Definition:Restriction/Relation

Definition
Let $\mathcal R$ be a relation on $S \times T$.

Let $X \subseteq S$, $Y \subseteq T$.

The restriction of $\mathcal R$ to $X \times Y$ is the relation on $X \times Y$ defined as:


 * $\mathcal R {\restriction_{X \times Y}}: = \mathcal R \cap \left({X \times Y}\right)$

If $Y = T$, then we simply call this the restriction of $\mathcal R$ to $X$, and denote it as $\mathcal R {\restriction_X}$.

A different way of saying the same thing is:


 * $\mathcal R {\restriction_X} = \left\{{\left({x, y}\right) \in \mathcal R: x \in X}\right\}$

Also known as
Some sources refer to $\mathcal R {\restriction_X}$ as the relation induced on $X$ by $\mathcal R$.

Also see

 * Definition:Extension of Relation


 * Definition:Restriction of Mapping
 * Definition:Restriction of Operation


 * Properties of Restriction of Relation‎

Technical Note
The expression:


 * $\mathcal R {\restriction_{X \times Y}} = \mathcal R \cap \left({X \times Y}\right)$

is produced by the following $\LaTeX$ code:

\mathcal R {\restriction_{X \times Y}} = \mathcal R \cap \left({X \times Y}\right)