Restriction (mathematics)

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For other uses, see Restriction (disambiguation).

In mathematics, the notion of restriction of a function is defined as follows:

If f : EF is a function from E to F, and A is a subset of E, then the restriction of f to A is the function

 {f|}_A \colon 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 \mathrm{dom} \, f.)

More generally, the restriction (or domain restriction or left-restriction) A ◁ R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G(AR) = {(x, y) ∈ G(R) | xA} . Similarly, one can define a right-restriction or range restriction RB. Indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of E×F for binary relations. These cases do not fit into the scheme of sheaves.[clarification needed]

The domain anti-restriction (or domain subtraction) of a function or binary relation R (with domain E and codomain F) by a set A may be defined as (E \ A) ◁ R; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R. Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as R ▷ (F \ B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.


  1. The restriction of the non-injective function  f: \mathbb R\to\mathbb R; x\mapsto x^2 to  \mathbb R_+=[0,\infty) is the injection  f: \mathbb R_+\to\mathbb R; x\mapsto x^2.
  2. The inclusion map of a set A into a superset E of A is the restriction of the identity function on E to A.

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