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Jumat, 28 Juli 2017

In mathematics, the restriction of a function f is a new function f | A {\displaystyle f\vert _{A}} obtained by choosing a smaller domain A for the original function f {\displaystyle f} . The notation f ↾ A {\displaystyle f{\upharpoonright _{A}}} is also used.

Formal definition



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Let f : E â†' F {\displaystyle f:E\to 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 {\displaystyle \mathrm {dom} \,f\subseteq E} ). If a set A is a subset of E, then the restriction of f {\displaystyle f} to A {\displaystyle A} is the function

f | A : A â†' F {\displaystyle {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 ∩ d o m f {\displaystyle A\cap \mathrm {dom} \,f} .

If the function f is thought of as a relation ( x , f ( x ) ) {\displaystyle (x,f(x))} on the Cartesian product E × F {\displaystyle E\times 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 ) {\displaystyle G({f|}_{A})=\{(x,f(x))\in G(f)\mid x\in A\}=G(f)\cap (A\times F)} , where the pairs ( x , f ( x ) ) {\displaystyle (x,f(x))} represent edges in the graph G.

Examples



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  1. The restriction of the non-injective function f : R â†' R ; x ↦ x 2 {\displaystyle f:\mathbb {R} \to \mathbb {R} ;x\mapsto x^{2}} to R + = [ 0 , ∞ ) {\displaystyle \mathbb {R} _{+}=[0,\infty )} is the injection f : R + â†' R ; x ↦ x 2 {\displaystyle 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



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  • Restricting a function f : X â†' Y {\displaystyle f:X\rightarrow Y} to its entire domain X {\displaystyle X} gives back the original function; i.e., f | X = f {\displaystyle f|_{X}=f} .
  • Restricting a function twice is the same as restricting it once; i.e. if A ⊆ B ⊆ d o m f {\displaystyle A\subseteq B\subseteq \mathrm {dom} f} , then ( f | B ) | A = f | A {\displaystyle (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.

Applications



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 {\displaystyle f(x)=x^{2}}

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

f âˆ' 1 ( y ) = y . {\displaystyle 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

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 ) {\displaystyle \sigma _{a\theta b}(R)} or σ a θ v ( R ) {\displaystyle \sigma _{a\theta v}(R)} where:

  • a {\displaystyle a} and b {\displaystyle b} are attribute names
  • θ {\displaystyle \theta } is a binary operation in the set { < , ≤ , = , ≠ , ≥ , > } {\displaystyle \{\;<,\leq ,=,\neq ,\geq ,\;>\}}
  • v {\displaystyle v} is a value constant
  • R {\displaystyle R} is a relation

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

The selection σ a θ v ( R ) {\displaystyle \sigma _{a\theta v}(R)} selects all those tuples in R {\displaystyle R} for which θ {\displaystyle \theta } holds between the a {\displaystyle a} attribute and the value v {\displaystyle 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 {\displaystyle X,Y} be both closed (or both open) subsets of a topological space A such that A = X ∪ Y {\displaystyle A=X\cup Y} , and let B also be a topological space. If f : A â†' B {\displaystyle 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

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

In sheaf theory, one assigns an object F ( U ) {\displaystyle F(U)} in a category to each open set U {\displaystyle 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 {\displaystyle V\subseteq U} , then there is a morphism resV,U : F(U) â†' 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) â†' F(U) is the identity morphism on F(U).
  • If we have three open sets W ⊆ V ⊆ U, 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,t ∈ F(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 si ∈ F(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|Ui∩Uj = sj|Ui∩Uj, then there is a section s ∈ F(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-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.

See also



  • Deformation retract
  • Function (mathematics) #Restrictions and extensions
  • Binary relation #Restriction
  • Relational algebra#Selection (σ)

References





 
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