Restriction (mathematics)

In mathematics, the restriction of a function f is a new function, denoted \( {\displaystyle f\vert _{A}} \) or \( f{\upharpoonright _{A}} \), obtained by choosing a smaller domain A for the original function f.

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[1]

\( {f|}_{A}\colon A\to F \)

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 \( {\displaystyle A\cap \operatorname {dom} f}. \)

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 \( {\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f)\mid x\in A\}=G(f)\cap (A\times F)} \) , where the pairs \( (x,f(x)) \) represent ordered pairs in the graph G.

Examples

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 \( {\displaystyle f:\mathbb {R} _{+}\to \mathbb {R} ,\ x\mapsto x^{2}} \).
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

Restricting a function \( f:X\rightarrow Y \) to its entire domain X gives back the original function, i.e., \( f|_{{X}}=f. \)
Restricting a function twice is the same as restricting it once, i.e. if \( {\displaystyle A\subseteq B\subseteq \operatorname {dom} f} \), then \( (f|_{B})|_{A}=f|_{A}. \)
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.[2]
The restriction of a continuous function is continuous.[3][4]

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 x2 = (−x)2 for any 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

\( f^{-1}(y)={\sqrt {y}}. \)

(If we instead restrict to the domain \( {\displaystyle (-\infty ,0]} \), then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we don't mind the inverse being 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 \( {\displaystyle \{<,\leq ,=,\neq ,\geq ,>\}}, \)
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 {\displaystyle X} X and Y {\displaystyle Y} 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; i.e., if \( V\subseteq U \), then there is a morphism resV,U : F(U) → 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 resU,U : F(U) → F(U) is the identity morphism on F(U).
If we have three open sets W ⊆ V ⊆ U, then the composite resW,V o resV,U = resW,U.
(Locality) If (Ui) is an open covering of an open set U, and if s,t ∈ F(U) are such that s|Ui = t|Ui for each set Ui of the covering, then s = t; and
(Gluing) If (Ui) is an open covering of an open set U, and if for each i a section si ∈ F(Ui) is given such that for each pair Ui,Uj of the covering sets the restrictions of si and sj agree on the overlaps: si|Ui∩Uj = sj|Ui∩Uj, then there is a section s ∈ F(U) such that s|Ui = si 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) ∈ G(R) | x ∈ 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 E × 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 \ A) ◁ R; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R.[5] Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as R ▷ (F \ B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.