Restriction (mathematics)

In mathematics, the notion of restriction finds a general definition in the context of sheaves.

Often, the following definition will be sufficient:

If "f: E -> F" is a (partial) function from "E" to "F", and "A" is a subset of "E", then the restriction of "f" to "A" is the (partial) function:$\{f\_A\; :\; A\; o\; F$ having the graph $G(\{f\_A)\; =\; \{\; (x,y)in\; G(f)\; mid\; xin\; A\; \}$.

(In rough words, it is "the same function", but only defined on $Acap\; D(f)$.)

More generally, the restriction of a binary relation is usually defined in the same way. (One could also define a restriction to a subset of "E" x "F", and the same applies to "n"-ary relations.These cases do not fit into the scheme of sheaves.)

Examples

# The restriction of the non injective function $f:\; mathbb\; R\; omathbb\; R;\; xmapsto\; x^2$ to $R\_+=\; [0,infty)$ is the injection $f:\; mathbb\; R\_+\; omathbb\; R;\; xmapsto\; x^2$.
# The canonical injection of a set "A" into a superset "E" of "A".