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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

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

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

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 $f: \mathbb R\to\mathbb R; x\mapsto x^2$ to $R_+=[0,\infty)$ is the injection $f: \mathbb R_+\to\mathbb R; x\mapsto x^2$ .
The canonical injection of a set A into a superset E of A.

Last updated: 05-29-2005 04:32:00

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10-26-2009 08:16:03
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