In
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 is a new function, denoted
or
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
* ...
for the original function
The function
is then said to extend
Formal definition
Let
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 ...
to a set
If a set
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
then the restriction of
to
is the function
[
]
given by
for
Informally, the restriction of
to
is the same function as
but is only defined on
.
If the function
is thought of as a
relation on the
Cartesian product then the restriction of
to
can be represented by its
graph where the pairs
represent
ordered pairs in the graph
Extensions
A function
is said to be an ' of another function
if whenever
is in the domain of
then
is also in the domain of
and
That is, if
and
A ''
'' (respectively, ''
'', etc.) of a function
is an extension of
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 function
to the domain
is the injection
f:\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_ = in which case
f^(y) = \sqrt .
(If we instead restrict to the domain (-\infty, 0">, \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