en>Jesin: /* Definition */ removed unnecessary line breaks

2013-12-08T23:22:28Z

Definition: removed unnecessary line breaks

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	~~Pleased~~ to ~~meet you~~! ~~Simple name~~ is ~~Eusebio on top~~ of that ~~I think~~ it to ~~become quite good~~ when ~~say~~ it. As a ~~man what My hubby~~ and ~~i really like~~ is ~~conducting but I~~'~~m thinking towards starting something new~~. ~~I work~~ as an ~~order clerk~~. ~~My house~~ is so ~~now~~ in ~~South Carolina~~ and ~~I don~~'~~t plan~~ on ~~changing it~~. ~~You can find my website here~~: http://~~prometeu~~.~~net<br><br>Also visit my web blog~~ :: [http://~~prometeu~~.~~net hack clash of clans no survey~~]		{{Other uses\|Closure (disambiguation)}}
			{{About\|the transitive closure of a binary relation\|the transitive closure of a set\|transitive set#Transitive closure}}
			In [[mathematics]], the '''transitive closure''' of a [[binary relation]] ''R'' on a [[Set (mathematics)\|set]] ''X'' is the [[transitive relation]] ''R''<sup>+</sup> on [[Set (mathematics)\|set]] ''X'' such that ''R''<sup>+</sup> contains ''R'' and ''R''<sup>+</sup> is minimal (Lidl and Pilz 1998:337). If the binary relation itself is [[transitive relation\|transitive]], then the transitive closure is that same binary relation; otherwise, the transitive closure is a different relation. For example, if ''X'' is a set of airports and ''x R y'' means "there is a direct flight from airport ''x'' to airport ''y''", then the transitive closure of ''R'' on ''X'' is the relation ''R''<sup>+</sup>: "it is possible to fly from ''x'' to ''y'' in one or more flights."

			== Transitive relations and examples ==

			A relation ''R'' on a set ''X'' is transitive if, for all ''x'',''y'',''z'' in ''X'', whenever ''x R y'' and ''y R z'' then ''x R z''. Examples of transitive relations include the equality relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "''x'' was born before ''y''" on the set of all people. Symbolically, this can be denoted as: if ''x'' < ''y'' and ''y'' < ''z'' then ''x'' < ''z''.

			One example of a non-transitive relation is "city ''x'' can be reached via a direct flight from city ''y''" on the set of all cities. Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to the third, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city ''x'' and ends at city ''y''". Every relation can be extended in a similar way to a transitive relation.

			==Existence and description==
			For any relation ''R'', the transitive closure of ''R'' always exists. To see this, note that the [[intersection (set theory)\|intersection]] of any [[indexed family\|family]] of transitive relations is again transitive. Furthermore, [[there exists]] at least one transitive relation containing ''R'', namely the trivial one: ''X'' × ''X''. The transitive closure of ''R'' is then given by the intersection of all transitive relations containing ''R''.

			For finite sets, we can construct the transitive closure step by step, starting from ''R'' and adding transitive edges.
			This gives the intuition for a general construction. For any set ''X'', we
			can prove that transitive closure is given by the following expression
			:<math>R^{+}=\bigcup_{i\in \{1,2,3,\ldots\}} R^i.</math>
			where <math>R^i</math> is the ''i''-th power of ''R'', defined inductively by
			:<math>R^1 = R\,\!</math>
			and, for <math>i>0</math>,
			:<math>R^{i+1} = R \circ R^i</math>
			where <math>\circ</math> denotes [[composition of relations]].

			To show that the above definition of ''R''<sup>+</sup> is the least transitive relation containing ''R'', we show that it contains ''R'', that it is transitive, and that it is the smallest set with both of those characteristics.

			* <math>R \subseteq R^{+}</math>: <math>\displaystyle R^+</math> contains all of the <math>\displaystyle R^i</math>, so in particular <math>\displaystyle R^+</math> contains <math>\displaystyle R</math>.
			* <math>\displaystyle R^{+}</math> is transitive: every element of <math>\displaystyle R^+</math> is in one of the <math>\displaystyle R^i</math>, so <math>\displaystyle R^+</math> must be transitive by the following reasoning: if <math>(s_1, s_2)\in R^j</math> and <math>(s_2, s_3)\in R^k</math>, then from composition's associativity, <math>(s_1, s_3)\in R^{j+k}</math> (and thus in <math>\displaystyle R^+</math>) because of the definition of <math>\displaystyle R^i</math>.
			* <math>\displaystyle R^{+}</math> is minimal: Let <math>\displaystyle G</math> be any transitive relation containing <math>\displaystyle R</math>, we want to show that <math>R^{+} \subseteq G</math>. It is sufficient to show that for every <math>i>0</math>, <math>R^i\subseteq G</math>. Well, since <math>\displaystyle G</math> contains <math>\displaystyle R</math>, <math>R^1\subseteq G</math>. And since <math>\displaystyle G</math> is transitive, whenever <math>R^i\subseteq G</math>, <math>R^{i+1}\subseteq G</math> according to the construction of <math>R^i\,\!</math> and what it means to be transitive. Therefore, by induction, <math>\displaystyle G</math> contains every <math>R^i\,\!</math>, and thus also <math>\displaystyle R^+</math>.

			== Properties ==
			The [[Intersection (set theory)\|intersection]] of two transitive relations is transitive.

			The [[union (set theory)\|union]] of two transitive relations need not be transitive. To preserve transitivity, one must take the transitive closure. This occurs, for example, when taking the union of two [[equivalence relation]]s or two [[preorder]]s. To obtain a new equivalence relation or preorder one must take the transitive closure (reflexivity and symmetry—in the case of equivalence relations—are automatic).

			==In graph theory==
			[[File:Transitive-closure.svg\|thumb\|right\|alt=Transitive closure constructs the output graph from the input graph.\|Transitive closure constructs the output graph from the input graph.]]
			In [[computer science]], the concept of transitive closure can be thought of as constructing a data structure that makes it possible to answer [[reachability]] questions. That is, can one get from node ''a'' to node ''d'' in one or more hops? A binary relation tells you only that node a is connected to node ''b'', and that node ''b'' is connected to node ''c'', etc. After the transitive closure is constructed, as depicted in the following figure, in an O(1) operation one may determine that node ''d'' is reachable from node ''a''. The data structure is typically stored as a matrix, so if matrix[1][4] = 1, then it is the case that node 1 can reach node 4 through one or more hops.

			The transitive closure of a [[directed acyclic graph]] (DAG) is the reachability relation of the DAG and a [[strict partial order]].

			== In logic and computational complexity ==
			The transitive closure of a binary relation cannot, in general, be expressed in [[first-order logic]] (FO).
			This means that one cannot write a formula using predicate symbols ''R'' and ''T'' that will be satisfied in
			any model if and only if ''T'' is the transitive closure of ''R''.
			In [[finite model theory]], first-order logic (FO) extended with a transitive closure operator is usually called '''transitive closure logic''', and abbreviated FO(TC) or just TC. TC is a sub-type of [[fixpoint logic]]s. The fact that FO(TC) is strictly more expressive than FO was discovered by [[Ronald Fagin]] in 1974; the result was then rediscovered by [[Alfred Aho]] and [[Jeffrey Ullman]] in 1979, who proposed to use fixpoint logic as a [[database query language]] (Libkin 2004:vii). With more recent concepts of finite model theory, proof that FO(TC) is strictly more expressive than FO follows immediately from the fact that FO(TC) is not [[Gaifman-local]] (Libkin 2004:49).

			In [[computational complexity theory]], the [[complexity class]] [[NL (complexity)\|NL]] corresponds precisely to the set of logical sentences expressible in TC. This is because the transitive closure property has a close relationship with the [[NL-complete]] problem [[STCON]] for finding [[directed path]]s in a graph. Similarly, the class [[L (complexity)\|L]] is first-order logic with the commutative, transitive closure. When transitive closure is added to [[second-order logic]] instead, we obtain [[PSPACE]].

			== In database query languages ==
			{{further\|Hierarchical and recursive queries in SQL}}
			Since the 1980s [[Oracle Database]] has implemented a proprietary [[SQL]] extension CONNECT BY... START WITH that allows the computation of a transitive closure as part of a declarative query. The [[SQL 3]] (1999) standard added a more general WITH RECURSIVE construct also allowing transitive closures to be computed inside the query processor; as of 2011 the latter is implemented in [[IBM DB2]], [[Microsoft SQL Server]], and [[PostgreSQL]], although not in [[MySQL]] (Benedikt and Senellart 2011:189).

			[[Datalog]] also implements transitive closure computations (Silberschatz et al. 2010:C.3.6).

			== Algorithms ==
			Efficient algorithms for computing the transitive closure of a graph can be found in Nuutila (1995). The simplest technique is probably the [[Floyd–Warshall algorithm]]. The fastest worst-case methods, which are not practical, reduce the problem to [[matrix multiplication]].

			More recent research has explored efficient ways of computing transitive closure on distributed systems based on the [[MapReduce]] paradigm (Afrati et al. 2011).

			== See also ==
			* [[Deductive closure]]
			* [[Transitive reduction]] (a smallest relation having the transitive closure of ''R'' as its transitive closure)
			* [[Symmetric closure]]
			* [[Reflexive closure]]
			* [[Ancestral relation]]

			== References ==
			* Lidl, R. and Pilz, G., 1998, ''Applied abstract algebra'', 2nd edition, Undergraduate Texts in Mathematics, Springer, ISBN 0-387-98290-6
			* Keller, U., 2004, ''[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.8266 Some Remarks on the Definability of Transitive Closure in First-order Logic and Datalog]'' (unpublished manuscript)
			* {{cite book\|author1=Erich Grädel\|author2=Phokion G. Kolaitis\|author3=Leonid Libkin\|coauthors=Maarten Marx, Joel Spencer, Moshe Y. Vardi, Yde Venema, Scott Weinstein\|title=Finite Model Theory and Its Applications\|year=2007\|publisher=Springer\|isbn=978-3-540-68804-4\|pages=151–152}}
			* {{citation\|first=Leonid\|last=Libkin\|title=Elements of Finite Model Theory\|year=2004\|publisher=Springer\|isbn=978-3-540-21202-7\|pages=}}
			* {{cite book\|author1=Heinz-Dieter Ebbinghaus\|author2=Jörg Flum\|title=Finite Model Theory\|year=1999\|publisher=Springer\|isbn=978-3-540-28787-2\|edition=2nd\|pages=123–124, 151–161, 220–235}}
			* {{cite doi\|10.1145/567752.567763}}
			* {{cite doi\|10.1007/978-1-4614-1168-0_10}}
			* Nuutila, E., [http://www.cs.hut.fi/~enu/thesis.html Efficient Transitive Closure Computation in Large Digraphs.] Acta Polytechnica Scandinavica, Mathematics and Computing in Engineering Series No. 74, Helsinki 1995, 124 pages. Published by the Finnish Academy of Technology. ISBN 951-666-451-2, ISSN 1237-2404, UDC 681.3.
			* {{cite book\|author1=Abraham Silberschatz\|author2=Henry Korth\|author3=S. Sudarshan\|title=Database System Concepts\|year=2010\|edition=6th\|publisher=McGraw-Hill\|isbn=978-0-07-352332-3}} [http://codex.cs.yale.edu/avi/db-book/db6/appendices-dir/c.pdf Appendix C] (online only)
			* Foto N. Afrati, Vinayak Borkar, Michael Carey, Neoklis Polyzotis, Jeffrey D. Ullman, [http://www.edbt.org/Proceedings/2011-Uppsala/papers/edbt/a1-afrati.pdf Map-Reduce Extensions and Recursive Queries], EDBT 2011, March 22–24, 2011, Uppsala, Sweden, ISBN 978-1-4503-0528-0

			== External links ==

			* "[http://www.cs.sunysb.edu/~algorith/files/transitive-closure.shtml Transitive closure and reduction]", The Stony Brook Algorithm Repository, Steven Skiena .

			[[Category:Mathematical relations]]
			[[Category:Closure operators]]
			[[Category:Graph algorithms]]

192.17.237.66: /* Multivariate differentiability classes */

2012-08-24T13:39:16Z

Multivariate differentiability classes

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Pleased to meet you! Simple name is Eusebio on top of that I think it to become quite good when say it. As a man what My hubby and i really like is conducting but I'm thinking towards starting something new. I work as an order clerk. My house is so now in South Carolina and I don't plan on changing it. You can find my website here: http://prometeu.net<br><br>Also visit my web blog :: [http://prometeu.net hack clash of clans no survey]

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en>Jesin: /* Definition */ removed unnecessary line breaks

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