Restriction (mathematics)

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

Let ${\displaystyle f:E\to F}${\displaystyle f:E\to F} be a function from a set ${\displaystyle E}${\displaystyle E} to a set ${\displaystyle F.}${\displaystyle F.} If a set ${\displaystyle A}${\displaystyle A} is a subset of ${\displaystyle E,}${\displaystyle E,} then the restriction of ${\displaystyle f}${\displaystyle f} to ${\displaystyle A}${\displaystyle A} is the function^[1] $${\displaystyle {f|}_{A}:A\to F}$${\displaystyle {f|}_{A}:A\to F} given by ${\displaystyle {f|}_{A}(x)=f(x)}${\displaystyle {f|}_{A}(x)=f(x)} for ${\displaystyle x\in A.}${\displaystyle x\in A.} Informally, the restriction of ${\displaystyle f}${\displaystyle f} to ${\displaystyle A}${\displaystyle A} is the same function as ${\displaystyle f,}${\displaystyle f,} but is only defined on ${\displaystyle A}${\displaystyle A}.

If the function ${\displaystyle f}${\displaystyle f} is thought of as a relation ${\displaystyle (x,f(x))}${\displaystyle (x,f(x))} on the Cartesian product ${\displaystyle E\times F,}${\displaystyle E\times F,} then the restriction of ${\displaystyle f}${\displaystyle f} to ${\displaystyle A}${\displaystyle A} can be represented by its graph,

${\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f):x\in A\}=G(f)\cap (A\times F),}${\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f):x\in A\}=G(f)\cap (A\times F),}

where the pairs ${\displaystyle (x,f(x))}${\displaystyle (x,f(x))} represent ordered pairs in the graph ${\displaystyle G.}${\displaystyle G.}

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

A linear extension (respectively, continuous extension , etc.) of a function ${\displaystyle f}${\displaystyle f} is an extension of ${\displaystyle f}${\displaystyle f} that is also a linear map (respectively, a continuous map, etc.).

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

• Restricting a function ${\displaystyle f:X\rightarrow Y}${\displaystyle f:X\rightarrow Y} to its entire domain ${\displaystyle X}${\displaystyle X} gives back the original function, that is, ${\displaystyle f|_{X}=f.}${\displaystyle f|_{X}=f.}
• Restricting a function twice is the same as restricting it once, that is, if ${\displaystyle A\subseteq B\subseteq \operatorname {dom} f,}${\displaystyle A\subseteq B\subseteq \operatorname {dom} f,} then ${\displaystyle \left(f|_{B}\right)|_{A}=f|_{A}.}${\displaystyle \left(f|_{B}\right)|_{A}=f|_{A}.}
• The restriction of the identity function on a set ${\displaystyle X}${\displaystyle X} to a subset ${\displaystyle A}${\displaystyle A} of ${\displaystyle X}${\displaystyle X} is just the inclusion map from ${\displaystyle A}${\displaystyle A} into ${\displaystyle X.}${\displaystyle X.}^[2]
• The restriction of a continuous function is continuous.^{[3] [4]}

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

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

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as ${\displaystyle \sigma _{a\theta b}(R)}${\displaystyle \sigma _{a\theta b}(R)} or ${\displaystyle \sigma _{a\theta v}(R)}${\displaystyle \sigma _{a\theta v}(R)} where:

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

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

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

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

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

Let ${\displaystyle X,Y}${\displaystyle X,Y} be two closed subsets (or two open subsets) of a topological space ${\displaystyle A}${\displaystyle A} such that ${\displaystyle A=X\cup Y,}${\displaystyle A=X\cup Y,} and let ${\displaystyle B}${\displaystyle B} also be a topological space. If ${\displaystyle f:A\to B}${\displaystyle f:A\to B} is continuous when restricted to both ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y,}${\displaystyle Y,} then ${\displaystyle f}${\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 provide a way of generalizing restrictions to objects besides functions.

In sheaf theory, one assigns an object ${\displaystyle F(U)}${\displaystyle F(U)} in a category to each open set ${\displaystyle U}${\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 ${\displaystyle V\subseteq U,}${\displaystyle V\subseteq U,} then there is a morphism ${\displaystyle \operatorname {res} _{V,U}:F(U)\to F(V)}${\displaystyle \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 ${\displaystyle U}${\displaystyle U} of ${\displaystyle X,}${\displaystyle X,} the restriction morphism ${\displaystyle \operatorname {res} _{U,U}:F(U)\to F(U)}${\displaystyle \operatorname {res} _{U,U}:F(U)\to F(U)} is the identity morphism on ${\displaystyle F(U).}${\displaystyle F(U).}
• If we have three open sets ${\displaystyle W\subseteq V\subseteq U,}${\displaystyle W\subseteq V\subseteq U,} then the composite ${\displaystyle \operatorname {res} _{W,V}\circ \operatorname {res} _{V,U}=\operatorname {res} _{W,U}.}${\displaystyle \operatorname {res} _{W,V}\circ \operatorname {res} _{V,U}=\operatorname {res} _{W,U}.}
• (Locality) If ${\displaystyle \left(U_{i}\right)}${\displaystyle \left(U_{i}\right)} is an open covering of an open set ${\displaystyle U,}${\displaystyle U,} and if ${\displaystyle s,t\in F(U)}${\displaystyle s,t\in F(U)} are such that ${\displaystyle s{\big \vert }_{U_{i}}=t{\big \vert }_{U_{i}}}${\displaystyle s{\big \vert }_{U_{i}}=t{\big \vert }_{U_{i}}} for each set ${\displaystyle U_{i}}${\displaystyle U_{i}} of the covering, then ${\displaystyle s=t}${\displaystyle s=t}; and
• (Gluing) If ${\displaystyle \left(U_{i}\right)}${\displaystyle \left(U_{i}\right)} is an open covering of an open set ${\displaystyle U,}${\displaystyle U,} and if for each ${\displaystyle i}${\displaystyle i} a section ${\displaystyle x_{i}\in F\left(U_{i}\right)}${\displaystyle x_{i}\in F\left(U_{i}\right)} is given such that for each pair ${\displaystyle U_{i},U_{j}}${\displaystyle U_{i},U_{j}} of the covering sets the restrictions of ${\displaystyle s_{i}}${\displaystyle s_{i}} and ${\displaystyle s_{j}}${\displaystyle s_{j}} agree on the overlaps: ${\displaystyle s_{i}{\big \vert }_{U_{i}\cap U_{j}}=s_{j}{\big \vert }_{U_{i}\cap U_{j}},}${\displaystyle s_{i}{\big \vert }_{U_{i}\cap U_{j}}=s_{j}{\big \vert }_{U_{i}\cap U_{j}},} then there is a section ${\displaystyle s\in F(U)}${\displaystyle s\in F(U)} such that ${\displaystyle s{\big \vert }_{U_{i}}=s_{i}}${\displaystyle s{\big \vert }_{U_{i}}=s_{i}} for each ${\displaystyle i.}${\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.

More generally, the restriction (or domain restriction or left-restriction) ${\displaystyle A\triangleleft R}${\displaystyle A\triangleleft R} of a binary relation ${\displaystyle R}${\displaystyle R} between ${\displaystyle E}${\displaystyle E} and ${\displaystyle F}${\displaystyle F} may be defined as a relation having domain ${\displaystyle A,}${\displaystyle A,} codomain ${\displaystyle F}${\displaystyle F} and graph ${\displaystyle G(A\triangleleft R)=\{(x,y)\in F(R):x\in A\}.}${\displaystyle G(A\triangleleft R)=\{(x,y)\in F(R):x\in A\}.} Similarly, one can define a right-restriction or range restriction ${\displaystyle R\triangleright B.}${\displaystyle R\triangleright B.} Indeed, one could define a restriction to ${\displaystyle n}${\displaystyle n}-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product ${\displaystyle E\times F}${\displaystyle E\times 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 ${\displaystyle R}${\displaystyle R} (with domain ${\displaystyle E}${\displaystyle E} and codomain ${\displaystyle F}${\displaystyle F}) by a set ${\displaystyle A}${\displaystyle A} may be defined as ${\displaystyle (E\setminus A)\triangleleft R}${\displaystyle (E\setminus A)\triangleleft R}; it removes all elements of ${\displaystyle A}${\displaystyle A} from the domain ${\displaystyle E.}${\displaystyle E.} It is sometimes denoted ${\displaystyle A}${\displaystyle A} ⩤ ${\displaystyle R.}${\displaystyle R.}^[5] Similarly, the range anti-restriction (or range subtraction) of a function or binary relation ${\displaystyle R}${\displaystyle R} by a set ${\displaystyle B}${\displaystyle B} is defined as ${\displaystyle R\triangleright (F\setminus B)}${\displaystyle R\triangleright (F\setminus B)}; it removes all elements of ${\displaystyle B}${\displaystyle B} from the codomain ${\displaystyle F.}${\displaystyle F.} It is sometimes denoted ${\displaystyle R}${\displaystyle R} ⩥ ${\displaystyle B.}${\displaystyle B.}

• Constraint – Condition of an optimization problem which the solution must satisfy
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• Binary relation § Restriction
• Relational algebra § Selection (σ)

• Sheaf theory

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