# Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.

In mathematical notation, this is:

$\forall a,b\in X,\ aRb\Rightarrow \;bRa.$ ## Examples

### In mathematics ### Outside mathematics

• "is married to" (in most legal systems)
• "is a fully biological sibling of"
• "is a homophone of"

## Relationship to asymmetric and antisymmetric relations

By definition, a relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").

Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actually independent of each other, as these examples show.

 Symmetric Not symmetric Antisymmetric equality "is less than or equal to" Not antisymmetric congruence in modular arithmetic "is divisible by", over the set of integers
 Symmetric Not symmetric Antisymmetric "is the same person as, and is married" "is the plural of" Not antisymmetric "is a full biological sibling of" "preys on"