Difference between revisions of "Functions:Restriction"

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(Created page with "thumb|The function ''x''<sup>2</sup> with domain '''R''' does not have an [[inverse function. If we restrict ''x''<sup>2</sup> to the non-neg...")
 
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[[File:Inverse square graph.svg|thumb|The function ''x''<sup>2</sup> with domain '''R''' does not have an [[inverse function]]. If we restrict ''x''<sup>2</sup> to the non-negative [[real number]]s, then it does have an inverse function, known as the [[square root]] of ''x''.]]
 
[[File:Inverse square graph.svg|thumb|The function ''x''<sup>2</sup> with domain '''R''' does not have an [[inverse function]]. If we restrict ''x''<sup>2</sup> to the non-negative [[real number]]s, then it does have an inverse function, known as the [[square root]] of ''x''.]]
  
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In [[mathematics]], the '''restriction''' of a [[Function (mathematics)|function]] <math>f</math> is a new function, denoted <math>f\vert_A</math> or <math>f {\restriction_A}</math>, obtained by choosing a smaller [[domain of a function|domain]] ''A'' for the original function <math>f</math>.
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The '''restriction''' of a function <math>f</math> is a new function, denoted <math>f\vert_A</math> or <math>f {\restriction_A}</math>, obtained by choosing a smaller domain ''A'' for the original function <math>f</math>.
  
 
==Formal definition==
 
==Formal definition==
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Let <math>f: E \to F</math> be a function from a [[Set (mathematics)|set]] {{mvar|E}} to a set {{mvar|F}}. If a set {{mvar|A}} is a [[subset]] of {{mvar|E}}, then the '''restriction of '''<math>f</math> '''to '''<math>A</math>  is the function<ref name="Stoll">
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Let <math>f: E \to F</math> be a function from a set {{mvar|E}} to a set {{mvar|F}}. If a set {{mvar|A}} is a [[subset]] of {{mvar|E}}, then the '''restriction of '''<math>f</math> '''to '''<math>A</math>  is the function
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{{Cite book  | last = Stoll  | first = Robert  | title = Sets, Logic and Axiomatic Theories  | publisher = W. H. Freeman and Company  | date = 1974  | location = San Francisco  | pages = [https://archive.org/details/setslogicaxiomat0000stol/page/5 5]  | edition = 2nd  | isbn = 0-7167-0457-9  | url = https://archive.org/details/setslogicaxiomat0000stol/page/5  }}</ref>
 
 
:<math> {f|}_A \colon A \to F</math>
 
:<math> {f|}_A \colon A \to F</math>
  
 
given by ''f''|<sub>''A''</sub>(''x'') = ''f''(''x'') for ''x'' in ''A''. Informally, the restriction of {{mvar|f}} to {{mvar|A}} is the same function as {{mvar|f}}, but is only defined on <math>A\cap \operatorname{dom} f</math>.
 
given by ''f''|<sub>''A''</sub>(''x'') = ''f''(''x'') for ''x'' in ''A''. Informally, the restriction of {{mvar|f}} to {{mvar|A}} is the same function as {{mvar|f}}, but is only defined on <math>A\cap \operatorname{dom} f</math>.
  
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If the function {{mvar|f}} is thought of as a [[relation (mathematics)|relation]] <math>(x,f(x))</math> on the [[Cartesian product]] <math>E \times F</math>, then the restriction of {{mvar|f}} to {{mvar|A}} can be represented by its [[Graph of a function|graph]] {{nowrap|<math>G({f|}_A) = \{ (x,f(x))\in G(f) \mid x\in A \} = G(f)\cap (A\times F)</math>,}} where the pairs <math>(x,f(x))</math> represent [[ordered pair]]s in the graph {{mvar|G}}.
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If the function {{mvar|f}} is thought of as a relation <math>(x,f(x))</math> on the [[Cartesian product]] <math>E \times F</math>, then the restriction of {{mvar|f}} to {{mvar|A}} can be represented by its graph {{nowrap|<math>G({f|}_A) = \{ (x,f(x))\in G(f) \mid x\in A \} = G(f)\cap (A\times F)</math>,}} where the pairs <math>(x,f(x))</math> represent ordered pairs in the graph {{mvar|G}}.
  
 
==Examples==
 
==Examples==
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# The restriction of the [[injective function|non-injective]] function<math>f: \mathbb R \to \mathbb R, \ x \mapsto x^2</math> to the domain <math>\mathbb R_{+} = [0,\infty) </math> is the injection<math>f:\mathbb R_+ \to \mathbb R, \ x \mapsto x^2</math>.
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# The restriction of the non-injective function<math>f: \mathbb R \to \mathbb R, \ x \mapsto x^2</math> to the domain <math>\mathbb R_{+} = [0,\infty) </math> is the injection<math>f:\mathbb R_+ \to \mathbb R, \ x \mapsto x^2</math>.
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# The [[factorial]] function is the restriction of the [[gamma function]] to the positive integers, with the argument shifted by one: <math>{\Gamma|}_{\mathbb Z^+}\!(n) = (n-1)!</math>
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# The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: <math>{\Gamma|}_{\mathbb Z^+}\!(n) = (n-1)!</math>
  
 
==Properties of restrictions==
 
==Properties of restrictions==

Revision as of 09:37, 13 October 2021

The function x2 with domain R does not have an inverse function. If we restrict x2 to the non-negative real numbers, then it does have an inverse function, known as the square root of x.

The restriction of a function is a new function, denoted or , obtained by choosing a smaller domain A for the original function .

Formal definition

Let be a function from a set E to a set F. If a set A is a subset of E, then the restriction of to is the function

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 .

If the function f is thought of as a relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,f(x))} on the Cartesian product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \times F} , then the restriction of f to A can be represented by its graph Template:Nowrap where the pairs Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,f(x))} represent ordered pairs in the graph G.

Examples

  1. The restriction of the non-injective functionFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \mathbb R \to \mathbb R, \ x \mapsto x^2} to the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R_{+} = [0,\infty) } is the injectionFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:\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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\Gamma|}_{\mathbb Z^+}\!(n) = (n-1)!}

Properties of restrictions

  • Restricting a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\rightarrow Y} to its entire domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} gives back the original function, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_{X}=f} .
  • Restricting a function twice is the same as restricting it once, i.e. if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\subseteq B \subseteq \operatorname{dom} f} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f|_B)|_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.[1]
  • The restriction of a continuous function is continuous.[2][3]

Applications

Inverse functions

Template:Main 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^2}

defined on the whole of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} is not one-to-one since x2 = (−x)2 for any x in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} . However, the function becomes one-to-one if we restrict to the domain Template:Nowrap in which case

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y) = \sqrt{y} . }

(If we instead restrict to the domain Template:Nowrap 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

Template:Main In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{a \theta b}( R )} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{a \theta v}( R )} where:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} are attribute names,
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is a binary operation in the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{<, \leq, =, \neq, \geq, >\}} ,
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is a value constant,
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a relation.

The selection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{a \theta b}( R )} selects all those tuples in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} for which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} holds between the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} attribute.

The selection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{a \theta v}( R )} selects all those tuples in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} for which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} holds between the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} attribute and the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} .

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

The pasting lemma

Template:Main The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X,Y} be two closed subsets (or two open subsets) of a topological space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = X \cup Y} , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} also be a topological space. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: A \to B} is continuous when restricted to both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Template:Main 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 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

  • For every open set U of X, the restriction morphism resU,U : F(U) → F(U) is the identity morphism on F(U).
  • If we have three open sets W ⊆ V ⊆ U, then the composite resW,V ∘ 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.Template:Clarify

  1. ↑ Template:Cite book Reprinted by Springer-Verlag, New York, 1974. Template:Isbn (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. Template:Isbn (Paperback edition).
  2. ↑ Template:Cite book
  3. ↑ Template:Cite book
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