Restriction (mathematics)

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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

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

Formal definition

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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[1]   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

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A function   is said to be an extension of another function   if whenever   is in the domain of   then   is also in the domain of   and   That is, if   and  

A linear extension (respectively, continuous extension, etc.) of a function   is an extension of   that is also a linear map (respectively, a continuous map, etc.).

Examples

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  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 restrictions

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  • 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  [2]
  • The restriction of a continuous function is continuous.[3][4]

Applications

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Inverse functions

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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

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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

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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

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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

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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.[clarification needed]

Anti-restriction

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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   ⩤  [5] 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

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References

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  1. ^ Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. [36]. 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)