Restriction (mathematics)



In mathematics, the restriction of a function $$f$$ is a new function, denoted $$f\vert_A$$ or $$f {\restriction_A},$$ obtained by choosing a smaller domain $$A$$ for the original function $$f.$$ The function $$f$$ is then said to extend $$f\vert_A.$$

Formal definition
Let $$f : E \to F$$ be a function from a set $$E$$ to a set $$F.$$ If a set $$A$$ is a subset of $$E,$$ then the restriction of $$f$$ to $$A$$ is the function $${f|}_A : A \to F$$ given by $${f|}_A(x) = f(x)$$ for $$x \in A.$$ Informally, the restriction of $$f$$ to $$A$$ is the same function as $$f,$$ but is only defined on $$A$$.

If the function $$f$$ is thought of as a relation $$(x,f(x))$$ on the Cartesian product $$E \times F,$$ then the restriction of $$f$$ to $$A$$ can be represented by its graph,
 * $$G({f|}_A) = \{ (x,f(x))\in G(f) : x\in A \} = G(f)\cap (A\times F),$$

where the pairs $$(x,f(x))$$ represent ordered pairs in the graph $$G.$$

Extensions
A function $$F$$ is said to be an  of another function $$f$$ if whenever $$x$$ is in the domain of $$f$$ then $$x$$ is also in the domain of $$F$$ and $$f(x) = F(x).$$ That is, if $$\operatorname{domain} f \subseteq \operatorname{domain} F$$ and $$F\big\vert_{\operatorname{domain} f} = f.$$

A Linear extension of a function (respectively, Continuous extension, etc.) of a function $$f$$ is an extension of $$f$$ that is also a linear map (respectively, a continuous map, etc.).

Examples

 * 1) The restriction of the non-injective function$$f: \mathbb{R} \to \mathbb{R}, \ x \mapsto x^2$$ to the domain $$\mathbb{R}_{+} = [0,\infty)$$ is the injection$$f:\mathbb{R}_+ \to \mathbb{R}, \ x \mapsto x^2.$$
 * 2) The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: $${\Gamma|}_{\mathbb{Z}^+}\!(n) = (n-1)!$$

Properties of restrictions

 * Restricting a function $$f:X\rightarrow Y$$ to its entire domain $$X$$ gives back the original function, that is, $$f|_X = f.$$
 * Restricting a function twice is the same as restricting it once, that is, if $$A \subseteq B \subseteq \operatorname{dom} f,$$ then $$\left(f|_B\right)|_A = f|_A.$$
 * The restriction of the identity function on a set $$X$$ to a subset $$A$$ of $$X$$ is just the inclusion map from $$A$$ into $$X.$$
 * The restriction of a continuous function is continuous.

Inverse functions
For a function to have an inverse, it must be one-to-one. If a function $$f$$ is not one-to-one, it may be possible to define a partial inverse of $$f$$ by restricting the domain. For example, the function $$f(x) = x^2$$ defined on the whole of $$\R$$ is not one-to-one since $$x^2 = (-x)^2$$ for any $$x \in \R.$$ However, the function becomes one-to-one if we restrict to the domain $$\R_{\geq 0} = [0, \infty),$$ in which case $$f^{-1}(y) = \sqrt{y} .$$

(If we instead restrict to the domain $$(-\infty, 0],$$ then the inverse is the negative of the square root of $$y.$$) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.

Selection operators
In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as $$\sigma_{a \theta b}(R)$$ or $$\sigma_{a \theta v}(R)$$ where:
 * $$a$$ and $$b$$ are attribute names,
 * $$\theta$$ is a binary operation in the set $$\{<, \leq, =, \neq, \geq, >\},$$
 * $$v$$ is a value constant,
 * $$R$$ is a relation.

The selection $$\sigma_{a \theta b}(R)$$ selects all those tuples in $$R$$ for which $$\theta$$ holds between the $$a$$ and the $$b$$ attribute.

The selection $$\sigma_{a \theta v}(R)$$ selects all those tuples in $$R$$ for which $$\theta$$ holds between the $$a$$ attribute and the value $$v.$$

Thus, the selection operator restricts to a subset of the entire database.

The pasting lemma
The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.

Let $$X,Y$$ be two closed subsets (or two open subsets) of a topological space $$A$$ such that $$A = X \cup Y,$$ and let $$B$$ also be a topological space. If $$f: A \to B$$ is continuous when restricted to both $$X$$ and $$Y,$$ then $$f$$ is continuous.

This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.

Sheaves
Sheaves provide a way of generalizing restrictions to objects besides functions.

In sheaf theory, one assigns an object $$F(U)$$ in a category to each open set $$U$$ of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested open sets; that is, if $$V\subseteq U,$$ then there is a morphism $$\operatorname{res}_{V,U} : F(U) \to F(V)$$ satisfying the following properties, which are designed to mimic the restriction of a function:
 * For every open set $$U$$ of $$X,$$ the restriction morphism $$\operatorname{res}_{U,U} : F(U) \to F(U)$$ is the identity morphism on $$F(U).$$
 * If we have three open sets $$W \subseteq V \subseteq U,$$ then the composite $$\operatorname{res}_{W,V} \circ \operatorname{res}_{V,U} = \operatorname{res}_{W,U}.$$
 * (Locality) If $$\left(U_i\right)$$ is an open covering of an open set $$U,$$ and if $$s, t \in F(U)$$ are such that $$s\big\vert_{U_i} = t\big\vert_{U_i}$$ for each set $$U_i$$ of the covering, then $$s = t$$; and
 * (Gluing) If $$\left(U_i\right)$$ is an open covering of an open set $$U,$$ and if for each $$i$$ a section $$x_i \in F\left(U_i\right)$$ is given such that for each pair $$U_i, U_j$$ of the covering sets the restrictions of $$s_i$$ and $$s_j$$ agree on the overlaps: $$s_i\big\vert_{U_i \cap U_j} = s_j\big\vert_{U_i \cap U_j},$$ then there is a section $$s \in F(U)$$ such that $$s\big\vert_{U_i} = s_i$$ for each $$i.$$

The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf.

Left- and right-restriction
More generally, the restriction (or domain restriction or left-restriction) $$A \triangleleft R$$ of a binary relation $$R$$ between $$E$$ and $$F$$ may be defined as a relation having domain $$A,$$ codomain $$F$$ and graph $$G(A \triangleleft R) = \{(x, y) \in F(R) : x \in A\}.$$  Similarly, one can define a right-restriction or range restriction $$R \triangleright B.$$ Indeed, one could define a restriction to $n$-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product $$E \times F$$ for binary relations. These cases do not fit into the scheme of sheaves.

Anti-restriction
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 $$(E \setminus A) \triangleleft R$$; 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 $$R \triangleright (F \setminus B)$$; it removes all elements of $$B$$ from the codomain $$F.$$ It is sometimes denoted $$R$$ ⩥ $$B.$$