Restriction (mathematics)

Restriction (mathematics)
In mathematics, the restriction of a function is a new function, denoted or obtained by choosing a smaller domain for the original function
The function is then said to extend
Formal definition
Let be a function from a set to a set If a set is a subset 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 (respectively, a continuous map, etc.).
Examples
The restriction of the non-injective function to the domain is the injection
The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one:
Properties of restrictions
Restricting a function to its entire domain gives back the original function, that is,
Restricting a function twice is the same as restricting it once, that is, if then
The restriction of the identity function on a set to a subset of is just the inclusion map from into
The restriction of a continuous function is continuous.
Applications
Inverse functions
For a function to have an inverse, it must be one-to-one. If a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function
defined on the whole of is not one-to-one since for any However, the function becomes one-to-one if we restrict to the domain in which case
(If we instead restrict to the domain then the inverse is the negative of the square root of ) 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
or where:
and are attribute names,
is a binary operation in the set
is a value constant,
is a relation.
The selection selects all those tuples in for which holds between the and the attribute.
The selection selects all those tuples in for which holds between the attribute and the value
Thus, the selection operator restricts to a subset of the entire database.
The 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 be two closed subsets (or two open subsets) of a topological space such that and let also be a topological space. If is continuous when restricted to both and then 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 in a category to each open set 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 then there is a morphism satisfying the following properties, which are designed to mimic the restriction of a function:
For every open set of the restriction morphism is the identity morphism on
If we have three open sets then the composite
(Locality) If is an open covering of an open set and if are such that for each set of the covering, then ; and
(Gluing) If is an open covering of an open set and if for each a section is given such that for each pair of the covering sets the restrictions of and agree on the overlaps: then there is a section such that for each
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) of a binary relation between and may be defined as a relation having domain codomain and graph Similarly, one can define a right-restriction or range restriction Indeed, one could define a restriction to -ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product 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 (with domain and codomain ) by a set may be defined as ; it removes all elements of from the domain It is sometimes denoted ⩤ Similarly, the range anti-restriction (or range subtraction) of a function or binary relation by a set is defined as ; it removes all elements of from the codomain It is sometimes denoted ⩥
See also
References
Sheaf theory