structural:summarization
Complement

If R {\displaystyle R} is a binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} then R ¯ = { (x, y) ∣ ¬ x R y } {\displaystyle {\bar {R}}=\{(x,y)\mid \neg xRy\}} (also denoted by ¬ R {\displaystyle \neg R}) is the complementary relation of R {\displaystyle R} over X {\displaystyle X} and Y {\displaystyle Y}.

For example, = {\displaystyle =} and ≠ {\displaystyle \neq } are each other's complement, as are ⊆ {\displaystyle \subseteq } and ⊈ {\displaystyle \not \subseteq }, ⊇ {\displaystyle \supseteq } and ⊉ {\displaystyle \not \supseteq }, ∈ {\displaystyle \in } and ∉ {\displaystyle \not \in }, and for total orders also < {\displaystyle <} and ≥ {\displaystyle \geq }, and > {\displaystyle >} and ≤ {\displaystyle \leq }.

The complement of the converse relation R T {\displaystyle R^{\textsf {T}}} is the converse of the complement: R T ¯ = R ¯ T. {\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}

If X = Y, {\displaystyle X=Y,} the complement has the following properties: