Editing Binary relation (section)

== Preorder R\R ==
Every relation <math>R</math> generates a [[preorder]] <math>R \backslash R</math> which is the [[Composition of relations#Quotients|left residual]].<ref>In this context, the symbol <math>\backslash</math> does not mean "[[set difference]]".</ref> In terms of converse and complements, <math>R \backslash R \equiv \overline{R^\textsf{T} \bar{R}}.</math> Forming the diagonal of <math>R^\textsf{T} \bar{R}</math>, the corresponding row of <math>R^{\textsf{T}}</math> and column of <math>\bar{R}</math> will be of opposite logical values, so the diagonal is all zeros. Then 
: <math>R^\textsf{T} \bar{R} \subseteq \bar{I} \implies I \subseteq \overline{R^\textsf{T} \bar{R}} = R \backslash R</math>, so that <math>R \backslash R</math> is a [[reflexive relation]].

To show [[Transitive relation|transitivity]], one requires that <math>(R\backslash R)(R\backslash R) \subseteq R \backslash R.</math> Recall that <math>X = R \backslash R</math> is the largest relation such that <math>R X \subseteq R.</math> Then 
: <math>R(R\backslash R) \subseteq R</math>
: <math>R(R\backslash R) (R\backslash R )\subseteq R</math> (repeat)
: <math>\equiv R^\textsf{T} \bar{R} \subseteq \overline{(R \backslash R)(R \backslash R)}</math> (Schröder's rule)
: <math>\equiv (R \backslash R)(R \backslash R) \subseteq \overline{R^\textsf{T} \bar{R}}</math> (complementation)
: <math>\equiv (R \backslash R)(R \backslash R) \subseteq R \backslash R.</math> (definition)

The [[inclusion (set theory)|inclusion]] relation &Omega; on the [[power set]] of <math>U</math> can be obtained in this way from the [[element (mathematics)|membership relation]] <math>\in</math> on subsets of <math>U</math>:
: <math>\Omega = \overline{\ni \bar{\in}} = \in \backslash \in .</math><ref name=GS11/>{{rp|283}}