Functions:Restriction - Revision history

Khanh at 20:00, 7 November 2021

2021-11-07T20:00:07Z

Lila at 15:50, 13 October 2021

2021-10-13T15:50:45Z

Lila: /* Left- and right-restriction */

2021-10-13T15:49:36Z

Left- and right-restriction

Lila: /* Left- and right-restriction */

2021-10-13T15:49:05Z

Left- and right-restriction

Lila: /* The pasting lemma */

2021-10-13T15:48:06Z

The pasting lemma

Lila: /* Applications */

2021-10-13T15:47:41Z

Applications

Lila: /* Applications */

2021-10-13T15:47:02Z

Applications

Lila at 15:45, 13 October 2021

2021-10-13T15:45:13Z

Lila: /* Properties of restrictions */

2021-10-13T15:39:27Z

Properties of restrictions

Lila: /* Formal definition */

2021-10-13T15:38:42Z

Formal definition

@@ Line 73: / Line 73: @@
 These cases do not fit into the scheme of sheaves.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Restriction_(mathematics) Restriction (mathematics)], Wikipedia
+Content obtained and/or adapted from:

← Older revision		Revision as of 15:50, 13 October 2021
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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 numbers, then it does have an inverse function, known as the square root of ''x''.]]

	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>.		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>.

@@ Line 70: / Line 70: @@
 ==Left- and right-restriction==
-More generally, the restriction (or '''domain restriction''' or  '''left-restriction''') {{math|''A'' ◁ ''R''}} of a binary relation {{mvar|R}} between {{mvar|E}} and {{mvar|F}} may be defined as a relation having domain {{mvar|A}}, codomain {{mvar|F}} and graph {{math|1=G(''A'' ◁ ''R'') = {(''x'', ''y'') ∈ G(''R'') {{!}} ''x'' ∈ ''A''} }}.  Similarly, one can define a '''right-restriction''' or '''range restriction''' {{math|''R'' ▷ ''B''}}. Indeed, one could define a restriction to [[arity|{{mvar|n}}-ary]] relations, as well as to subsets understood as relations, such as ones of {{math|''E'' [[Cartesian product|×]] ''F''}} for binary relations.
+More generally, the restriction (or '''domain restriction''' or  '''left-restriction''') {{math|''A'' ◁ ''R''}} of a binary relation {{mvar|R}} between {{mvar|E}} and {{mvar|F}} may be defined as a relation having domain {{mvar|A}}, codomain {{mvar|F}} and graph {{math|1=G(''A'' ◁ ''R'') = {(''x'', ''y'') ∈ G(''R'') {{!}} ''x'' ∈ ''A''} }}.  Similarly, one can define a '''right-restriction''' or '''range restriction''' {{math|''R'' ▷ ''B''}}. Indeed, one could define a restriction to {{mvar|n}}-ary relations, as well as to subsets understood as relations, such as ones of {{math|''E'' [[Cartesian product|×]] ''F''}} for binary relations.
 These cases do not fit into the scheme of sheaves.
 ==Resources==
 * [https://en.wikipedia.org/wiki/Restriction_(mathematics) Restriction (mathematics)], Wikipedia

@@ Line 70: / Line 70: @@
 ==Left- and right-restriction==
-More generally, the restriction (or '''domain restriction''' or  '''left-restriction''') {{math|''A'' ◁ ''R''}} of a [[binary relation]] {{mvar|R}} between {{mvar|E}} and {{mvar|F}} may be defined as a relation having domain {{mvar|A}}, codomain {{mvar|F}} and graph {{math|1=G(''A'' ◁ ''R'') = {(''x'', ''y'') ∈ G(''R'') {{!}} ''x'' ∈ ''A''} }}.  Similarly, one can define a '''right-restriction''' or '''range restriction''' {{math|''R'' ▷ ''B''}}. Indeed, one could define a restriction to [[arity|{{mvar|n}}-ary]] relations, as well as to [[subset]]s understood as relations, such as ones of {{math|''E'' [[Cartesian product|×]] ''F''}} for binary relations.
+More generally, the restriction (or '''domain restriction''' or  '''left-restriction''') {{math|''A'' ◁ ''R''}} of a binary relation {{mvar|R}} between {{mvar|E}} and {{mvar|F}} may be defined as a relation having domain {{mvar|A}}, codomain {{mvar|F}} and graph {{math|1=G(''A'' ◁ ''R'') = {(''x'', ''y'') ∈ G(''R'') {{!}} ''x'' ∈ ''A''} }}.  Similarly, one can define a '''right-restriction''' or '''range restriction''' {{math|''R'' ▷ ''B''}}. Indeed, one could define a restriction to [[arity|{{mvar|n}}-ary]] relations, as well as to subsets understood as relations, such as ones of {{math|''E'' [[Cartesian product|×]] ''F''}} for binary relations.
-These cases do not fit into the scheme of [[sheaf (mathematics)|sheaves]].{{clarify|date=July 2013}}
+These cases do not fit into the scheme of sheaves.
 ==Resources==
 * [https://en.wikipedia.org/wiki/Restriction_(mathematics) Restriction (mathematics)], Wikipedia

@@ Line 50: / Line 50: @@
 ===The 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.

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 :<math>f(x) = x^2</math>
-defined on the whole of <math>\R</math> is not one-to-one since ''x''<sup>2</sup> = (−''x'')<sup>2</sup> for any ''x'' in <math>\R</math>.  However, the function becomes one-to-one if we restrict to the domain {{nowrap|<math>\R_{\geq 0} = [0, \infty)</math>,}} in which case
+defined on the whole of <math>\R</math> is not one-to-one since ''x''<sup>2</sup> = (−''x'')<sup>2</sup> for any ''x'' in <math>\R</math>.  However, the function becomes one-to-one if we restrict to the domain <math>\R_{\geq 0} = [0, \infty)</math>, in which case
 :<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 <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===
-{{main|Selection (relational algebra)}}
 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:

@@ Line 23: / Line 23: @@
 ==Applications==
 ===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>
@@ Line 32: / Line 32: @@
 :<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===
 {{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>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>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>.
@@ Line 51: / Line 51: @@
 ===The 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.
@@ Line 58: / Line 58: @@
 ===Sheaves===
-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.
 * 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''.

← Older revision		Revision as of 15:45, 13 October 2021
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	More generally, the restriction (or '''domain restriction''' or '''left-restriction''') {{math\|''A'' ◁ ''R''}} of a [[binary relation]] {{mvar\|R}} between {{mvar\|E}} and {{mvar\|F}} may be defined as a relation having domain {{mvar\|A}}, codomain {{mvar\|F}} and graph {{math\|1=G(''A'' ◁ ''R'') = {(''x'', ''y'') ∈ G(''R'') {{!}} ''x'' ∈ ''A''} }}. Similarly, one can define a '''right-restriction''' or '''range restriction''' {{math\|''R'' ▷ ''B''}}. Indeed, one could define a restriction to [[arity\|{{mvar\|n}}-ary]] relations, as well as to [[subset]]s understood as relations, such as ones of {{math\|''E'' [[Cartesian product\|×]] ''F''}} for binary relations.		More generally, the restriction (or '''domain restriction''' or '''left-restriction''') {{math\|''A'' ◁ ''R''}} of a [[binary relation]] {{mvar\|R}} between {{mvar\|E}} and {{mvar\|F}} may be defined as a relation having domain {{mvar\|A}}, codomain {{mvar\|F}} and graph {{math\|1=G(''A'' ◁ ''R'') = {(''x'', ''y'') ∈ G(''R'') {{!}} ''x'' ∈ ''A''} }}. Similarly, one can define a '''right-restriction''' or '''range restriction''' {{math\|''R'' ▷ ''B''}}. Indeed, one could define a restriction to [[arity\|{{mvar\|n}}-ary]] relations, as well as to [[subset]]s understood as relations, such as ones of {{math\|''E'' [[Cartesian product\|×]] ''F''}} for binary relations.
	These cases do not fit into the scheme of [[sheaf (mathematics)\|sheaves]].{{clarify\|date=July 2013}}		These cases do not fit into the scheme of [[sheaf (mathematics)\|sheaves]].{{clarify\|date=July 2013}}
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Restriction_(mathematics) Restriction (mathematics)], Wikipedia

@@ Line 18: / Line 18: @@
 * Restricting a function <math>f:X\rightarrow Y</math> to its entire domain <math>X</math> gives back the original function, i.e., <math>f|_{X}=f</math>.
 *Restricting a function twice is the same as restricting it once, i.e. if <math>A\subseteq B \subseteq \operatorname{dom} f</math>, then <math>(f|_B)|_A=f|_A</math>.
-*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''.<ref>{{cite book |author-link=Paul Halmos |last=Halmos |first=Paul |title=[[Naive Set Theory (book)|Naive Set Theory]] |location=Princeton, NJ |publisher=D. Van Nostrand |year=1960 }} Reprinted by Springer-Verlag, New York, 1974. {{isbn|0-387-90092-6}} (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. {{isbn|978-1-61427-131-4}} (Paperback edition).</ref>
+*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.<ref>{{cite book |last=Munkres |first=James R. |title=Topology |edition=2nd |location=Upper Saddle River |publisher=Prentice Hall |year=2000 |isbn=0-13-181629-2 }}</ref><ref>{{cite book |last=Adams |first=Colin Conrad |first2=Robert David |last2=Franzosa |title=Introduction to Topology: Pure and Applied |publisher=Pearson Prentice Hall |year=2008 |isbn=978-0-13-184869-6 }}</ref>
+*The restriction of a continuous function is continuous.
 ==Applications==

@@ Line 9: / Line 9: @@
 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 <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}}.
+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 <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==