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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the restriction of a function f is a new function, denoted f\vert_A or f , obtained by choosing a smaller
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
A for the original function f. The function f is then said to extend f\vert_A.


Formal definition

Let f : E \to F be a function from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
E to a set F. If a set A is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of E, then the restriction of f to A is the function _A : A \to F given by _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 A. 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 where the pairs (x,f(x)) represent ordered pairs in the graph G.


Extensions

A function F is said to be an ' of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname f \subseteq \operatorname F and F\big\vert_ = f. A '' '' (respectively, '' '', etc.) of a function f is an extension of f that is also a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
(respectively, a continuous map, etc.).


Examples

# The restriction of the non-injective functionf: \mathbb \to \mathbb, \ x \mapsto x^2 to the domain \mathbb_ = ,\infty) is the injectionf:\mathbb_+ \to \mathbb, \ x \mapsto x^2. # The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: _\!(n) = (n-1)!


Properties of restrictions

* Restricting a function f:X\rightarrow Y to its entire domain X gives back the original function, that is, f, _X = f. * Restricting a function twice is the same as restricting it once, that is, if A \subseteq B \subseteq \operatorname f, then \left(f, _B\right), _A = f, _A. * The restriction of the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on a set X to a subset A of X is just the inclusion map from A into X. * The restriction of a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
is continuous.


Applications


Inverse functions

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 \R is not one-to-one since x^2 = (-x)^2 for any x \in \R. However, the function becomes one-to-one if we restrict to the domain \R_ = , \infty), in which case f^(y) = \sqrt . (If we instead restrict to the domain (-\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 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 \sigma_(R) or \sigma_(R) where: * a and b are attribute names, * \theta is a binary operation in the set \, * v is a value constant, * R is a relation. The selection \sigma_(R) selects all those
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s in R for which \theta holds between the a and the b attribute. The selection \sigma_(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

The pasting lemma is a result in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 and 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

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 In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
U of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, 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 V\subseteq U, then there is a morphism \operatorname_ : F(U) \to 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 \operatorname_ : F(U) \to F(U) is the identity morphism on F(U). * If we have three open sets W \subseteq V \subseteq U, then the
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
\operatorname_ \circ \operatorname_ = \operatorname_. * (Locality) If \left(U_i\right) is an open covering of an open set U, and if s, t \in F(U) are such that s\big\vert_ = t\big\vert_''s'', ''U''''i'' = ''t'', ''U''''i'' for each set U_i of the covering, then s = t; and * (Gluing) If \left(U_i\right) is an open covering of an open set U, and if for each i a section x_i \in F\left(U_i\right) is given such that for each pair U_i, U_j of the covering sets the restrictions of s_i and s_j agree on the overlaps: s_i\big\vert_ = s_j\big\vert_, then there is a section s \in F(U) such that s\big\vert_ = s_i 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 \triangleleft 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 \triangleleft R) = \. Similarly, one can define a right-restriction or range restriction R \triangleright B. Indeed, one could define a restriction to n-ary relations, as well as to
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s understood as relations, such as ones of the Cartesian product 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 R (with domain E and codomain F) by a set A may be defined as (E \setminus A) \triangleleft R; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R.Dunne, S. and Stoddart, Bill ''Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 5–7, 2006, Revised Selected ... Computer Science and General Issues)''. Springer (2006) Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as R \triangleright (F \setminus B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.


See also

* * * * * *


References

{{DEFAULTSORT:Restriction (Mathematics) Sheaf theory