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).