Restriction (mathematics)

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 subset of ${\displaystyle E,}$  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 ${\displaystyle G.}$ 

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 ${\displaystyle F{\big \vert }_{\operatorname {domain} f}=f.}$ 

A linear 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.).

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: ${\displaystyle {\Gamma |}_{\mathbb {Z} ^{+}}\!(n)=(n-1)!}$ 

• 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]

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 ${\displaystyle \mathbb {R} _{\geq 0}=[0,\infty ),}$  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.

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

The selection ${\displaystyle \sigma _{a\theta b}(R)}$  selects all those 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 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 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 ${\displaystyle \operatorname {res} _{V,U}:F(U)\to F(V)}$  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.

More generally, the restriction (or domain restriction or left-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-restriction or range 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]}

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 ${\displaystyle (E\setminus A)\triangleleft R}$ ; 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 (or range 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 ${\displaystyle R}$  ⩥ ${\displaystyle B.}$ 

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• Function (mathematics) § Restriction and extension
• Binary relation § Restriction
• Relational algebra § Selection (σ)