Theorem of corresponding states - Revision history

128.227.98.159: /* See also */

2014-12-05T15:00:48Z

See also

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	~~In [[mathematics]]~~, ~~the composition of [[binary relation]]s~~ is ~~a concept of forming a new relation {{nowrap\|''S'' ∘ ''R''}} from two given relations ''R''~~ and ~~''S'', having as its most well-known special case the [[composition of functions]]~~.		Nice to meet you, my title is Ling and I totally dig that title. Delaware is the location I love most but I require to move for my family members. Bookkeeping is what I do for a living. Playing crochet is a factor that I'm totally addicted to.<br><br>Feel free to visit my web blog: [http://Ghaziabadmart.com/oxwall/blogs/post/16946 http://Ghaziabadmart.com]

	~~== Definition ==~~
	~~If <math>R\subseteq X\times Y</math> and <math>S\subseteq Y\times Z</math> are two binary relations, then~~
	~~their composition <math>S\circ R</math>~~ is the ~~relation~~
	~~:<math>S\circ R = \{ (x,z)\in X\times Z\mid \exists y\in Y: (x,y)\in R\land (y,z)\in S \}~~.~~</math>~~

	~~In other words, <math>S\circ R\subseteq X\times Z</math> is defined by the rule that says <math>(x,z)\in S\circ R</math> if and only if there~~ is ~~an element <math>y\in Y</math> such that <math>x\,R\,y\,S\,z</math> (i.e. <math>(x,y)\in R</math> and <math>(y,z)\in S</math>).~~

	~~In particular fields, authors might denote by {{nowrap\|''R'' ∘ ''S''}}~~ what ~~is defined here to be {{nowrap\|''S'' ∘ ''R''}}~~.
	~~The convention chosen here~~ is ~~such~~ that ~~[[function composition]] (with the usual notation) is obtained as a special case, when ''R~~'~~' and ''S'' are [[functional relation]]s. Some authors<ref>Kilp, Knauer & Mikhalev, p. 7</ref> prefer~~ to ~~write <math>\circ_l</math> and <math>\circ_r</math> explicitly when necessary, depending whether the left or the right relation is the first one applied~~.

	~~A further variation encountered in computer science is the [[Z notation]]:~~ <~~math~~>~~\circ~~<~~/math~~> ~~is used~~ to ~~denote the traditional (right) composition, but ⨾ <span style="font-size~~:~~180%; font-weight: 600;">;</span> (a fat open semicolon with Unicode code point U+2A3E<ref>~~http://~~www.fileformat~~.~~info/info/unicode~~/~~char~~/~~2a3e~~/~~index.htm<~~/ref>) denotes left composition. This use of semicolon [[Function_composition#Alternative_notations\|coincides]] with the notation for function composition used (mostly by computer scientists) in [[Category theory]], as well as the notation for dynamic conjunction within linguistic dynamic semantics.<ref>http://~~plato.stanford~~.~~edu/entries/dynamic-semantics/#EncDynTypLog</ref>~~

	The binary relations <math>R\subseteq X\times Y</math> are sometimes regarded as the morphisms <math>R\colon X\to Y</math> in a [[category (mathematics)\|category]] '''[[Category of relations\|Rel]]''' which has the sets as objects. In '''Rel''', composition of morphisms is exactly composition of relations as defined above. The category '''[[category of sets\|Set]]''' of sets is a subcategory of '''Rel''' that has the same objects but fewer morphisms. A generalization of this is found in the theory of [[allegory (category theory)\|allegories]].

	~~==Properties==~~

	~~Composition of relations is [[associative]].~~

	~~The [[inverse relation]] of {{nowrap\| ''S'' ∘ ''R''}} is~~
	~~{{nowrap\|1= (''S'' ∘ ''R'')<sup>-1</sup> = ''R''<sup>−1</sup> ∘ ''S''<sup>−1</sup>}}. This property makes the set of all binary relations on a set a [[semigroup with involution]].~~

	~~The compose of [[partial function\|(partial) function]]s (i.e. functional relations) is again a (partial) function.~~

	~~If ''R'' and ''S'' are [[injective]], then {{nowrap\| ''S'' ∘ ''R''}} is injective, which conversely implies only the injectivity of ''R''.~~

	~~If ''R'' and ''S'' are [[surjective]], then {{nowrap\| ''S'' ∘ ''R''}} is surjective, which conversely implies only the surjectivity of ''S''.~~

	The set of binary relations on a set ''X'' (i.e. relations from ''X'' to ''X'') together with (left or right) relation composition forms a [[monoid]] with zero, where the identity map on ''X'' is the [[neutral element]], and the empty set is the [[Absorbing element\|zero element]].

	~~== Join: another form of composition ==~~
	~~{{main\|Join (relational algebra)}}~~

	Other forms of composition of relations, which apply to general ''n''-place relations instead of binary relations, are found in the ''[[Join (relational algebra)\|join]]'' operation of [[relational algebra]]. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.

	~~==See also==~~
	* [[Binary relation]]
	* [[Relation algebra]]
	* [[Demonic composition]]
	* [[Function composition]]
	* [[Join (SQL)]]
	* [[Logical matrix]]

	~~== Notes ==~~
	~~{{reflist}}~~

	~~==References==~~
	*M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

	~~[[Category:Mathematical relations]~~]

en>Monkbot: /* Compressibility factor at the critical point */Fix CS1 deprecated date parameter errors

2014-01-30T21:19:35Z

Compressibility factor at the critical point: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 23:19, 30 January 2014
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	~~I would like to introduce myself to you~~, ~~I am Jayson Simcox but I don~~'~~t like when people use my full title~~. ~~Invoicing~~ is what ~~I do for a living but I~~'~~ve always wanted my own business. For a while I~~'~~ve been in Mississippi but now I~~'~~m contemplating other options~~. ~~One of~~ the ~~issues she enjoys most~~ is ~~canoeing~~ and ~~she~~'s ~~been performing it for quite a while~~.<br><br>~~Feel~~ [http://~~modenpeople~~.co.kr/~~modn~~/~~qna~~/~~292291 free tarot readings] to surf to my web-site~~ ... ~~telephone [~~http://~~myoceancounty~~.~~net~~/~~groups~~/~~apply~~-~~these-guidelines-when-gardening-and-grow~~/ ~~real psychic~~] ([~~http:~~//~~Www~~.~~edmposts~~.~~com/build~~-a-~~beautiful~~-~~organic~~-~~garden-using-these-ideas/ please click the next page~~])		In [[mathematics]], the composition of [[binary relation]]s is a concept of forming a new relation {{nowrap\|''S'' ∘ ''R''}} from two given relations ''R'' and ''S'', having as its most well-known special case the [[composition of functions]].

			== Definition ==
			If <math>R\subseteq X\times Y</math> and <math>S\subseteq Y\times Z</math> are two binary relations, then
			their composition <math>S\circ R</math> is the relation
			:<math>S\circ R = \{ (x,z)\in X\times Z\mid \exists y\in Y: (x,y)\in R\land (y,z)\in S \}.</math>

			In other words, <math>S\circ R\subseteq X\times Z</math> is defined by the rule that says <math>(x,z)\in S\circ R</math> if and only if there is an element <math>y\in Y</math> such that <math>x\,R\,y\,S\,z</math> (i.e. <math>(x,y)\in R</math> and <math>(y,z)\in S</math>).

			In particular fields, authors might denote by {{nowrap\|''R'' ∘ ''S''}} what is defined here to be {{nowrap\|''S'' ∘ ''R''}}.
			The convention chosen here is such that [[function composition]] (with the usual notation) is obtained as a special case, when ''R'' and ''S'' are [[functional relation]]s. Some authors<ref>Kilp, Knauer & Mikhalev, p. 7</ref> prefer to write <math>\circ_l</math> and <math>\circ_r</math> explicitly when necessary, depending whether the left or the right relation is the first one applied.

			A further variation encountered in computer science is the [[Z notation]]: <math>\circ</math> is used to denote the traditional (right) composition, but ⨾ <span style="font-size:180%; font-weight: 600;">;</span> (a fat open semicolon with Unicode code point U+2A3E<ref>http://www.fileformat.info/info/unicode/char/2a3e/index.htm</ref>) denotes left composition. This use of semicolon [[Function_composition#Alternative_notations\|coincides]] with the notation for function composition used (mostly by computer scientists) in [[Category theory]], as well as the notation for dynamic conjunction within linguistic dynamic semantics.<ref>http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog</ref>

			The binary relations <math>R\subseteq X\times Y</math> are sometimes regarded as the morphisms <math>R\colon X\to Y</math> in a [[category (mathematics)\|category]] '''[[Category of relations\|Rel]]''' which has the sets as objects. In '''Rel''', composition of morphisms is exactly composition of relations as defined above. The category '''[[category of sets\|Set]]''' of sets is a subcategory of '''Rel''' that has the same objects but fewer morphisms. A generalization of this is found in the theory of [[allegory (category theory)\|allegories]].

			==Properties==

			Composition of relations is [[associative]].

			The [[inverse relation]] of {{nowrap\| ''S'' ∘ ''R''}} is
			{{nowrap\|1= (''S'' ∘ ''R'')<sup>-1</sup> = ''R''<sup>−1</sup> ∘ ''S''<sup>−1</sup>}}. This property makes the set of all binary relations on a set a [[semigroup with involution]].

			The compose of [[partial function\|(partial) function]]s (i.e. functional relations) is again a (partial) function.

			If ''R'' and ''S'' are [[injective]], then {{nowrap\| ''S'' ∘ ''R''}} is injective, which conversely implies only the injectivity of ''R''.

			If ''R'' and ''S'' are [[surjective]], then {{nowrap\| ''S'' ∘ ''R''}} is surjective, which conversely implies only the surjectivity of ''S''.

			The set of binary relations on a set ''X'' (i.e. relations from ''X'' to ''X'') together with (left or right) relation composition forms a [[monoid]] with zero, where the identity map on ''X'' is the [[neutral element]], and the empty set is the [[Absorbing element\|zero element]].

			== Join: another form of composition ==
			{{main\|Join (relational algebra)}}

			Other forms of composition of relations, which apply to general ''n''-place relations instead of binary relations, are found in the ''[[Join (relational algebra)\|join]]'' operation of [[relational algebra]]. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.

			==See also==
			* [[Binary relation]]
			* [[Relation algebra]]
			* [[Demonic composition]]
			* [[Function composition]]
			* [[Join (SQL)]]
			* [[Logical matrix]]

			== Notes ==
			{{reflist}}

			==References==
			*M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

			[[Category:Mathematical relations]]

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2012-05-08T08:30:24Z

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I would like to introduce myself to you, I am Jayson Simcox but I don't like when people use my full title. Invoicing is what I do for a living but I've always wanted my own business. For a while I've been in Mississippi but now I'm contemplating other options. One of the issues she enjoys most is canoeing and she's been performing it for quite a while.<br><br>Feel [http://modenpeople.co.kr/modn/qna/292291 free tarot readings] to surf to my web-site ... telephone [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ real psychic] ([http://Www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ please click the next page])