Definitions

# 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

$\left\{f|\right\}_A : A to F$ having the graph $G\left(\left\{f|\right\}_A\right) = \left\{ \left(x,y\right)in G\left(f\right) mid xin A \right\}$.

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

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

1. The restriction of the non injective function $f: mathbb Rtomathbb R; xmapsto x^2$ to $R_+=\left[0,infty\right)$ is the injection $f: mathbb R_+tomathbb R; xmapsto x^2$.
2. The canonical injection of a set A into a superset E of A.

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