restriction of a function

Definition

Let $f:X\to Y$ be a function from a set $X$ to a set $Y$.If $A$ is a subset of $X$,then the restriction of $f$ to $A$ is the function

${f|}_A:A$$\to$$Y$

$x$$\mapsto$$f(x).$

Some authors write $f\upharpoonright A$ instead of ${f|}_A$.

Properties

If $A\subseteq X$, and $B\subseteq Y$, then

${({f|}_A)}^{-1}(B)$$=$$A\cap f^{-1}(B).$