Restriction (mathematics)

For other uses, see Restriction (disambiguation).

File:Inverse square graph.svg

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 $x.$

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

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

Properties of restrictions

• Restricting a function $f:X\to Y$ 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 $A\subseteq B\subseteq \mathrm{dom}f,$ 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 ${\mathbb{ℝ}}_{\ge 0}=[0,\mathrm{\infty }),$ in which case $$f^{-1}(y)=\sqrt{y}.$$ (If we instead restrict to the domain $(-\mathrm{\infty },0],$ 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 $\{<,\le ,=,\ne ,\ge ,>\},$
• $v$ 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 ${\mathrm{res}}_{V,U}:F(U)\to F(V)$ satisfying the following properties, which are designed to mimic the restriction of a function:

• For every open set $U$ of $X,$ the restriction morphism ${\mathrm{res}}_{U,U}:F(U)\to F(U)$ is the identity morphism on $F(U).$
• If we have three open sets $W\subseteq V\subseteq U,$ then the composite ${\mathrm{res}}_{W,V}\circ {\mathrm{res}}_{V,U}={\mathrm{res}}_{W,U}.$
• (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 $s{|}_{U_i}=t{|}_{U_i}$ 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: $s_i{|}_{U_i\cap U_j}=s_j{|}_{U_i\cap U_j},$ then there is a section $s\in F(U)$ such that $s{|}_{U_i}=s_i$ 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) $A◃R$ of a binary relation $R$ between $E$ and $F$ may be defined as a relation having domain $A,$ codomain $F$ and graph $G(A◃R)=\{(x,y)\in F(R):x\in A\}.$ Similarly, one can define a right-restriction or range restriction $R▹B.$ 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 $(E\setminus A)◃R$; 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 $R▹(F\setminus B)$; 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
• Function (mathematics) § Restriction and extension
• Binary relation § Restriction
• Relational algebra § Selection (σ)