Restriction (mathematics)

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For other uses, see Restriction (disambiguation).
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 f is a new function f|A obtained by choosing a smaller domain A for the original function f. The notation f {\restriction_A} is also used.

Formal definition[edit]

Let f : EF be a function from a set E to a set F, so that the domain of f is in E (\mathrm{dom} \, f \subseteq E). If a set A is a subset of E, then the restriction of f to A is the function[1]

 {f|}_A \colon A \to F.

Informally, the restriction of f to A is the same function as f, but is only defined on A\cap \mathrm{dom} \, f.

If the function f is thought of as a relation (x,f(x)) on the Cartesian product E \times F, then the restriction of f to A can be represented by the graph G({f|}_A) = \{ (x,f(x))\in G(f) \mid x\in A \}, where the pairs (x,f(x)) represent edges in the graph G.

Examples[edit]

  1. The restriction of the non-injective function  f: \mathbb R\to\mathbb R; x\mapsto x^2 to  \mathbb R_+=[0,\infty) is the injection  f: \mathbb R_+\to\mathbb R; x\mapsto x^2.
  2. The factorial function is the restriction of the gamma function to the integers.

Properties of restrictions[edit]

Applications[edit]

Inverse functions[edit]

Main article: Inverse function

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

f(x) = x^2

is not one-to-one, since x2 = (−x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case

f^{-1}(y) = \sqrt{y} .

(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 a multivalued function:

Selection operators[edit]

In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as \sigma_{a \theta b}( R ) or \sigma_{a \theta v}( R ) where:

The selection \sigma_{a \theta b}( R ) selects all those tuples in R for which \theta holds between the a and the b attribute.

The selection \sigma_{a \theta v}( R ) selects all those tuples in R for which \theta holds between the a attribute and the value v.

Thus, the selection operator restricts to a subset of the entire database.

The Pasting Lemma[edit]

Main article: 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 X,Y be both closed (or both open) subsets of a topological space A such that A = X \cup Y, and let B also be a topological space. If f: A \to B is continuous when restricted to both X 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[edit]

Main article: Sheaf theory

Sheaves provide a way of generalizing restrictions to objects besides functions.

In sheaf theory, one assigns an object F(U) in a category to each open set U 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 V\subseteq U, 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:

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[edit]

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-restriction[edit]

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 also[edit]

References[edit]

  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 978-1-61427-131-4 (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, 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)