Restriction (mathematics)

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".


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