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Topic: Disjoint sets

	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.
		The data structure used is disjoint set forests with contribution from the path compression heuristic.
	www.cs.mcgill.ca /~cs251/OldCourses/1997/topic24 (1992 words)

	Disjoint sets - Wikipedia, the free encyclopedia
		In mathematics, two sets are said to be disjoint if they have no element in common.
		For example, {1, 2, 3} and {4, 5, 6} are disjoint sets.
		Formally, two sets A and B are disjoint if their intersection is the empty set, i.e.
	en.wikipedia.org /wiki/Disjoint_sets (204 words)

	Almost disjoint sets - Wikipedia, the free encyclopedia
		In mathematics, two sets are almost disjoint if their intersection is small in some sense.
		However, the unit interval [0, 1] and the set of rational numbers Q are not almost disjoint, because their intersection is infinite.
		Sometimes "almost disjoint" is used in some other sense, or in the sense of measure theory or topological category.
	en.wikipedia.org /wiki/Almost_disjoint_sets (438 words)

	PlanetMath: disjoint union
		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 (236 words)

	Part 6: Efficiently Representing Sets
		The set {1, 1, 1, 2, 3} and the set {1, 2, 3} are equivalent.
		When unioning together two disjoint sets, the correctness of the algorithm is not affected by which of the two sets is appended to the other.
		When unioning two sets, the set with the smaller rank is appended as a child of the root with the larger rank.
	msdn2.microsoft.com /en-us/library/Aa289153(VS.71).aspx (6105 words)

	Part 6: Efficiently Representing Sets
		The set {1, 1, 1, 2, 3} and the set {1, 2, 3} are equivalent.
		When unioning together two disjoint sets, the correctness of the algorithm is not affected by which of the two sets is appended to the other.
		When unioning two sets, the set with the smaller rank is appended as a child of the root with the larger rank.
	msdn.microsoft.com /library/en-us/dv_vstechart/html/datastructures_guide6.asp?frame=true (5831 words)

	Sets
		are sets and the objects in the set denoted by
		is a subset of the set denoted by
		Sets are mutually-disjoint if and only if there is no object that is a member of all of the sets.
	logic.stanford.edu /kif/hypertext/node23.html (511 words)

	Ontolingua Theory KIF-SETS
		A proper set is a set that cannot be a member of another set.
		Sets are pairwise-disjoint if and only if every set is disjoint from every other set.
		Sets are mutually-disjoint if and only if there is no object that is a member of all of the sets.
	www-ksl.stanford.edu /htw/dme/thermal-kb-tour/kif-sets.html (1034 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Recall that the general undirected graph G = (V, E, g): a 3-tuple whose components are the set of vertices, the set of edges, and the relation function.
		The worst case of Union() is where the set being appended X is the larger set; a larger than necessary number of elements are having to have their representative reset.
		Previously, the disjoint set was a linked list sequence of nodes where one node pointed to a next node and each node had a pointer field to its representative.
	ranger.uta.edu /~cook/aa/transcript/ln15f (1271 words)

	Station Information - Disjoint sets
		Given several sets, we say they are mutually disjoint or pairwise disjoint if any two of the sets in question are disjoint.
		For example, the sets {1,2,3}, {4,5,6}, and {7,8,9} are mutually disjoint.
		We also say that a set U whose elements are themselves sets is mutually disjoint if its members are mutually disjoint.
	www.stationinformation.com /encyclopedia/d/di/disjoint_sets.html (153 words)

	Sets (Site not responding. Last check: 2007-10-16)
		Important operations on disjoint sets are finding which set a given element belongs to, and creating a union of two sets (why is intersection not very important for disjoint sets?).
		That’s one of the objects that belongs to the set.
		If we represent sets as trees where the root is the representative and every node points to its parent while the root points to itself, and apply a couple of heuristics, we can get asymptotically faster disjoint-set implementation.
	www.owlnet.rice.edu /~comp314/05spring/lec/week5/Sets.htm (693 words)

	Disjoint sets -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-16)
		Formally, two sets A and B are disjoint if their (A junction where one street or road crosses another) intersection is (A container that has been emptied) empty, i.e.
		For example, the collection of sets is pairwise disjoint.
		A collection of sets is a (A vertical structure that divides or separates (as a wall divides one room from another)) partition of the set X if is a pairwise disjoint collection not containing the empty set, and if
	www.absoluteastronomy.com /encyclopedia/d/di/disjoint_sets.htm (201 words)

	[No title]
		DISJOINT SETS ============= A _disjoint_sets_ data structure represents a collection of sets that are _disjoint_: that is, no item is found in more than one set.
		The collection of disjoint sets is called a _partition_, because the items are partitioned among the sets.
		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.
	www.cs.berkeley.edu /~jrs/61bf06/lec/33 (1447 words)

	Lecture 16 for Math 232
		Furthermore, these two sets are disjoint, and (A TIMES {x}) has size m because it has a bijection with A (taking a to (a,x)).
		The set of bit strings of length n+1 is in a bijection with the direct product of the bit strings of length n with the set {0,1}, because for any string w we can map the pair (w,0) or (w,1) to the string w0 or w1.
		This is the union of the strings beginning with 0 and the strings ending with 1, but as these two sets are not disjoint in general we cannot use the sum rule directly (unless n < 2).
	www.mtholyoke.edu /courses/barring/232/lecture/16.htm (1210 words)

	Set theory
		The union of sets A and B is the set of all elements which belong to A, to B or to both.
		The intersection of sets A and B is the set of elements that are common to sets A and B. It is denoted by A
		The intersection ofsets A and B is the null set.
	library.thinkquest.org /C0126820/algebra.html (1608 words)

	Peter Suber, "Glossary of First-Order Logic"
		A property possessed by all the wffs in a set is logically hereditary iff the accepted rules of inference pass it on (transmit it) to all the conclusions derivable from that set by those rules.
		The set of all the subsets of a set.
		A set N is represented in a system iff there is some propositional function with exactly one free variable, Px, such that Px is a theorem whenever x is instantiated to a member of the set, and a non-theorem otherwise.
	www.earlham.edu /~peters/courses/logsys/glossary.htm (9715 words)

	Disjoint nondenumerable sets of irrationals
		We know the set of h functions is infinite and we presume that that set, and hence the set of f functions, is denumerable.
		Since g is aperiodic and cannot be indexed, we have established a nondenumerable set of irrationals, which we denote R. In general, we will write R_a to mean the set R derived from the rational q_a.
		Remark: It is curious that the set of writable functions is considered denumerable because each expression is composed of a finite string of symbols.
	angelfire.com /az3/nfold/qcomp.html (610 words)

	5.7.1 Set Objects
		Two sets are equal if and only if every element of each set is contained in the other (each is a subset of the other).
		A set is less than another set if and only if the first set is a proper subset of the second set (is a subset, but is not equal).
		A set is greater than another set if and only if the first set is a proper superset of the second set (is a superset, but is not equal).
	python.org /doc/current/lib/set-objects.html (182 words)

	Part 6: Efficiently Representing Sets
		The set of positive integers, for example, is an infinite set because there is no bounds to the number of elements in the set.
		is the set of real numbers, which are all rational numbers, plus irrational numbers as well (numbers that cannot be expressed as a fraction of two integers, such as pi, and the square root of 2).
		For instance, the set of positive integers has the same cardinality as the set of all integers, but the set of real numbers has a larger cardinality than the set of all integers.
	msdn.microsoft.com /library/en-us/dnvs05/html/datastructures20_6.asp (5766 words)

	Tuples of disjoint NP-sets — Institut für Informatik
		Disjoint NP-pairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity.
		In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k>1.
		Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable.
	www.informatik.hu-berlin.de /forschung/gebiete/algorithmenII/mitarbeiter/beyersdorff/Abstracts/dnpp06_2_html?month:int=10&year:int=2006 (177 words)

	Theorem 7.3.8: Properties of Lebesgue measure
		Lemma 1: Suppose O is a collection of sets such that the union of two elements and the complement of every element from O is again part of O (such a collection is called an Algebra of Sets).
		That countable intersections of measurable sets are measurable follows from de Morgan laws and because complements of measurable sets are measurable.
		By (1) intervals are measurable and by (3) countable unions of measurable sets are measurable.
	pirate.shu.edu /~wachsmut/ira/integ/proofs/propmsr.html (635 words)

	Hausdorff and Other Separation Axioms for Topological Spaces (Site not responding. Last check: 2007-10-16)
		The fully general, abstract topological spaces (S,T), where S is a set and T is a collection of subsets of S that is closed under arbitrary unions and finite intersections and includes both the null set ∅ and the whole set S, is of limited interest.
		Disjoint sets means that their closures are separate.
		For any pair of sets G and H such that neither have any points in common with the closure of the other there is a pair of disjoint open sets, one containing G and the other containing H. Some Theorems That Apply for the Various T
	www.applet-magic.com /separation.htm (285 words)

	Programming Assignment #4 (Site not responding. Last check: 2007-10-16)
		In an undirected graph, a connected component indicates a set of vertices in which each pair of vertices is connected by a path.
		One of them is to use the disjoint sets data structure.
		In this assignment, we implement a program to find the connected components for an undirected graph using a disjoint sets data structure.
	carbon.cudenver.edu /~kmlee/prog4.htm (148 words)

	2D Regularized Boolean Set-Operations
		The set depicted on the right is represented as a single polygon having a triangular outer boundary with a single triangluar hole in its interior - and not as three triangles that have no holes at all.
		The central class-template General_polygon_set_2 is used to represent point sets that are comprised of a finite number of general polygons with holes that are pairwise disjoint, and provides various Boolean set-operations on such sets.
		Note that as the four circles are disjoint, their union is computed with the insert method, while the union with the rectangles is computed with the join operator.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Boolean_set_operations_2/Chapter_main.html (4756 words)

	Disjoint critical sets in Latin squares - Adams, Bean, Khodkar (ResearchIndex) (Site not responding. Last check: 2007-10-16)
		Abstract: : A critical set in a Latin square is a subset of the Latin square containing just enough information to determine the complete Latin square.
		It has been conjectured that the smallest possible critical set in a Latin square is of size b (Update)
		A new bound on the size of the largest critical set in a..
	citeseer.ist.psu.edu /441794.html (485 words)

	[No title]
		The intersection of two disjoint disjoint sets is the empty set.
		Place the contrasting ones in a disjoint set and the similar ones in an intersection set.
		The range is the image set of a relation.Given the domain and rule of a relation to find the range and vice versa.
	www.sdnp.org.gy /ncerd/resources/maths/maths2.doc (3210 words)

	[No title]
		Each stage of separation of the elements in a set which is a node in the abstract predecessor tree results in the formation of a leaf node set and two sets which are parental to nodes in the next generation.
		The leaf subset is a subset of the parental set.
		Subsequent generations of the abstract tree result in further subsetting of the sets established higher in the tree; once subsets are established as disjoint at any level of the predecessor tree, there is no possibility that an overlap will develop at later generations.
	www-personal.ksu.edu /~kconrow/disjoint.html (530 words)

	Boost Disjoint Sets
		This is class that provides disjoint sets operations with union by rank and path compression.
		Each set is identified by a representative which is some member of of the set.
		Sets are represented by rooted trees which are encoded in the
	www.boost.org /libs/disjoint_sets/disjoint_sets.html (442 words)

	[No title]
		CS61B: Lecture 33 Wednesday, April 17 DISJOINT SETS ============= A _disjoint_sets_ data structure represents a collection of sets that are _disjoint_: that is, no item is found in more than one set.
		The solution is to maintain a separate table that maps root items to set names, and perhaps vice versa (depending on the application's needs).
		Often, however, we don't care about the name of a set at all; we only want to know if two items x and y are in the same set.
	www.cs.berkeley.edu /~jrs/61bs02/lec/33 (1490 words)

	ASPN : Python Cookbook : Grouping objects into disjoint sets (Site not responding. Last check: 2007-10-16)
		This recipe provides a lightweight way to group arbitrary objects together into disjoint sets when a full-blown graph data structure would be overkill.
		Objects can be joined using.join(), tested for connectedness using.joined(), and all disjoint sets can be retreived using.get().
		I've used this class to group vector objects on page images, and group sets of webpages by their links, but there are many other conceivable applications.
	aspn.activestate.com /ASPN/Cookbook/Python/Recipe/387776 (453 words)

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