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Topic: Singleton set

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	NationMaster - Encyclopedia: Singleton set
		Note that a set such as {{1,2,3}} is also a singleton: the only element is a set (which itself is however not a singleton).
		In axiomatic set theory, the existence of singletons is a consequence of the axiom of empty set and the axiom of pairing: the former yields the empty set {}, and the latter, applied to the pairing of {} and {}, yields the singleton {{}}.
		If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the one element of S.
	www.nationmaster.com /encyclopedia/Singleton-set (447 words)

	Singleton (mathematics) - Wikipedia, the free encyclopedia
		Note that a set such as {{1,2,3}} is also a singleton: the only element is a set (which itself is however not a singleton).
		In axiomatic set theory, the existence of singletons is a consequence of the axiom of empty set and the axiom of pairing: the former yields the empty set {}, and the latter, applied to the pairing of {} and {}, yields the singleton {{}}.
		If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the one element of S.
	en.wikipedia.org /wiki/Singleton_set (292 words)

	Math Notes - Relations
		Suppose that the element ‘a’ is a member of the universal set U. The set {a} is a singleton set. A singleton set represents a set such that all members of the set equal the same element. Thus, a singleton set may be defined as the following axiom.
		Some properties of singleton sets are not generally true for all sets. Theorem 3 is an equivalence expression for singleton sets, but only an implication if generalized to all sets.
		Theorem 3: {a} Í P Ú {a} Í Q º {a} Í (P È Q) A singleton set is a member of a set or the set’s complement. This property is not generally true of sets with two or more elements.
	home.att.net /~p.konieczko/mathsing1.html (312 words)

	Singleton (mathematics) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: )
		If A is any set and S is any singleton, then there exists precisely one (A mathematical relation such that each element of one set is associated with at least one element of another set) function from A to S, the function sending every element of A to the one element of S.
		Any singleton can be turned into a ((mathematics) any set of points that satisfy a set of postulates of some kind) topological space in just one way (all subsets are open).
		Any singleton can be turned into a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group in just one way (the unique element serving as (An operator that leaves unchanged the element on which it operates) identity element).
	www.absoluteastronomy.com /encyclopedia/s/si/singleton_(mathematics).htm (397 words)

	Drugging for Death, Singleton, death penalty, forced drugging
		Singleton then petitioned the district court for a stay of execution and writ of habeas corpus, raising claims including ineffective assistance of counsel, use of invalid aggravating factors, and that he was incompetent and thus ineligible for execution under Ford v.
		Singleton went on to tell Dr. Mrad that he was penning a book at the request of God, that he and St. John were on a mission to fight homosexuals, and that Sylvester Stallone and Arnold Schwarzenegger were somewhere between this universe and another one and were trying to save him.
		Singleton was examined at Springfield in the summer of 2000, and the [*56] district court used this evaluation in making its findings about his mental condition and the status of his treatment.
	www.cognitiveliberty.org /dll/singleton_8circ2.htm (10325 words)

	PBS - THE WEST - Benjamin "Pap" Singleton
		Singleton was born in 1809 in Nashville, Tennessee, where he was several times sold as a slave but always managed to escape.
		Those who answered Singleton's call to head west became known as "Exodusters," and Singleton himself was described as the "Father of the Exodus." But the massive migration of African Americans from the South that reached a peak in 1879 was not inspired by Singleton alone.
		In 1880, Singleton was called to testify at Congressional hearings on the alarming migration of fls from the South.
	www.pbs.org /weta/thewest/people/s_z/singleton.htm (619 words)

	Set (Java 2 Platform SE 5.0)
		The behavior of a set is not specified if the value of an object is changed in a manner that affects equals comparisons while the object is an element in the set.
		A special case of this prohibition is that it is not permissible for a set to contain itself as an element.
		The hash code of a set is defined to be the sum of the hash codes of the elements in the set, where the hashcode of a
	java.sun.com /j2se/1.5.0/docs/api/java/util/Set.html (1608 words)

	Data.Set
		If the set already contains an element equal to the given value, it is replaced with the new value.
		The union of two sets, preferring the first set when equal elements are encountered.
		Partition the set into two sets, one with all elements that satisfy the predicate and one with all elements that don't satisfy the predicate.
	www.haskell.org /ghc/dist/current/docs/libraries/base/Data-Set.html (422 words)

	[No title]
		The sets {a,b}, {a,c}, and {b,c} can be "fused" into the set {a,b,c} which is the smallest set which has all of them as "parts" (subsets), but the "fusion" of {a,b} and {a,c} gives the same result.
		My suggestion is that the role of the singleton construction is _semantic_; a singleton is a kind of _name_ for its element; a way of putting it which is perhaps better is that the singleton is a _token_ used to replace its element in the construction of sets.
		This formulation of the notion of "set" avoids (and perhaps illuminates) Russell's paradox: Let R be the "set of all sets which are not elements of themselves".
	math.boisestate.edu /~holmes/holmes/fomletter1.txt (1642 words)

	Singleton (Site not responding. Last check: )
		The Singleton pattern ensures that a class has only one instance and provides a global point of access to that instance.
		It is named after the singleton set, which is defined to be a set containing one element.
		The office of the President of the United States is a Singleton.
	home.earthlink.net /~huston2/dp/singleton.html (571 words)

	T1 space (Site not responding. Last check: )
		In other words, the singleton sets {x} and {y} are separated unless x = y.
		Given any x in X, the singleton set {x} is a closed set.
		Alternatively, the set of even integers is compact but not closed, which would be impossible in a Hausdorff space.
	www.bidprobe.com /en/wikipedia/t/t1/t1_space.html (820 words)

	Sets and formal languages
		The empty set is a subset of every set, including itself, though of course it is not a proper subset of itself.
		Singleton subsets of a given set are distinct from the members of that set.
		For example, a is a member of the set S = {a, b, c}, whereas the singleton set {a} is a subset of that set; that is a
	minimalism.linguistics.arizona.edu /~langendoen/LING501/LING501set.htm (745 words)

	Glossary « WordPress Codex
		A character set is a collection of symbols (letters, numbers, punctuation, and special characters), when used together, represent meaningful words in a language.
		A.htaccess file is a granular configuration file for the Apache web server software, used to set or alter the server's configuration settings for the directory in which it is present, and/or its child directories.
		Singletons are objects that exist globally and are the only one of their class; if one instance of a class is created per request, this instance is a singleton object.
	codex.wordpress.org /Glossary (5966 words)

	[No title] (Site not responding. Last check: )
		SET MEMBERSHIP Set membership is denoted by EMBED Equation.3 which is read as EMBED Equation.3 is a member of, or is an element of, or belongs to e.g.
		TYPED SETS Recall our (informal) definition of a set:A set is a well-defined, unordered collection of similar items where each item is clearly identifiable and distinct from the other items Well-defined means that given a ‘value’ we are able to decide whether it is a member of the set.
		Sets are homogeneous in the sense that all members of a set are in some way similar All possible values that a set may have as members is said to define the type of the set.
	www.eng.nene.ac.uk /~scott/SETS.doc (825 words)

	High Performance Debugging Forum: Process Control Working Group (Site not responding. Last check: )
		,if present,must be a singleton set; if not present,the current process/thread set must be a singleton set.
		If the target-set is not a singleton set, the operation fails, producing the error message: "???".
		Set the signal handling behavior for each process/thread in the current process/thread set for signal 2 to stop the process/thread and print a message.
	www.ptools.org /ptools_www/hpdf/meetings/jun97/wgs/proc/working.3.0.html (6598 words)

	Glossary
		Set relational functions are true if the relation in question is true for the endpoints of the intervals.
		Set theoretic is the means of or pertaining to the algebra of sets.
		supremum of a set of numbers is the set's least upper bound, which is either the largest number in the set or the smallest number that is greater than all the numbers in the set.
	docs.sun.com /source/817-5077/iapgCglossary.html (1481 words)

	Singleton era set to begin in MU softball
		Missouri softball Coach Ty Singleton said his top goal is for the Tigers to improve from Day One to the end of the season.
		Singleton, in his first season at MU, takes over a team that went 29-33 last season and lost in the first round of the Big 12 Tournament as the No. 9 seed.
		Singleton was 75-45 in two seasons as head coach at Northwestern State in Louisiana and spent four years as an assistant coach at Texas.
	www.showmenews.com /2003/Feb/20030206Spor007.asp (695 words)

	Sets
		As usual, we assume that all of the elements of a set are of the same data type, which is part of the definition of the type of the set.
		Some axiomatizations of set theory provide for a complementation operation constructs the set of all values that are not members of a given set.
		Sundering is a generalization of this idea to give the set of all members of a set that meet some condition that is expressed as a Boolean function.
	www.math.grin.edu /~stone/courses/fundamentals/sets.html (2381 words)

	Feature Structures
		Sets and bags are thus distinguished from lists in that the order in which the values are specified does not matter for the former, but does matter for the latter, while sets are distinguished from bags and lists in that repetitions of values do not count for the former but do count for the latter.
		The set, bag or list which has no members is known as the null (or empty) set, bag or list.
		Finally, the fourth example states that the ‘extras’ feature has as its value a set of which the null set is not a subset.
	www.tei-c.org /P4X/FS.html (9510 words)

	sciforums.com - The First Axioms of Complementary Set Theory
		If you mean the set of integers, then the set has cardinality 4, which is obviously greater than the cardinality of the set of two integers {3,4}.
		A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
		A singleton set is denoted by {a} and is the simplest example of a nonempty set.
	www.sciforums.com /showthread.php?t=39203 (4673 words)

	Solution 4 (Site not responding. Last check: )
		Thus since every singleton is open and any subset A is the union of all the singleton sets of points in A we get the result that every subset is open.
		The closed sets are the finite sets {1, 2, 3,...
		The union of two countable sets is countable and since the intersection of (arbitrarily many!) countable sets is a subset of any one of them, it too is countable.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Solutions/S4.html (414 words)

	ipedia.com: Cofinite Article (Site not responding. Last check: )
		In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set.
		The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operations of union, intersection, and complementation.
		In this case, the non-principal ultrafilter is the set of all cofinite sets.
	www.ipedia.com /cofinite.html (339 words)

	The Foundations of a Non-Naive Mathematics - Physics Help and Math Help - Physics Forums (Site not responding. Last check: )
		If a content of a set is a singleton and a urelement and has no directions, then it is a point.
		If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
		An answer to the request would look something like: if S is the set of reals as defined by you then implies it is the set of reals accoridng to the proper definitions, conversely, if you take the proper definitions then which implies my construction.
	www.physicsforums.com /showthread.php?t=31836 (2103 words)

	CS 212 Section Notes (02/23/98)
		One approach is to filter out all of the duplicates in the first set before appending the two sets together.
		The definition also specifies that values contain one slot (also known as an instance variable or field).
		For example, we can define empty-set and singleton as follows: (define (empty-set ) (make elements: '())) (define (singleton ) (method ((i )) (make elements: (cons i '())))) To define member, we need some way to get at the list of elements in a set.
	www.cs.cornell.edu /courses/cs212/1998sp/sections/sec0223.html (868 words)

	singleton - OneLook Dictionary Search
		Singleton, singleton : Encarta® World English Dictionary, North American Edition [home, info]
		Singleton, singleton : LookWAYup Translating Dictionary/Thesaurus [home, info]
		Phrases that include singleton: john singleton mosby, hoffman singleton graph, lyndhurst john singleton copley baron, singleton set, alexander singleton, more...
	www.onelook.com /?w=singleton&ls=a (270 words)

	Monoid Article, Monoid Information (Site not responding. Last check: )
		The set of all n by n matrices over a given ring, with matrix addition or matrix multiplication as the operation.
		The set of all finite strings over some fixed alphabet Σ forms a monoid with string concatenation as the operation.
		The axiomsrequired of a monoid operation are exactly those required of morphism compositionwhen restricted to the set of all morphisms which start and end at a given object (i.e.
	www.anoca.org /operation/group/monoid.html (1069 words)

	Re: Web application licenses
		I'm saying that the GPL has the same >>clause as the hypothetical license would, which is true by definition, >>since in the hypothetical license, I incorporated the GPL clause in >>question (3) by reference; I did not intend to suggest that "on the same >>medium" was the only option.
		There would be a full set of clauses >>that would parallel GPL clauses 3 a-c, so one could either accompany the >>binary with source, or distribute an offer.
		Sets of articles are interesting on their own; often, a > singleton set is interesting.
	lists.debian.org /debian-legal/2004/10/msg00091.html (4304 words)

	[No title] (Site not responding. Last check: )
		The set is implemented using pointer based linked list.
		Comment: The set is done as a template class.
		Else the operation fails and // the set is not changed.
	www.cse.fau.edu /~roy/cop3530.98f/programs/set.h (1807 words)

	Zermelo-Frankel Set Theory
		Thus the set that the Axiom of Infinity declares to exist is the set:
		However, recall that we are also using sets to model the natural numbers.
		The idea is that if P(x,y) corresponds to a function from members of a given set to objects, then we can form a new set by replacing every member of the given set with the object the function maps it to.
	www.trinity.edu /cbrown/topics_in_logic/sets/node4.html (514 words)

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