Editing Binary relation (section)

=== Complement ===
{{main|Complementary relation}}
If <math>R</math> is a binary relation over sets <math>X</math> and <math>Y</math> then <math>\bar{R} = \{ (x, y) \mid \neg xRy \}</math> (also denoted by <math>\neg R</math>) is the {{em|complementary relation}} of <math>R</math> over <math>X</math> and <math>Y</math>.

For example, <math>=</math> and <math>\neq</math> are each other's complement, as are <math>\subseteq</math> and <math>\not \subseteq</math>, <math>\supseteq</math> and <math>\not \supseteq</math>, <math>\in</math> and <math>\not \in</math>, and for [[total order]]s also <math><</math> and <math>\geq</math>, and <math>></math> and <math>\leq</math>.

The complement of the [[converse relation]] <math>R^\textsf{T}</math> is the converse of the complement: <math>\overline{R^\mathsf{T}} = \bar{R}^\mathsf{T}.</math>

If <math>X = Y,</math> the complement has the following properties:
* If a relation is symmetric, then so is the complement.
* The complement of a reflexive relation is irreflexive—and vice versa.
* The complement of a [[strict weak order]] is a total preorder—and vice versa.