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