1. What an Irreflexive Relation Means
An irreflexive relation never relates an element to itself. This means no self-pair (a,a) appears in the relation for any element in the set.
Even if the relation contains many pairs, it must avoid all self-pairs to be irreflexive.
2. Formal Condition
The condition for a relation R on a set A to be irreflexive is:
2.1. Definition
(a,a) \notin R \quad \text{for every } a \in A
3. Examples of Irreflexive Relations
Here are some relations that satisfy the irreflexive condition:
3.1. Example 1
Let A = \{1,2,3\}. Then the relation:
R = \{(1,2),(2,3)\}
is irreflexive because (1,1), (2,2), and (3,3) are all missing.
3.2. Example 2
The empty relation is always irreflexive:
R = \emptyset
It contains no self-pairs.
4. How to Check Irreflexivity
Check each element of A to see if any self-pair (a,a) appears in R. If even one self-pair is present, the relation is not irreflexive.
4.1. Illustration
For A = \{a,b\}:
R = \{(a,b),(b,a)\}
This relation is irreflexive because neither (a,a) nor (b,b) appears.
5. Non-Examples (Not Irreflexive)
Any relation containing at least one self-pair fails to be irreflexive.
5.1. Example
R = \{(1,1),(2,3)\}
Because (1,1) is present, the relation is not irreflexive.
6. Where Irreflexive Relations Appear
Irreflexive relations are common when elements are being compared in a way that excludes equality, such as “less than” ( < ) or “greater than” ( > ). These comparisons never relate a number to itself.