Restriction (mathematics)

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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.)

[edit] Examples

  1. 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.
  2. The canonical injection of a set A into a superset E of A.
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