Difference between revisions of "Functions:Restriction"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(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...")
 
Line 1: Line 1:
 
[[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''.]]
  
−
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>.
+
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==
−
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">
+
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
−
{{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>.
  
−
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}}.
+
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==
−
# 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>.
+
# 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>.
−
# 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>
+
# 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

Error creating thumbnail:
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 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 a new function, denoted 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\vert_A} or , obtained by choosing a smaller domain A for the original 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} .

Formal definition

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 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 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} to is 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|}_A \colon 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 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\cap \operatorname{dom} f} .

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 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 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., .
  • 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 . 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 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,
  • 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 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 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 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 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 , then 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 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(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 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
Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Functions:Restriction&oldid=2238"