Asymmetric Relation

What is an Asymmetric Relation?

In discrete Mathematics, the opposite of symmetric relation is asymmetric relation. In a set X, if one element is less than another element, agrees the one relation, then the other element will not be less than the first one. Therefore, less than (>), greater than (<) and minus (-) are examples of asymmetric relation. We can even say that the ordered pair of set X agrees with the condition of asymmetric only if the reverse of the ordered pair does not agree with the condition. This makes it identical from symmetric relation, where even the exact opposite of their orders are reversed, the condition is satisfied. There are 8 types of relations, these are :

  • Empty Relation

  • Universal Relation

  • Identity Relation

  • Inverse Relation

  • Reflexive Relation

  • Symmetric Relation

  • Transitive Relation

  • Equivalence Relation

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Domain and Range

If there are two relations A and B and relation for A and B is R (a,b), then the domain is stated as the set { a | (a,b) ∈ R for some b in B} and range is stated as the set {b | (a,b) ∈ R for some a in A}.

Asymmetric Relation Definition

Asymmetric relation is the opposite of symmetric relation but not considered as equivalent to antisymmetric relation.

In Set theory, A relation R on set A is known as asymmetric relation if no (b,a) ∈ R when (a,b) ∈ R or we can even say that relation R on set A is symmetric if only if (a,b) ∈ R⟹(b,a) ∉R.

For example: If R is a relation on set A= (18,9) then (9,18) ∈ R indicates 18>9 but (9,18) R,  Since 9 is  not greater than 18. 

Note- Asymmetric relation is the opposite of symmetric relation but not considered as equivalent to antisymmetric relation.

The mathematical operators -,< and > are asymmetric examples whereas  =, ≥, ≤, are considered as the twins of ()  and do not agree with the asymmetric condition. 

Asymmetrical Relation Properties

Some basic asymmetrical relation properties  are :

  • A relation is considered as an asymmetric if it is both antisymmetric and irreflexive or else it is not.

  • Limitations and opposite of asymmetric relation are considered as asymmetric relation. For example- the inverse of less than is also an asymmetric relation.

  • Every asymmetric relation is not strictly partial order.

  • Subsequently, if a relation is of a strict partial order, then it will be considered as transitive and symmetric.

  • An asymmetric relation should not have the convex property. For example, the strict subset relation is regarded as asymmetric and neither of the assets such as  {3,4} and  {5,6} is a strict subset of others.

  • A transitive relation is considered as asymmetric if it is irreflexive or else it is not. For example: if aRb and bRa , transitivity gives aRa contradicting ir-reflexivity.

Asymmetric Relation Solved Examples

1. If X= (3,4) and Relation R on set  X is (3,4), then Prove that the Relation is Asymmetric.

Solution: Give X= {3,4} and {3,4}∈ R

Clearly, we can see that 3 is less than 4 but 4 is not less than 3, hence

{3,4} ∈ R  ⇒ {4,3}∉ R

Hence, it is proved that the relation on set X is symmetric

Quiz Time

2. Let R be the Relation Between x and y. R is Asymmetric if and only if

  1. Intersect of D (X) and R is empty, where D(X) indicates diagonals of sets.

  2. The intersection of R and R-1 is D(X)

  3. D(X) is a subset of R, where D(X) indicates diagonals of set

  4. R-1 is a subset of R, where R-1 indicates the inverse of R.

3. The Range of the Function f(x) = x is |x|

  1. ( 0, ¥)

  2. (- ¥, 0)

  1. (0, ¥)

  2. None of these

FAQ (Frequently Asked Questions)

1. Explain the Comparison Between Asymmetric and Antisymmetric Relation?

Ans.

Asymmetric Relation

An  asymmetric relation says R agrees to the following condition 

If (a,b) is in R, the (b,a ) will not be in R.

Hence, if element a is related to element b through some rules, then b will not be related to a through that same rule.

Antisymmetric Relation 

If (a,b), and (b,a) are in set Z, then a = b.

Hence, if an element a is related to element b, and element b is also related to element a, then a and b should be a similar element. Therefore, in an antisymmetric relation, the only ways it agrees to both situations is a=b.

Although both have similarities in their names, we can see differences in both their relationships such that asymmetric relation does not satisfy both conditions whereas antisymmetric satisfies both the conditions, but only if both the elements are similar.