1. What an Asymmetric Relation Means
An asymmetric relation never allows a reverse pair. If (a,b) is in the relation, then (b,a) must never appear. This must be true for all pairs of distinct elements.
Asymmetry is stronger than antisymmetry. It does not allow both (a,b) and (b,a) even when a = b, so self-pairs are also not allowed.
2. Formal Condition
A relation R on a set A is asymmetric if:
2.1. Definition
(a,b) \in R \; \Rightarrow \; (b,a) \notin R
and also:
(a,a) \notin R
3. Key Ideas Behind Asymmetry
Asymmetric relations describe strictly one-way relationships. They do not loop back and do not allow equality-based pairs. Every connection points in only one direction.
4. Examples of Asymmetric Relations
Here are some relations that satisfy the asymmetric condition:
4.1. Example 1 (Simple)
Let A = \{1,2,3\}. Consider:
R = \{(1,2),(2,3)\}
No reverse pair appears, and no self-pairs appear, so the relation is asymmetric.
4.2. Example 2 (Strict Ordering)
The relation “<” on numbers is asymmetric:
a < b \Rightarrow b ¬< a
4.3. Example 3
If R = \{(a,b),(b,c),(c,d)\}, no pair reverses, so the relation is asymmetric.
5. How to Check Asymmetry
To test an asymmetric relation:
- Look for any (a,b) in R.
- Check whether (b,a) also appears.
- If even one reverse pair appears, the relation is not asymmetric.
- Also ensure no self-pairs (a,a) appear.
5.1. Illustration
For A = \{x,y,z\}:
R = \{(x,y),(y,x)\}
This relation is not asymmetric because both directions appear for x and y.
6. Non-Examples (Not Asymmetric)
A relation fails to be asymmetric if it contains either a reverse pair or a self-pair.
6.1. Example 1
R = \{(1,2),(2,1)\}
Reverse pair present → not asymmetric.
6.2. Example 2
R = \{(3,3)\}
Contains self-pair → not asymmetric.
7. Where Asymmetric Relations Appear
Asymmetric relations appear in situations with strict direction and no possibility of reversing, such as:
- “is taller than”
- “is older than”
- “flows into” (for rivers or networks)
- “is the parent of”