antisymmetric if either (p,q) does not belong to relation L or (q, p) does not belong to relation L whenever p is not equal to y.

A relation L denies being antisymmetric if p,q∈S exists such that (p,q) ∈ L and (q,p) ∈ L but p ≠ q.

In simpler words, It is not mandatory that if a relation is not symmetric, that means it is definitely antisymmetric or vice versa. 

Solved Examples

Q1.  Which of these pairs of elements is antisymmetric?

(i) L = {(1,5),(1,2),(5,4),(3,3),(3,4),(4,5),(7,9)}

(ii) L = {(1,2),(2,4),(3,1)}

(iii) L = {(1,3),(1,6),(1,1),(6,1),(2,7),(7,3),(3,1),(5,9)}

Solution:

(i) L is not antisymmetric here because of (5,4) ∈ L and (4,5) ∈ L, but 5 ≠ 4.

(ii) L is antisymmetric here because neither (1,2) or (2,4) nor (3,1) have (2,1), (4,2) or (1,3) to form a symmetric relation.

(iii) L is not antisymmetric here because of (1,3) ∈ L and (3,1) ∈ L, but 1 ≠ 3 and also (1,6) ∈ L and (6,1) ∈ L but 1 ≠ 6.

Q2. If S = {2,4,6,9} and L is the relation on set S, find the antisymmetric relation within the set S.

Solution: The antisymmetric relation on set S = {2,4,6,9} can be;

L = {(2,2), (4,4),(6,6),(9,9)} because 2=2, 4=4, 6=6, 9=9.

Q3. Determine if the following relation L is antisymmetric on set S = {2,4,5,7,9}.

L={(2,2),(4,4),(5,5),(7,7),(9,9)}

Solution: There is no pair of numerical elements of S that can be related to each other by L.

Hence, L is an antisymmetric relation.

Conclusion

In antisymmetric relation, if the element p=the element p, then element q = the element of q has to satisfy. In other words, a relation L is antisymmetric only if (q, p) belonging to L, and when (p and q the elements of set S should never belong to L. Let’s say we have an ordered pair set  S = {1,4,5}. L = {(1,3), (3,7), (7,1)}. So, now, if we interchange (p, q) in the relation L, we will not get (q, p) such as (3,1),(7,3), and (1,7)  for any pair (1,3), (3,7), and (7,1). Therefore, you can easily claim that in the given three possible elements presented in ordered pairs cases, not even a single pair has symmetric relation with the other pairs; hence, this is an antisymmetric relation.