Difference between revisions of "Functions:Restriction"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 23: Line 23:
 
==Applications==
 
==Applications==
 
===Inverse functions===
 
===Inverse functions===
−
{{main|Inverse function}}
+
 
−
For a function to have an inverse, it must be [[one-to-one function|one-to-one]]. If a function {{mvar|f}} is not one-to-one, it may be possible to define a '''partial inverse''' of {{mvar|f}} by restricting the domain.  For example, the function
+
For a function to have an inverse, it must be one-to-one. If a function {{mvar|f}} is not one-to-one, it may be possible to define a '''partial inverse''' of {{mvar|f}} by restricting the domain.  For example, the function
  
 
:<math>f(x) = x^2</math>
 
:<math>f(x) = x^2</math>
Line 32: Line 32:
 
:<math>f^{-1}(y) = \sqrt{y} . </math>
 
:<math>f^{-1}(y) = \sqrt{y} . </math>
  
−
(If we instead restrict to the domain {{nowrap|<math>(-\infty, 0]</math>,}} then the inverse is the negative of the square root of {{mvar|y}}.)  Alternatively, there is no need to restrict the domain if we allow the inverse to be a [[multivalued function]].
+
(If we instead restrict to the domain {{nowrap|<math>(-\infty, 0]</math>,}} then the inverse is the negative of the square root of {{mvar|y}}.)  Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.
  
 
===Selection operators===
 
===Selection operators===
 
{{main|Selection (relational algebra)}}
 
{{main|Selection (relational algebra)}}
−
In [[relational algebra]], a [[selection (relational algebra)|selection]] (sometimes called a restriction to avoid confusion with [[SQL]]'s use of SELECT) is a [[unary operation]] written as
+
In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as
 
<math>\sigma_{a \theta b}( R )</math> or <math>\sigma_{a \theta v}( R )</math> where:
 
<math>\sigma_{a \theta b}( R )</math> or <math>\sigma_{a \theta v}( R )</math> where:
 
* <math>a</math> and <math>b</math> are attribute names,
 
* <math>a</math> and <math>b</math> are attribute names,
−
* <math>\theta</math> is a [[binary operation]] in the set <math>\{<, \leq, =, \neq, \geq, >\}</math>,
+
* <math>\theta</math> is a binary operation in the set <math>\{<, \leq, =, \neq, \geq, >\}</math>,
 
* <math>v</math> is a value constant,
 
* <math>v</math> is a value constant,
−
* <math>R</math> is a [[Relation (database)|relation]].
+
* <math>R</math> is a relation.
  
−
The selection <math>\sigma_{a \theta b}( R )</math> selects all those [[tuple]]s in <math>R</math> for which <math>\theta</math> holds between the <math>a</math> and the <math>b</math> attribute.
+
The selection <math>\sigma_{a \theta b}( R )</math> selects all those tuples in <math>R</math> for which <math>\theta</math> holds between the <math>a</math> and the <math>b</math> attribute.
  
 
The selection <math>\sigma_{a \theta v}( R )</math> selects all those tuples in <math>R</math> for which <math>\theta</math> holds between the <math>a</math> attribute and the value <math>v</math>.
 
The selection <math>\sigma_{a \theta v}( R )</math> selects all those tuples in <math>R</math> for which <math>\theta</math> holds between the <math>a</math> attribute and the value <math>v</math>.
Line 51: Line 51:
 
===The pasting lemma===
 
===The pasting lemma===
 
{{main|Pasting lemma}}
 
{{main|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.
+
The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.
  
 
Let <math>X,Y</math> be two closed subsets (or two open subsets) of a topological space <math>A</math> such that <math>A = X \cup Y</math>, and let <math>B</math> also be a topological space. If <math>f: A \to B</math> is continuous when restricted to both <math>X</math> and <math>Y</math>, then <math>f</math> is continuous.
 
Let <math>X,Y</math> be two closed subsets (or two open subsets) of a topological space <math>A</math> such that <math>A = X \cup Y</math>, and let <math>B</math> also be a topological space. If <math>f: A \to B</math> is continuous when restricted to both <math>X</math> and <math>Y</math>, then <math>f</math> is continuous.
Line 58: Line 58:
  
 
===Sheaves===
 
===Sheaves===
−
{{main|Sheaf theory}}
 
−
[[Sheaf theory|Sheaves]] provide a way of generalizing restrictions to objects besides functions.
 
  
−
In [[sheaf theory]], one assigns an object <math>F(U)</math> in a [[category (category theory)|category]] to each [[open set]] {{mvar|U}} of a [[topological space]], and requires that the objects satisfy certain conditions. The most important condition is that there are ''restriction [[morphism]]s'' between every pair of objects associated to nested open sets; i.e., if <math>V\subseteq U</math>, then there is a morphism res<sub>''V'',''U''</sub> : ''F''(''U'') → ''F''(''V'') satisfying the following properties, which are designed to mimic the restriction of a function:
+
Sheaves provide a way of generalizing restrictions to objects besides functions.
 +
 
 +
In sheaf theory, one assigns an object <math>F(U)</math> in a category to each open set {{mvar|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 <math>V\subseteq U</math>, then there is a morphism res<sub>''V'',''U''</sub> : ''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<sub>''U'',''U''</sub> : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
 
* For every open set ''U'' of ''X'', the restriction morphism res<sub>''U'',''U''</sub> : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
−
* If we have three open sets {{math|1=''W'' ⊆ ''V'' ⊆ ''U''}}, then the [[function composition|composite]] {{math|1=res<sub>''W'',''V''</sub> ∘ res<sub>''V'',''U''</sub> = res<sub>''W'',''U''</sub>}}.
+
* If we have three open sets {{math|1=''W'' ⊆ ''V'' ⊆ ''U''}}, then the composite {{math|1=res<sub>''W'',''V''</sub> ∘ res<sub>''V'',''U''</sub> = res<sub>''W'',''U''</sub>}}.
−
* (Locality) If (''U''<sub>''i''</sub>) is an open [[cover (topology)|covering]] of an open set ''U'', and if ''s'',''t'' ∈ ''F''(''U'') are such that <span class="texhtml">''s''|<sub>''U''<sub>''i''</sub></sub> = ''t''|<sub>''U''<sub>''i''</sub></sub></span> for each set ''U''<sub>''i''</sub> of the covering, then ''s'' = ''t''; and
+
* (Locality) If (''U''<sub>''i''</sub>) is an open covering of an open set ''U'', and if ''s'',''t'' ∈ ''F''(''U'') are such that <span class="texhtml">''s''|<sub>''U''<sub>''i''</sub></sub> = ''t''|<sub>''U''<sub>''i''</sub></sub></span> for each set ''U''<sub>''i''</sub> of the covering, then ''s'' = ''t''; and
 
* (Gluing) If (''U''<sub>''i''</sub>) is an open covering of an open set ''U'', and if for each ''i'' a section {{math|1=''s''<sub>''i''</sub> ∈ ''F''(''U''<sub>''i''</sub>)}} is given such that for each pair ''U''<sub>''i''</sub>,''U''<sub>''j''</sub> of the covering sets the restrictions of ''s''<sub>''i''</sub> and ''s''<sub>''j''</sub> agree on the overlaps: <span class="texhtml">''s''<sub>''i''</sub>|<sub>''U''<sub>''i''</sub>∩''U''<sub>''j''</sub></sub> =  ''s''<sub>''j''</sub>|<sub>''U''<sub>''i''</sub>∩''U''<sub>''j''</sub></sub></span>, then there is a section {{math|1=''s'' ∈ ''F''(''U'')}} such that <span class="texhtml">''s''|<sub>''U''<sub>''i''</sub></sub> = ''s''<sub>''i''</sub></span> for each ''i''.
 
* (Gluing) If (''U''<sub>''i''</sub>) is an open covering of an open set ''U'', and if for each ''i'' a section {{math|1=''s''<sub>''i''</sub> ∈ ''F''(''U''<sub>''i''</sub>)}} is given such that for each pair ''U''<sub>''i''</sub>,''U''<sub>''j''</sub> of the covering sets the restrictions of ''s''<sub>''i''</sub> and ''s''<sub>''j''</sub> agree on the overlaps: <span class="texhtml">''s''<sub>''i''</sub>|<sub>''U''<sub>''i''</sub>∩''U''<sub>''j''</sub></sub> =  ''s''<sub>''j''</sub>|<sub>''U''<sub>''i''</sub>∩''U''<sub>''j''</sub></sub></span>, then there is a section {{math|1=''s'' ∈ ''F''(''U'')}} such that <span class="texhtml">''s''|<sub>''U''<sub>''i''</sub></sub> = ''s''<sub>''i''</sub></span> for each ''i''.
  

Revision as of 09:47, 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 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 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 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 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} 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 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 G({f|}_A) = \{ (x,f(x))\in G(f) \mid x\in A \} = G(f)\cap (A\times F)} , 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.
  • 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

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

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

Resources

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Functions:Restriction&oldid=2243"