Reflexive Relation

Definition

A relation is reflexive when every element is related to itself.

For a set:

A = {1, 2, 3}

a reflexive relation must include:

(1, 1)
(2, 2)
(3, 3)

Reflexive relation = every element has a self-pair.

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Simple Example

The relation:

less than or equal to

is reflexive over numbers because every number is less than or equal to itself.

Example:

1 <= 1
2 <= 2
3 <= 3

Reflexivity captures self-connection.

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Non-Example

The relation:

less than

is not reflexive.

Why?

Because:

1 < 1

is false.

No number is less than itself.

A relation is not reflexive if even one element lacks a self-pair.

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Reflexive On A Set

Reflexivity is always relative to a set.

Example:

A = {1, 2}
R = {(1, 1), (2, 2)}

R is reflexive on A.

But if the set is:

B = {1, 2, 3}

then R is not reflexive on B, because (3, 3) is missing.

Reflexivity depends on the set being considered.

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Final Summary

• A reflexive relation relates every element to itself.
• Reflexivity requires every self-pair.
• "Less than or equal to" is reflexive over numbers.
• "Less than" is not reflexive.
• Reflexivity depends on the set being considered.