Restriction_(mathematics).pdf 2p4l4f

Restriction (mathematics) In mathematics, the restriction of a function f is a new function

obtained by

choosing a smaller domain A for the original function . The notation

is also

used.

Contents 1

Formal definition

2

Examples

3

Properties of restrictions

4

Applications 4.1 Inverse functions 4.2 Selection operators 4.3 The Pasting Lemma 4.4 Sheaves

5

Left- and right-restriction

6

Anti-restriction

7

See also

8

References

The function x2 with domain R does not have an inverse. If we restrict x2 to the non-negative real numbers, then it does have an inverse, known as the square root of x.

Formal definition Let

be a function from a set E to a set F, so that the domain of f is in E (

the restriction of to

is the

). If a set A is a subset of E, then

function[1]

. Informally, the restriction of f to A is the same function asf, but is only defined on If the function f is thought of as a relation

on the Cartesian product

represented by the graph

. , then the restriction of f to A can be , where the pairs

represent

edges in the graph G.

Examples 1. The restriction of the non-injective function

to

is the injection

. 2. The factorial function is the restriction of thegamma function to the integers.

Properties of restrictions Restricting a function

to its entire domain

gives back the original function; i.e.,

Restricting a function twice is the same as restricting it once; i.e. if

, then

. .

The restriction of the identity function on a space X to a subset A of X is just the inclusion map of A into X.[2] The restriction of a continuous function is continuous.[3][4]

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

is not one-to-one, sincex2 = (−x)2. However, the function becomes one-to-one if we restrict to the domainx ≥ 0 , in which case

(If we instead restrict to the domain x ≤ 0 , then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with the inverse being amultivalued 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

and

where:

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 intopology that relates the continuity of a function with the continuity of its restrictions to subsets. Let

be both closed (or both open) subsets of a topological spaceA such that

If

, and let B also be a topological space.

is continuous when restricted to bothX 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

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; i.e., if

, then there is a morphism resV,U : F(U) → 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 resU,U : F(U) → F(U) is the identity morphism onF(U). If we have three open setsW ⊆ V ⊆ U, then the composite resW,V o resV,U = resW,U. (Locality) If (Ui) is an open covering of an open set U, and if s,t ∈ F(U) are such that s|Ui = t|Ui for each set Ui of the covering, then s = t; and (Gluing) If (Ui) is an open covering of an open setU, and if for each i a section si ∈ F(Ui) is given such that for each pair Ui,Uj of the covering sets the restrictions ofsi and sj agree on the overlaps:si|Ui∩Uj = sj|Ui∩Uj, then there is a section s ∈ F(U) such that s|Ui = si for each i. The collection of all such objects is called asheaf. If only the first two properties are satisfied, it is apre-sheaf.

Left- and right-restriction More generally, the restriction (or domain restriction or left-restriction) A ◁ R of a binary relation R between E and F may be defined as a relation having domainA, codomain F and graph G(A ◁ R) = {(x, y) ∈ G(R) | x ∈ A} . Similarly, one can define a right-restriction or range restriction R ▷ B. Indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones ofE × F for binary relations. These cases do not fit into the scheme ofsheaves.

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 \ A) ◁ R; it removes all elements of A from the domain E. It is sometimes denoted A R.[5] Similarly, the range anti-restriction (or range subtraction) of a function or binary relationR by a set B is defined as R ▷ (F \ B); it removes all elements of B from the codomain F. It is sometimes denotedR B.

See also Deformation retract Function (mathematics) #Restrictions and extensions Binary relation #Restriction Relational algebra#Selection (σ)

References 1. Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5. 2. Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 9781-61427-131-4 (Paperback edition). 3. Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000. 4. Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008. 5. Dunne, S. and Stoddart, BillUnifying Theories of Programming: First International Symposium, UTP 2006, W alworth Castle, County Durham, UK, February 5-7, 2006, Revised Selected ... Computer Science and General Issues) . Springer (2006) Retrieved from "https://en.wikipedia.org/w/index.php?title=Restriction_(mathematics)&oldid=786593531 " This page was last edited on 20 June 2017, at 11:51. Text is available under theCreative Commons Attribution-ShareAlike License ; additional may apply. By using this site, you agree to the of Use and . Wikipedia® is a ed trademark of theWikimedia Foundation, Inc., a non-profit organization.