Contents

Restriction (mathematics)

Function
x ↦ f (x)
History of the function concept
Examples of domains and codomains
${\displaystyle X}$ → ${\displaystyle \mathbb {B} }$, ${\displaystyle \mathbb {B} }$ → ${\displaystyle X}$, ${\displaystyle \mathbb {B} ^{n}}$ → ${\displaystyle X}$ ${\displaystyle X}$ → ${\displaystyle \mathbb {Z} }$, ${\displaystyle \mathbb {Z} }$ → ${\displaystyle X}$ ${\displaystyle X}$ → ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {R} }$ → ${\displaystyle X}$, ${\displaystyle \mathbb {R} ^{n}}$ → ${\displaystyle X}$ ${\displaystyle X}$ → ${\displaystyle \mathbb {C} }$, ${\displaystyle \mathbb {C} }$ → ${\displaystyle X}$, ${\displaystyle \mathbb {C} ^{n}}$ → ${\displaystyle X}$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
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Inmathematics, the restriction of a function ${\displaystyle f}$ is a new function, denoted ${\displaystyle f\vert _{A}}$or${\displaystyle f{\upharpoonright _{A}},}$ obtained by choosing a smaller domain ${\displaystyle A}$ for the original function ${\displaystyle f.}$ The function ${\displaystyle f}$ is then said to extend ${\displaystyle f\vert _{A}.}$

Formal definition[edit]

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

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

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

Extensions[edit]

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

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

Examples[edit]

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

Properties of restrictions[edit]

• Restricting a function ${\displaystyle f:X\rightarrow Y}$ 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 ${\displaystyle A\subseteq B\subseteq \operatorname {dom} f,}$ 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 ${\displaystyle X.}$^[2]
• The restriction of a continuous function is continuous.^[3][4]

Applications[edit]

Inverse functions[edit]

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 inverseof${\displaystyle f}$ 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 ${\displaystyle \mathbb {R} _{\geq 0}=[0,\infty ),}$ in which case $${\displaystyle f^{-1}($$y)={\sqrt {y}}.}

(If we instead restrict to the domain ${\displaystyle (-\infty ,0],}$ 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[edit]

Inrelational 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 ${\displaystyle \{<,\leq ,=,\neq ,\geq ,>\},}$
• ${\displaystyle v}$ is a value constant,
• ${\displaystyle R}$ is a relation.

The selection ${\displaystyle \sigma _{a\theta b}($R)} selects all those tuplesin${\displaystyle R}$ 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[edit]

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[edit]

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

Insheaf 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 ${\displaystyle \operatorname {res} _{V,U}:F($U)\to F(V)} satisfying the following properties, which are designed to mimic the restriction of a function:

• For every open set ${\displaystyle U}$of${\displaystyle X,}$ the restriction morphism ${\displaystyle \operatorname {res} _{U,U}:F($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 ${\displaystyle \operatorname {res} _{W,V}\circ \operatorname {res} _{V,U}=\operatorname {res} _{W,U}.}$
• (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 ${\displaystyle s{\big \vert }_{U_{i}}=t{\big \vert }_{U_{i}}}$ 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: ${\displaystyle s_{i}{\big \vert }_{U_{i}\cap U_{j}}=s_{j}{\big \vert }_{U_{i}\cap U_{j}},}$ then there is a section ${\displaystyle s\in F($U)} such that ${\displaystyle s{\big \vert }_{U_{i}}=s_{i}}$ 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[edit]

More generally, the restriction (ordomain restrictionorleft-restriction) ${\displaystyle A\triangleleft R}$ 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 ${\displaystyle G(A\triangleleft R)=\{(x,y)\in F($R):x\in A\}.} Similarly, one can define a right-restrictionorrange restriction ${\displaystyle R\triangleright B.}$ 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.^{[clarification needed]}

Anti-restriction[edit]

The domain anti-restriction (ordomain 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 ${\displaystyle (E\setminus A)\triangleleft R}$; it removes all elements of ${\displaystyle A}$ from the domain ${\displaystyle E.}$ It is sometimes denoted ${\displaystyle A}$ ⩤ ${\displaystyle R.}$^[5] Similarly, the range anti-restriction (orrange subtraction) of a function or binary relation ${\displaystyle R}$ by a set ${\displaystyle B}$ is defined as ${\displaystyle R\triangleright (F\setminus B)}$; it removes all elements of ${\displaystyle B}$ from the codomain ${\displaystyle F.}$ It is sometimes denoted ${\displaystyle R}$ ⩥ ${\displaystyle B.}$

See also[edit]

• 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)

References[edit]