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Restriction (mathematics)

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.

In mathematics, the restriction of a function is a new function, denoted or , obtained by choosing a smaller domain A for the original function .

Formal definitionEdit

Let   be a function from a set E to a set F. If a set A is a subset of E, then the restriction of   to   is the function[1]

 

given by f|A(x) = f(x) for x in A. Informally, the restriction of f to A is the same function as f, but is only defined on  .

If the function f is thought of as a relation   on the Cartesian product  , then the restriction of f to A can be represented by its graph , where the pairs   represent ordered pairs in the graph G.

ExamplesEdit

  1. The restriction of the non-injective function  to the domain   is the injection .
  2. The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one:  

Properties of restrictionsEdit

  • 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 set X to a subset A of X is just the inclusion map from A into X.[2]
  • The restriction of a continuous function is continuous.[3][4]

ApplicationsEdit

Inverse functionsEdit

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

 

defined on the whole of   is not one-to-one since x2 = (−x)2 for any x in  . 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 y.) Alternatively, there is no need to restrict the domain if we don't mind the inverse being a multivalued function.

Selection operatorsEdit

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 lemmaEdit

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.

SheavesEdit

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 on F(U).
  • If we have three open sets WVU, 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,tF(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 set U, and if for each i a section siF(Ui) is given such that for each pair Ui,Uj of the covering sets the restrictions of si and sj agree on the overlaps: si|UiUj = sj|UiUj, then there is a section sF(U) such that s|Ui = si 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-restrictionEdit

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 domain A, codomain F and graph G(AR) = {(x, y) ∈ G(R) | xA} . Similarly, one can define a right-restriction or range restriction RB. Indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of E×F for binary relations. These cases do not fit into the scheme of sheaves.[clarification needed]

Anti-restrictionEdit

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 relation R by a set B is defined as R ▷ (F \ B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.

See alsoEdit

ReferencesEdit

  1. ^ Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. p. 5. ISBN 0-7167-0457-9.
  2. ^ Halmos, Paul (1960). Naive Set Theory. Princeton, NJ: D. Van Nostrand. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
  3. ^ Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2.
  4. ^ Adams, Colin Conrad; Franzosa, Robert David (2008). Introduction to Topology: Pure and Applied. Pearson Prentice Hall. ISBN 978-0-13-184869-6.
  5. ^ 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)