Symmetric Relation

Definition

A relation is symmetric when every related pair also appears in reverse.

If:

(a, b)

is in the relation, then:

(b, a)

must also be in the relation.

Symmetric relation = every connection works both ways.

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Simple Example

The relation:

is sibling of

is symmetric.

If:

Ada is sibling of Ben

then:

Ben is sibling of Ada

The relationship works in both directions.

Symmetry captures mutual connection.

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Non-Example

The relation:

is parent of

is not symmetric.

If:

Ada is parent of Ben

it does not follow that:

Ben is parent of Ada

A one-way relationship is not symmetric.

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Symmetry In Ordered Pairs

Example:

R = {(1, 2), (2, 1), (2, 3), (3, 2)}

This relation is symmetric for the pairs shown.

Each pair has its reverse.

But:

R = {(1, 2), (2, 3)}

is not symmetric, because the reverse pairs are missing.

Symmetry requires reverse pairs.

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Final Summary

• A symmetric relation works in both directions.
• If (a, b) is present, (b, a) must also be present.
• "Is sibling of" is symmetric.
• "Is parent of" is not symmetric.
• Symmetry is about reverse ordered pairs.