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 functiongiven 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 andThat is, if and
A Linear extension of a function| of a function is an extension of that is also a linear map.
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 functiondefined 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
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 is a unary operation written asor 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 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 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 subsets of a topological space and create a new one.
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
- If is an open covering of an open set and if are such that for each set of the covering, then ; and
- 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
Left- and right-restriction
More generally, the 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.