Restriction (mathematics)

For other uses, see Restriction (disambiguation).

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

The function $x^{2}$ with domain $\mathbb {R}$ does not have an inverse function. If we restrict $x^{2}$ to the non-negative real numbers, then it does have an inverse function, known as the square root of $x.$

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 extension 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 (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.

Applications

Inverse functions

Main article: Inverse function

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 $\mathbb {R}$ is not one-to-one since $x^{2}=(-x)^{2}$ for any $x\in \mathbb {R} .$ However, the function becomes one-to-one if we restrict to the domain $\mathbb {R} _{\geq 0}=[0,\infty ),$ 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

Main article: Selection (relational algebra)

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

Main article: 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

Main article: Sheaf theory

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 $R$ ⩥ $B.$

See also

• Constraint – Condition of an optimization problem which the solution must satisfy
• Deformation retract – Continuous, position-preserving mapping from a topological space into a subspacePages displaying short descriptions of redirect targets
• Local property – property which occurs on sufficiently small or arbitrarily small neighborhoods of pointsPages displaying wikidata descriptions as a fallback
• Function (mathematics) § Restriction and extension
• Binary relation § Restriction
• Relational algebra § Selection (σ)