Restriction (mathematics)

Examples

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

Properties of restrictions

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

Applications

Inverse functions

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

(If we instead restrict to the domain ${\displaystyle (-\mathrm{\infty },0],}$ then the inverse is the negative of the square root of ${\displaystyle y.}$) 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 ${\displaystyle {\sigma }_{a\theta b}(R)}$ or ${\displaystyle {\sigma }_{a\theta v}(R)}$ where:

• ${\displaystyle a}$ and ${\displaystyle b}$ are attribute names,
• ${\displaystyle \theta }$ is a binary operation in the set ${\displaystyle \{<,\le ,=,\ne ,\ge ,>\},}$
• ${\displaystyle v}$ is a value constant,
• ${\displaystyle R}$ is a relation.

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

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

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 ${\displaystyle X,Y}$ be two closed subsets (or two open subsets) of a topological space ${\displaystyle A}$ such that ${\displaystyle A=X\cup Y,}$ and let ${\displaystyle B}$ also be a topological space. If ${\displaystyle f:A\to B}$ is continuous when restricted to both ${\displaystyle X}$ and ${\displaystyle Y,}$ then ${\displaystyle f}$ 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 ${\displaystyle F(U)}$ in a category to each open set ${\displaystyle U}$ 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 ${\displaystyle V\subseteq U,}$ then there is a morphism ${\displaystyle {\mathrm{res}}_{V,U}:F(U)\to F(V)}$ satisfying the following properties, which are designed to mimic the restriction of a function:

• For every open set ${\displaystyle U}$ of ${\displaystyle X,}$ the restriction morphism ${\displaystyle {\mathrm{res}}_{U,U}:F(U)\to F(U)}$ is the identity morphism on ${\displaystyle F(U).}$
• If we have three open sets ${\displaystyle W\subseteq V\subseteq U,}$ then the composite ${\displaystyle {\mathrm{res}}_{W,V}\circ {\mathrm{res}}_{V,U}={\mathrm{res}}_{W,U}.}$
• (Locality) If ${\displaystyle \left(U_i\right)}$ is an open covering of an open set ${\displaystyle U,}$ and if ${\displaystyle s,t\in F(U)}$ are such that ${\displaystyle s{|}_{U_i}=t{|}_{U_i}}$ for each set ${\displaystyle U_i}$ of the covering, then ${\displaystyle s=t}$; and
• (Gluing) If ${\displaystyle \left(U_i\right)}$ is an open covering of an open set ${\displaystyle U,}$ and if for each ${\displaystyle i}$ a section ${\displaystyle x_i\in F\left(U_i\right)}$ is given such that for each pair ${\displaystyle U_i,U_j}$ of the covering sets the restrictions of ${\displaystyle s_i}$ and ${\displaystyle s_j}$ agree on the overlaps: ${\displaystyle s_i{|}_{U_i\cap U_j}=s_j{|}_{U_i\cap U_j},}$ then there is a section ${\displaystyle s\in F(U)}$ such that ${\displaystyle s{|}_{U_i}=s_i}$ for each ${\displaystyle i.}$

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) ${\displaystyle A◃R}$ of a binary relation ${\displaystyle R}$ between ${\displaystyle E}$ and ${\displaystyle F}$ may be defined as a relation having domain ${\displaystyle A,}$ codomain ${\displaystyle F}$ and graph ${\displaystyle G(A◃R)=\{(x,y)\in F(R):x\in A\}.}$ Similarly, one can define a right-restriction or range restriction ${\displaystyle R▹B.}$ Indeed, one could define a restriction to ${\displaystyle n}$-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product ${\displaystyle E\times F}$ 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 ${\displaystyle R}$ (with domain ${\displaystyle E}$ and codomain ${\displaystyle F}$) by a set ${\displaystyle A}$ may be defined as ${\displaystyle (E\setminus A)◃R}$; it removes all elements of ${\displaystyle A}$ from the domain ${\displaystyle E.}$ It is sometimes denoted ${\displaystyle A}$ ⩤ ${\displaystyle R.}$ Similarly, the range anti-restriction (or range subtraction) of a function or binary relation ${\displaystyle R}$ by a set ${\displaystyle B}$ is defined as ${\displaystyle R▹(F\setminus B)}$; it removes all elements of ${\displaystyle B}$ from the codomain ${\displaystyle F.}$ It is sometimes denoted ${\displaystyle R}$ ⩥ ${\displaystyle B.}$