Editing Symmetric relation (section)

== Relationship to asymmetric and antisymmetric relations ==

[[File:Symmetric-and-or-antisymmetric.svg|thumb|Symmetric and antisymmetric relations|340x340px]]

By definition, a nonempty relation cannot be both symmetric and [[asymmetric relation|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 relation|antisymmetric]] (where the only way ''a'' can be related to ''b'' and ''b'' be related to ''a'' is if {{nowrap|1=''a'' = ''b''}}) are actually independent of each other, as these examples show.

{| class="wikitable"
|+Mathematical examples
|-
| || '''Symmetric''' || '''Not symmetric'''
|-
| '''Antisymmetric''' || [[equality (mathematics)|equality]] || [[Divisor|divides]], less than or equal to
|-
| '''Not antisymmetric''' || [[congruence relation|congruence]] in [[modular arithmetic]] || [[integer division|//]] (integer division), most nontrivial [[Permutation|permutations]]
|}

{| class="wikitable"
|+Non-mathematical examples
|-
| || '''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
|}