Restriction (mathematics) - Alchetron, the free social encyclopedia

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 ↾ A is also used.

Formal definition

Let f : E → F be a function from a set E to a set F, so that the domain of f is in E ( d o m f ⊆ E ). If a set A is a subset of E, then the restriction of f to A is the function

f | A : A → F .

Informally, the restriction of f to A is the same function as f, but is only defined on A ∩ d o m f .

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

Examples

The restriction of the non-injective function f : R → R ; x ↦ x 2 to R + = [ 0 , ∞ ) is the injection f : R + → R ; x ↦ x 2 .
The factorial function is the restriction of the gamma function to the integers.

Properties of restrictions

Restricting a function f : X → Y to its entire domain X gives back the original function; i.e., f | X = f .

Restricting a function twice is the same as restricting it once; i.e. if A ⊆ B ⊆ d o m f , then ( f | B ) | A = f | A .

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.

The restriction of a continuous function is continuous.

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

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 ) = 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

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

a and b are attribute names

θ is a binary operation in the set { < , ≤ , = , ≠ , ≥ , > }

v is a value constant

R is a relation

The selection σ a θ b ( R ) selects all those tuples in R for which θ holds between the a and the b attribute.

The selection σ a θ v ( R ) selects all those tuples in R for which θ holds between the a attribute and the value v .

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

The 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 ∪ Y , and let B also be a topological space. If f : A → 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

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 ⊆ U , then there is a morphism res_V,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 res_U,U : F(U) → F(U) is the identity morphism on F(U).

If we have three open sets W ⊆ V ⊆ U, then the composite res_W,V o res_V,U = res_W,U.

(Locality) If (U_i) is an open covering of an open set U, and if s,t ∈ F(U) are such that s|_{U_i} = t|_{U_i} for each set U_i of the covering, then s = t; and

(Gluing) If (U_i) is an open covering of an open set U, and if for each i a section s_i ∈ F(U_i) 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|_{U_i∩U_j} = s_j|_{U_i∩U_j}, then there is a section s ∈ F(U) such that s|_{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 ◁ 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 ◁ 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 of E × 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 A) ◁ 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 ▷ (F B); it removes all elements of B from the codomain F. It is sometimes denoted R ⩥ B.

References

Restriction (mathematics) Wikipedia

(Text) CC BY-SA

Contents