Restriction (mathematics)

Formal definition

Let f : E \to F be a function from a set E to a set F. If a set A is a subset of E, then the **restriction of **f **to **A is the function {f|}_A : A \to F 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 A.

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 its graph, :G({f|}_A) = { (x,f(x))\in G(f) : x\in A } = G(f)\cap (A\times F), where the pairs (x,f(x)) represent ordered pairs in the graph G.

Extensions

A function F is said to be an Extension of a function of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname{domain} f \subseteq \operatorname{domain} F and F\big\vert_{\operatorname{domain} f} = f.

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

Examples

1. The restriction of the non-injective functionf: \mathbb{R} \to \mathbb{R}, \ x \mapsto x^2 to the domain \mathbb{R}{+} = 0,\infty) is the injectionf:\mathbb{R}+ \to \mathbb{R}, \ x \mapsto x^2.
2. The [factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: {\Gamma|}_{\mathbb{Z}^+}!(n) = (n-1)!

Properties of restrictions

• Restricting a function f:X\rightarrow Y to its entire domain X gives back the original function, that is, f|_X = f.
• Restricting a function twice is the same as restricting it once, that is, if A \subseteq B \subseteq \operatorname{dom} f, then \left(f|_B\right)|_A = f|_A.
• 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.
• The restriction of a continuous function is continuous.

Applications

Inverse functions

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 defined on the whole of \R is not one-to-one since x^2 = (-x)^2 for any x \in \R. However, the function becomes one-to-one if we restrict to the domain \R_{\geq 0} = [0, \infty), in which case f^{-1}(y) = \sqrt{y} .

(If we instead restrict to the domain (-\infty, 0], then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.

Selection operators

Main article: Selection (relational algebra)

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:

• a and b are attribute names,
• \theta is a binary operation in the set {},
• v is a value constant,
• R is a relation.

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

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 two closed subsets (or two 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

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; that is, if V\subseteq U, then there is a morphism \operatorname{res}_{V,U} : F(U) \to 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 \operatorname{res}_{U,U} : F(U) \to F(U) is the identity morphism on F(U).
• If we have three open sets W \subseteq V \subseteq U, then the composite \operatorname{res}{W,V} \circ \operatorname{res}{V,U} = \operatorname{res}_{W,U}.
• (Locality) If \left(U_i\right) is an open covering of an open set U, and if s, t \in F(U) are such that s\big\vert_{U_i} = t\big\vert_{U_i} for each set U_i of the covering, then s = t; and
• (Gluing) If \left(U_i\right) is an open covering of an open set U, and if for each i a section x_i \in F\left(U_i\right) is given such that for each pair U_i, U_j of the covering sets the restrictions of s_i and s_j agree on the overlaps: s_i\big\vert_{U_i \cap U_j} = s_j\big\vert_{U_i \cap U_j}, then there is a section s \in F(U) such that s\big\vert_{U_i} = s_i 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-restriction

More generally, the restriction (or domain restriction or left-restriction) A \triangleleft R of a binary relation R between E and F may be defined as a relation having domain A, codomain F and graph G(A \triangleleft R) = {(x, y) \in F(R) : x \in A}. Similarly, one can define a right-restriction or range restriction R \triangleright B. Indeed, one could define a restriction to n-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product E \times F 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 R (with domain E and codomain F) by a set A may be defined as (E \setminus A) \triangleleft R; it removes all elements of A from the domain E. It is sometimes denoted A ⩤ R. Similarly, the range anti-restriction (or range subtraction) of a function or binary relation R by a set B is defined as R \triangleright (F \setminus B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.

References

References

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