restriction of a function

Definition

Let $f\colon X\to Y$ 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

	$\displaystyle f\|_{A}\colon A$	$\displaystyle\to$	$\displaystyle Y$
	$\displaystyle x$	$\displaystyle\mapsto$	$\displaystyle f(x).$

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

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

$\displaystyle(f|_{A})^{-1}(B)$

$\displaystyle=$

$\displaystyle A\cap f^{-1}(B).$