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.

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