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 with 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 relations.
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 the 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 an 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 a 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 asymmetric if it is both antisymmetric and irreflexive or else it is not.
Limitations and opposite of asymmetric relations are considered as asymmetric relations. For example- the inverse of less than is also an asymmetric relation.
Every asymmetric relation is not strictly in the 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 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
Conclusion
This is all about the definition and explanation of asymmetric relation and its different forms. Focus on how the concept has been explained. Understand the concept well so that you can answer questions judiciously.
FAQs on Asymmetric Relation
1. Explain the Comparison Between Asymmetric and Antisymmetric Relation?
Asymmetric Relation
An asymmetric relation says R agrees to the following condition
If (a,b) is in R, then (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 element a is related to element b, and element b is also related to element a, then a and b should be similar elements. Therefore, in an antisymmetric relation, the only way 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.
2. What makes a relation asymmetric?
An asymmetric relation is totally opposite to symmetric relation but not equal to antisymmetric. A relation is called asymmetric only if it is both antisymmetric and irreflexive or else it won't be called asymmetric. It is not necessary that all asymmetric relations are strict partial orders. Connex property should be absent in an asymmetric relation. There are a few more types of relation, which are :
Equivalence Relation, Transitive Relation, Symmetric Relation, Reflexive Relation, Inverse Relation, Identity Relation, Universal Relation, and Empty Relation.
3. How do you prove antisymmetric relations?
We need to assume that (a, b) and (b, a) are in a relationship and then we need to show that A is equal to B. Now to prove A=B we need to show that A is divisible by B and B is divisible by A to prove that our relation R is antisymmetric.
For example, The number of biscuits is divisible by the number of students in the class.
The number of students in the class is divisible by the number of biscuits.
4. What makes a relation symmetric?
A symmetric relation is classified as a binary relation. For example, if the relationship is equal to because if A is equal to B is not a false statement then B is equal to A is also true. Similarly, a binary relation R over a set X is symmetric if: If RT represents the converse of R, then R is symmetric if and only if R is equal to RT. Asymmetric relations will always remain flexible.