What is Antisymmetric Relation?
An antisymmetric relation is a relation in which is two elements of set are related with relation R i.e., first element R second element and second element R first element then, first element is equal to second element.
In other words, antisymmetric relation is defined as if aRb and bRa then, a = b. A relation R = {(a, b) → R | a ≤ b} is an asymmetric relation since a ≤ b and b ≤ a implies a = b.
Antisymmetric Relation Definition
The relation is said to be an antisymmetric relation if in a set S the two elements p and q are related with relation R then, p = q. Also, if for every (p, q) ∈ R, (q, p) ∉ R then, R is antisymmetric. Mathematically, the antisymmetric relation is defined as:
If x and y are two elements in set X and R is a relation then, conditions for relation to be antisymmetric:
(xRy and yRx) ⇒ (x = y) ∀ x, y ∈ X
or
(x, y) ∈ R then, (y, x) ∉ R
Antisymmetric Relation
Antisymmetric Relation is a type of binary relation on a set where any two distinct elements related to each other in one direction cannot be related in the opposite direction. For example, consider the relation “less than or equal to” (≤) on the set of integers. This relation is antisymmetric because if a ≤ b and b ≤ a, then a must be equal to b. This article deals with Antisymmetric Relation including their definition, examples, as well as properties.
Table of Content
- What is Antisymmetric Relation?
- Examples of Antisymmetric Relations
- How to Check Relation is Antisymmetric or not?
- Number of Antisymmetric Relations
- Properties of Antisymmetric Relations
- Symmetric and Antisymmetric Relations