Editing Binary relation (section)

=== Restriction ===
{{main|Restriction (mathematics)}}
If <math>R</math> is a binary [[homogeneous relation]] over a set <math>X</math> and <math>S</math> is a subset of <math>X</math> then <math>R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \text{ and } y \in S \}</math> is the {{em|{{visible anchor|restriction relation|Restriction relation|Restriction of a homogeneous relation}}}} of <math>R</math> to <math>S</math> over <math>X</math>.

If <math>R</math> is a binary relation over sets <math>X</math> and <math>Y</math> and if <math>S</math> is a subset of <math>X</math> then <math>R_{\vert S} = \{ (x, y) \mid xRy \text{ and } x \in S \}</math> is the {{em|{{visible anchor|left-restriction relation|Left-restriction relation}}}} of <math>R</math> to <math>S</math> over <math>X</math> and <math>Y</math>.{{clarify|reason=Introduce notational distinction between restriction and left restriction.|date=November 2022}}

If a relation is [[Reflexive relation|reflexive]], irreflexive, [[Symmetric relation|symmetric]], [[Antisymmetric relation|antisymmetric]], [[Asymmetric relation|asymmetric]], [[Transitive relation|transitive]], [[Serial relation|total]], [[Trichotomy (mathematics)|trichotomous]], a [[partial order]], [[total order]], [[strict weak order]], [[Strict weak order#Total preorders|total preorder]] (weak order), or an [[equivalence relation]], then so too are its restrictions.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "<math>x</math> is parent of <math>y</math>" to females yields the relation "<math>x</math> is mother of the woman <math>y</math>"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of [[Completeness (order theory)|completeness]] (not to be confused with being "total") do not carry over to restrictions. For example, over the [[real number]]s a property of the relation <math>\leq</math> is that every [[Empty set|non-empty]] subset <math>S \subseteq \R</math> with an [[upper bound]] in <math>\R</math> has a [[Supremum|least upper bound]] (also called supremum) in <math>\R.</math> However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation <math>\leq</math> to the rational numbers.

<!---This definition is needed by the closure defs, too, but maybe should better given in an earlier section(?):--->
A binary relation <math>R</math> over sets <math>X</math> and <math>Y</math> is said to be {{em|{{visible anchor|contained in|Containment of relations}}}} a relation <math>S</math> over <math>X</math> and <math>Y</math>, written <math>R \subseteq S,</math> if <math>R</math> is a subset of <math>S</math>, that is, for all <math>x \in X</math> and <math>y \in Y,</math> if <math>xRy</math>, then <math>xSy</math>. If <math>R</math> is contained in <math>S</math> and <math>S</math> is contained in <math>R</math>, then <math>R</math> and <math>S</math> are called {{em|equal}} written <math>R = S</math>. If <math>R</math> is contained in <math>S</math> but <math>S</math> is not contained in <math>R</math>, then <math>R</math> is said to be {{em|{{visible anchor|smaller|Smaller relation}}}} than <math>S</math>, written <math>R \subsetneq S.</math> For example, on the [[rational number]]s, the relation <math>></math> is smaller than <math>\geq</math>, and equal to the composition <math>> \circ ></math>.