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Topic: Disjoint union


  
  Disjoint union - Wikipedia, the free encyclopedia
In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint.
This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.
In the language of category theory, the disjoint union is the coproduct in the category of sets.
en.wikipedia.org /wiki/Disjoint_union   (339 words)

  
 PlanetMath: disjoint union   (Site not responding. Last check: 2007-11-07)
This is also often called being pairwise disjoint and is a much stronger condition than that the intersection of all the images is empty.
Of course, there are many categories where this usage is unnatural: In the category of pointed sets, the coproduct will be the disjoint union with the distinguished points identified.
This is version 4 of disjoint union, born on 2004-03-01, modified 2004-03-23.
planetmath.org /encyclopedia/DisjointUnion.html   (214 words)

  
 Disjoint union - Encyclopedia, History, Geography and Biography   (Site not responding. Last check: 2007-11-07)
In set theory, a disjoint union or discriminated union is a set union in which each element of the resulting union is disjoint from each of the others; the intersection over a disjoint union is the empty set.
The term is also used to refer to a modified union operation which indexes the elements according to which set they originated in, ensuring that the result is a disjoint union.
As mentioned, one can take the disjoint union of sets that are not in fact disjoint by using an indexing device.
www.arikah.net /encyclopedia/Discriminated_union   (292 words)

  
 Encyclopedia: Disjoint union   (Site not responding. Last check: 2007-11-07)
is a disjoint union if and only if for all A and B in C In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if.
The disjoint union of this family is the set In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set.
In computer science, a tagged union, also called a variant, variant record, or disjoint union, is a data structure used to hold a value that could take on several different, but fixed types.
www.nationmaster.com /encyclopedia/Disjoint-union   (856 words)

  
 [No title]
However, union operations are slow, because when two sets are united, we must walk through one set and relabel all the items so that they reference the other set.
The quick- union data structure comprises a _forest_ (a collection of trees), in which each item is initially the root of its own tree; then trees are merged by union operations.
Union is a simple O(1) time operation: we simply make the root of one set become a child of the root of the other set.
www.cs.berkeley.edu /~jrs/61bs02/lec/33   (1490 words)

  
 Disjoint Union (topology) Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-07)
In topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology.
The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e.
This shows that the disjoint union is the coproduct in the category of topological spaces.
www.karr.net /encyclopedia/Disjoint_union_%28topology%29   (526 words)

  
 Class notes CS251B -- Winter 1997
Disjoint sets structures are yet another new abstract data type (A.D.T.) for which we must now define a mathematical model and a series of related operations.
Disjoint sets, sometimes called partitions, are the model of this A.D.T. A partition is a collection of disjoint sets such that the union of the sets is the entire space.
When it is subject to a Union operation where its representative pointer changes, the size of the new set in which x now lives in is at least twice as large.
www.cs.mcgill.ca /~cs251/OldCourses/1997/topic24   (1992 words)

  
 Research Interests, Darren A. Narayan
Given a tournament that has a disjoint union of directed paths as a feedback arc set we establish necessary and sufficient conditions for this feedback arc set to have minimum size [1].
We show that if a tournament has a feedback arc set consisting of a disjoint union of directed paths, then this feedback arc set is minimum-sized if and only if the tournament does not contain one of two types of forbidden digraphs.
In [1] as a consequence of our classification of all tournaments having a disjoint union of directed paths as a feedback arc set, we obtained precise reversing numbers for all digraphs that are a disjoint union of directed paths.
www.rit.edu /~dansma/research.html   (1142 words)

  
 Disjoint union -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-07)
For example given and, which may have common elements, with union, the disjoint union as a subset of is the union of and.
The disjoint union of a collection of sets is often denoted
In the language of (Click link for more info and facts about category theory) category theory, the disjoint union is the (Click link for more info and facts about coproduct) coproduct in the (Click link for more info and facts about category of sets) category of sets.
www.absoluteastronomy.com /encyclopedia/d/di/disjoint_union.htm   (251 words)

  
 Disjoint union   (Site not responding. Last check: 2007-11-07)
In set theory, a disjoint union or discriminatedunion is a set union in which each element of theresulting union is disjoint from each of the others; the intersection over a disjoint union is the empty set.
The term is also used to refer to a modified union operation which indexes the elements according to which set they originatedin, ensuring that the result is a disjoint union.
In computerscience, this concept is important to construction of many datastructures and is implemented directly by tagged unions and algebraic data types.
www.therfcc.org /disjoint-union-69142.html   (166 words)

  
 No Title
When writing routines that operate on disjoint union types, we generally write conditionals, which check which kind of thing we've got, of the multiple kinds of things that count as members of the larger type.
Notice that the main structure of this definition is a disjoint union; the set of empty lists and the set of pairs don't overlap, because an empty list is never a pair and vice versa.
And within the branch for the recursive case, we have recursion for each part that is recursive in the definition--a binary infix-expr has two subexpressions that are also infix-expr's, so when we compute the value of a binary infix-expr, we must compute the values of those infix-exprs (the car and the caddr).
www.cs.utexas.edu /~bogo/cs307/asg07/asg07.html   (2410 words)

  
 Common Systems of Coset Representatives - Ashay Dharwadker
Thus G is the disjoint union of the left cosets of H.
Similarly, G is the disjoint union of the right cosets of H and lemma 1 below proves that G is the disjoint union of the double cosets of H.
Hence HgH is the disjoint union of the left cosets of H in G that are contained in HgH.
www.geocities.com /dharwadker/coset.html   (2321 words)

  
 Read about Disjoint union at WorldVillage Encyclopedia. Research Disjoint union and learn about Disjoint union here!   (Site not responding. Last check: 2007-11-07)
In set theory, a disjoint union (or discriminated union) is a union of a collection of
In this case one can show that the disjoint union of this family is the
In the language of category theory, the disjoint union is the coproduct in the
encyclopedia.worldvillage.com /s/b/Disjoint_union   (323 words)

  
 Disjoint - Data structures and algorithms for disjoint set union problems
Two sets are disjoint iff they share no members, ie iff their intersection is the null set.
The edge disjoint paths problem is, given an undirected graph G=(V,A) and K If we put all the uij to 1, the SCP problem becomes a shortest disjoint
For example, the sets A = {a,b,c} and B = {d,e,f} are disjoint.
www.vnyu.com /?q=disjoint   (180 words)

  
 Equivalence Relation
Equivalence relation is defined in a branch of mathematics called set theory, a vital branch underpinning all branches of mathematics and those fields that use mathematics.
The power of an equivalence relation lies in its ability to partition a set into the disjoint union of subsets called equivalence classes.
Because of its power to partition a set, an equivalence relation is one of the most used and pervasive tools in mathematics.
www.iscid.org /encyclopedia/Equivalence_Relation   (2634 words)

  
 The Disjoint Sum
Given two sets, A and B, we define their disjoint sum, A+B, to be the collection of symbols of form (a,) with a in A or of form (,b) with b in B. If A and B are ordered, we can extend their orders to A+B by deeming every (a,) to come before every (,b).
their intersection is empty), which is where the disjoint sum (or disjoint union) gets its name.
It should be noted that the standard embeddings of A and B in A+B compose with this to yield f and g as appropriate: in this sense, A+B is a minimal set via embeddings of A and B in which one may factorise any pair of functions, as above.
www.chaos.org.uk /~eddy/math/disjoint.html   (603 words)

  
 Disjoint Union   (Site not responding. Last check: 2007-11-07)
A   union operator represents another basic way of combining concepts.
In general if A and B are types, then so is their disjoint union, AB.
Semantically, not only is the union disjoint, but given an element of AB, it must be possible to decide which component it is in.
www.cs.cornell.edu /Info/Projects/NuPrl/book/node39.html   (171 words)

  
 DISJOINT UNION   (Site not responding. Last check: 2007-11-07)
Disjoint union In domain theory, a union (or sum) which results in a domain without a least element.
Source: compiled by the editor from various references; see credits.
To take the disjoint union of sets that are not in fact disjoint, one can use an indexing device.
www.websters-online-dictionary.org /di/disjoint+union.html   (263 words)

  
 Coproduct   (Site not responding. Last check: 2007-11-07)
The coproduct indexed by the empty set (that is, an empty coproduct) is the same as an initial object in C.
In the case of topological spaces coproducts are disjoint unions on the underlying sets, and the open sets are sets open in each of the spaces, in a rather evident sense (see disjoint union (topology)).
In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
www.worldhistory.com /wiki/C/Coproduct.htm   (515 words)

  
 Citations: A Linear-Time Algorithm for a Special Case of Disjoint Set Union - Gabow, Tarjan (ResearchIndex)   (Site not responding. Last check: 2007-11-07)
For details concerning the set union algorithms, see [19] Example 1 Consider a sample fa 1 a 2 a 2 a 2 a 3 ; a 1 a 2 a 4 g from a Szilard language of a regular grammar.
The operations allowed on R are a contraction of an edge of R that results in merging the sets corresponding to the two end points,....
The asymptotically fastest available algorithm is that of Micali and Vazirani [MV80] with a bound of O(p nm) Following the footsteps of Crocker [Cro93] and Mattingly and Ritchie[MR93] whose studies appeared on the First DIMACS Implementation Challenge workshop in 1991, we have been....
citeseer.ist.psu.edu /cs?q=dbnum=1,GID=86040,DID=0,start=50,cluster=none,qtype=context:   (788 words)

  
 Disjoint Union   (Site not responding. Last check: 2007-11-07)
The disjoint union operation is used to form a new domain from two existing domains.
The disjoint union of Id and Article contains pairs of values, the second item in each pair is a "1" if the first item came from Id and a "2" if the first item came from article.
Special functions exist, that given a value from one of the original domains can be used to form the equivalent value in the disjoint union domain.
scom.hud.ac.uk /scomtlm/cas810/notes/djoin.html   (378 words)

  
 Brainstorms: Virtues of Scientists
Because the arithmetical operation of addition corresponds to disjoint union of sets, and because the proposed experiment involves taking a disjoint union of sets - that's what adding objects to a set of other objects is necessarily about - addition qualifies as an irreducible category of perception with respect to it.
In other words, to add marbles to a set of other marbles is to create a disjoint union of sets with cardinality the sum of their cardinalities, and one can do this entirely in one’s head.
The methods of actually checking our claims carefully with observation in various forms has served very well -- that is why science is so successful and armchair philosophy is not so successful in subject matters of the actual workings of the actual physical world.
www.iscid.org /boards/ubb-get_topic-f-6-t-000337-p-5.html   (4134 words)

  
 Description of disjoint_union
disjoint_union(g,h,k) --- overwrite g with the disjoint union of the
disjoint_union(g,h,k) --- overwrite g with the disjoint union of the graphs h and k.
0001 function disjoint_union(g,h,k) 0002 % disjoint_union(g,h,k) --- overwrite g with the disjoint union of the 0003 % graphs h and k.
www.mts.jhu.edu /~ers/matgraph/html/matgraph/@graph/disjoint_union.html   (221 words)

  
 Disjoint Union   (Site not responding. Last check: 2007-11-07)
Disjoint union is like set union, but each element is ``tagged'' with the set that it originally came from.
The disjoint union of two types is a type defined by:
are predicates that tell which ``side'' of the union an element belongs to.
www.cs.cornell.edu /Info/Projects/NuPrl/cs611/fall94notes/cn4/paragraph3_1_1_2_2.html   (110 words)

  
 disjoint union - OneLook Dictionary Search
Tip: Click on the first link on a line below to go directly to a page where "disjoint union" is defined.
Disjoint Union : Eric Weisstein's World of Mathematics [home, info]
disjoint union : FOLDOP - Free On Line Dictionary Of Philosophy [home, info]
www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=disjoint+union   (104 words)

  
 [No title]   (Site not responding. Last check: 2007-11-07)
op union _ _ :: set, set -> set.
re variables of (~x and ~y) = union (variables of ~x, variables of ~y).
re variables of (~x and then ~y) = union (variables of ~x, variables of ~y).
www.brics.dk /~btools/ABACO/020315/languages.prj   (4182 words)

  
 On the Modularity of Termination of Term Rewriting Systems - Ohlebusch (ResearchIndex)   (Site not responding. Last check: 2007-11-07)
Abstract: It is well-known that termination is not a modular property of term rewriting systems, i.e., it is not preserved under disjoint union.
The objective of this paper is to provide a "uniform framework" for sufficient conditions which ensure the modularity of termination.
Whenever the disjoint union of two terminating term rewriting systems is non-terminating, then one of the systems is not C E -terminating (i.e., it looses its termination property when extended...
citeseer.csail.mit.edu /541.html   (613 words)

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