Symmetric relation

Y indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed
in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,
is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require the homogeneous relation${\displaystyle R}$ be transitive: for all ${\displaystyle a,b,c,}$ if ${\displaystyle aRb}$ and ${\displaystyle bRc}$ then ${\displaystyle aRc.}$
A term's definition may require additional properties that are not listed in this table.

A symmetric relation is a type of binary relation. A homogeneous relation${\displaystyle R}$ on a set${\displaystyle X}$ is symmetric if:

for all ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle X}$, ${\displaystyle aRb}$ if and only if ${\displaystyle bRa}$,

where the notation ${\displaystyle aRb}$ means that the ordered pair${\displaystyle (a,b)}$ is in the relation ${\displaystyle R}$.

An example is the relation "is equal to", because if a = b is true then b = a is also true. If R represents the converse of R, then R is symmetric if and only if R = R.

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.