1. What an Antisymmetric Relation Means
An antisymmetric relation allows pairs to reverse only when the two elements are actually the same. This means if (a,b) and (b,a) both appear in the relation, then a and b must be equal. If a ≠ b, the relation cannot contain both (a,b) and (b,a) at the same time.
2. Formal Condition
The condition for a relation R on a set A to be antisymmetric is:
2.1. Definition
(a,b),(b,a) \in R \; \Rightarrow \; a = b
3. Understanding the Idea
Antisymmetry does not mean the relation forbids reverse pairs entirely. It only forbids reverse pairs between different elements. Self-pairs like (a,a) never violate antisymmetry because a = b.
4. Examples of Antisymmetric Relations
Here are relations that satisfy the antisymmetric condition:
4.1. Example 1 (Simple)
Let A = \{1,2,3\}. Consider:
R = \{(1,1),(2,2),(3,3),(1,2)\}
Since (2,1) does not appear, the relation is antisymmetric.
4.2. Example 2 (All Self-Pairs)
R = \{(a,a),(b,b),(c,c)\}
This is antisymmetric because self-pairs never break the condition.
4.3. Example 3 (Ordering Type)
The relation “≤” on numbers is antisymmetric because:
a \le b \text{ and } b \le a \Rightarrow a = b
5. How to Check Antisymmetry
Look for any two different elements a and b such that both (a,b) and (b,a) appear. If such a pair exists and a ≠ b, the relation is not antisymmetric.
5.1. Illustration
Let A = \{x,y,z\} and:
R = \{(x,y),(y,x),(z,z)\}
Here (x,y) and (y,x) both appear, and x ≠ y, so this relation is not antisymmetric.
6. Non-Examples (Not Antisymmetric)
A relation fails to be antisymmetric when two different elements relate to each other both ways.
6.1. Example
R = \{(1,2),(2,1)\}
Since 1 ≠ 2, this breaks antisymmetry.
7. Where Antisymmetric Relations Appear
Antisymmetric relations commonly represent ordering or comparison where equality is the only allowed reversal. Examples include:
- “less than or equal to”
- “subset of” (⊆)
- “divides” relation in numbers