1. What a Reflexive Relation Means
A reflexive relation ensures that every element of the set is related to itself. This means each element must appear in a self-pair.
If the set is A, then every element a in A must have the pair (a,a) in the relation.
2. Formal Condition
The condition for a relation R on a set A to be reflexive is:
2.1. Definition
(a,a) \in R \quad \text{for every } a \in A
3. Examples of Reflexive Relations
Here are some relations that satisfy the reflexive condition:
3.1. Example 1
Let A = \{1,2,3\}. A reflexive relation must include:
(1,1),(2,2),(3,3)
One possible reflexive relation is:
R = \{(1,1),(2,2),(3,3)\}
3.2. Example 2
Another reflexive relation on the same set can have extra pairs:
R = \{(1,1),(2,2),(3,3),(1,2),(2,3)\}
Extra pairs do not affect reflexivity as long as all self-pairs are present.
4. How to Check Reflexivity
To check whether a relation is reflexive:
- Look at the set A.
- List all required self-pairs (a,a).
- Check that each self-pair is present in R.
4.1. Illustration
For A = \{x,y,z\}:
Required pairs: (x,x), (y,y), (z,z)
If even one of these is missing, the relation is not reflexive.
5. Non-Examples (Not Reflexive)
A relation fails to be reflexive if it misses even one self-pair.
5.1. Example
R = \{(1,1),(2,2)\}
On A = \{1,2,3\}, the pair (3,3) is missing. So this relation is not reflexive.
6. Why Reflexive Relations Matter
Reflexive relations help identify structures where every element naturally connects to itself. This idea appears in equality, similarity, and many logical comparisons.