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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
* Do ...
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 (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
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
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
on the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
then the restriction of
to
can be represented by its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
where the pairs
represent
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s 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 Map (mathematics), mapping V \to W between two vect ...
(respectively, a
continuous map
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 value ...
, 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
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
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 value ...
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
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
.
Selection operators
In
relational algebra
In database theory, relational algebra is a theory that uses algebraic structures with a well-founded semantics for modeling data, and defining queries on it. The theory was introduced by Edgar F. Codd.
The main application of relational algebra ...
, a
selection
Selection may refer to:
Science
* Selection (biology), also called natural selection, selection in evolution
** Sex selection, in genetics
** Mate selection, in mating
** Sexual selection in humans, in human sexuality
** Human mating strategie ...
(sometimes called a restriction to avoid confusion with
SQL's use of SELECT) is a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on ...
written as
\sigma_(R) or
\sigma_(R) where:
*
a and
b are attribute names,
*
\theta is a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
in the set
\,
*
v is a value constant,
*
R is a
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
.
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 language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, one assigns an object
F(U) in a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
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 suf ...
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 points ...
, and requires that the objects satisfy certain conditions. The most important condition is that there are ''restriction
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s'' 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
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
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 (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
s understood as relations, such as ones of the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
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