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Topic: Countable set

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	Countable set - Biocrawler
		In mathematics the term countable set is used to describe the size of a set, e.g.
		The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision, both from the beginning the distinguishing faculties of mathematicians.
		The set of real numbers is uncountable, and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument).
	www.biocrawler.com /encyclopedia/Countable_set (1054 words)

	Reference.com/Encyclopedia/Countable set
		In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers.
		Note that countable set is sometimes given an alternate definition: a set with as many elements as the set of natural numbers.
		The set of real numbers is uncountable (see Cantor's first uncountability proof), and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument).
	www.reference.com /browse/wiki/Countable (1946 words)

	Countable set
		In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers.
		Note that countable set is sometimes given a more specific definition: sometimes, it is defined as a set with the same cardinality as the set of natural numbers.
		The set of real numbers is uncountable, and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument).
	www.brainyencyclopedia.com /encyclopedia/c/co/countable_set.html (1602 words)

	NationMaster - Encyclopedia: Separable space
		In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
		In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
		The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number.
	www.nationmaster.com /encyclopedia/Separable-space (2087 words)

	Science Fair Projects - Countable set
		In mathematics the term countable set is used to describe the size of a set, e.g.
		The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision, both from the beginning the distinguishing faculties of mathematicians.
		However, not all sets are finite: for instance, the set of all integers or the set of all real numbers.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Countable (1188 words)

	NationMaster - Encyclopedia: Choice function
		In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range).
		The axiom of countable choice or axiom of denumerable choice is an axiom of set theory similar to the axiom of choice.
		A choice function is a mathematical function f whose domain X is a collection of nonempty sets such that for every S in X, f(S) is in S. A weaker form of axiom of Choice, the axiom of countable choice (CC) states that every countable set of nonempty sets has a choice function.
	www.nationmaster.com /encyclopedia/Choice-function (883 words)

	countable@Everything2.com
		When dealing with infinite sets, the way to define the property of "having the same number of elements" is to look for a bijection (one to one correspondance) between the two sets.
		The algebraics: this is the set of roots of polynomial equations with integer coefficients (or equivalently rational coefficients).
		Sets like the set of all integers or the set of all positive integers that are both infinite and enumerable are called enumerably infinite sets.
	www.everything2.com /index.pl?node=countable (1709 words)

	countable set
		A set that is either finite or countably infinite.
		A countably infinite set is one that can be put in one-to-one correspondence with the natural numbers and thus has a cardinal number ("size") of aleph-null.
		Examples of countable sets include the set of all people on Earth and the set of all fractions.
	www.daviddarling.info /encyclopedia/C/countable_set.html (79 words)

	Countable set Summary
		A set, which should be thought of simply as a collection of objects, is called countable if it either consists of a finite number of objects or if the objects can be put in one to one correspondence with the integers.
		Thus infinite sets that are countable have the same cardinality as the integers.
		In this situation the set theoretic notion of cardinality is an important way to think about the size of infinite sets and the notion of countability captures the idea that an infinite set has the same size as the set of integers.
	www.bookrags.com /Countable_set (2497 words)

	Set Theory (Stanford Encyclopedia of Philosophy)
		Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.
		For instance, it is desirable to have the “set of all integers that are divisible by number 3,” the “set of all straight lines in the Euclidean plane that are parallel to a given line”, the “set of all continuous real functions of two real variables” etc.
		Cantor's discovery of uncountable sets led him to the subsequent development of ordinal and cardinal numbers, with their underlying order and arithmetic, as well as to a plethora of fundamental questions that begged to be answered (such as the Continuum Hypothesis).
	plato.stanford.edu /entries/set-theory (3292 words)

	Sets : Software Foundations : Thomas Alspaugh : UCI
		For example, {1,2,3} is the set whose elements are 1, 2, and 3; the extension of "the positive integers no greater than 3" is {1,2,3}.
		We can also use diagonalization to show that the powerset of a countable set is not countable.
		We avoid Russell's Paradox by restricting sets to be constructed only from sets that already exist (specifically, when naming a set by intension, we require that its elements be drawn from some other already-existing set E).
	www.ics.uci.edu /~alspaugh/foundations/set.html (1486 words)

	PlanetMath: countable basis
		The archetypical example of a countable basis is the Fourier series of a function: every continuous real-valued periodic function
		Note: A countable basis is a countable set, but it is not usually a basis.
		This is version 4 of countable basis, born on 2002-01-07, modified 2002-06-20.
	www.planetmath.org /encyclopedia/CountableBasis.html (132 words)

	countable - Search Results - MSN Encarta
		Finite, limited or having a countable number of elements, the opposite of infinite.
		Countable and uncountable nouns and quantifier expressions - Grammar...
		OWL at Purdue University: Adjectives with Countable and Uncountable...
	encarta.msn.com /countable.html (188 words)

	All Elementary Mathematics - Study Guide - Sets - Basic notions. Examples of sets...
		A set and an element of a set concern with category of primary notions, for which it's impossible to formulate the strict definitions.
		For instance, a set of books in a library, a set of cars on a parking lot, a set of stars in the sky, a world of plants, a world of animals – these are examples of sets.
		An uncountable set is a set, elements of which can't be numbered.
	www.bymath.com /studyguide/sets/sec/sets1.htm (417 words)

	Countable - Remarkably It Works., Are All Infinite Sets Countable?
		An infinite set of numbers, points, or other elements is said to be "countable" (also called denumerable) if its elements can be paired one-to-one with the natural numbers, 1, 2, 3, etc. The term countable is somewhat misleading because, of course, it is humanly impossible actually to count infinitely many things.
		The set of even numbers is an example of a countable set, as the pairing in Table 1 shows.
		One might guess that it is uncountable because the set of natural numbers is a proper subset of it.
	science.jrank.org /pages/1841/Countable.html (388 words)

	Prove the following statements
		A countable union of countable sets is countable.
		The set of polynomials with integer coefficients is countable.
		Suppose A is an unknown set and B = { 7 }.
	www.math.toronto.edu /jkorman/Math246Y/problemsincardinality.htm (351 words)

	Countable set
		The difference between the two definitions is that the former defines finite sets to be countable, while the latter does not.
		Technically, a countably infinite set is any set which, in spite of its boundlessness, can be shown equinumerous to the natural numbers â” nothing more, nothing less.
		This makes it possible to set apart elements of a countably infinite set using natural numbers as indices, and in turn puts the logic associated with them in very close proximity to the logic associated with the natural numbers themselves; and this makes such sets easily logically tractable.
	www.dejavu.org /cgi-bin/get.cgi?ver=93&url=http%3A%2F%2Farticles.gourt.com%2F%3Farticle%3Dcountable%26type%3Den (1563 words)

	COUNTABLE - Definition
		A term describing a set which is isomorphic to a subet of the natural numbers.
		If the isomorphism is stated explicitly then the set is called "a counted set" or "an enumeration".
		Examples of countable sets are any finite set, the natural numbers, integers, and rational numbers.
	www.hyperdictionary.com /dictionary/countable (64 words)

	PlanetMath: countable
		See Also: proof that the rationals are countable, Cantor set
		This is version 2 of countable, born on 2001-11-16, modified 2002-02-27.
		Object id is 880, canonical name is Countable.
	planetmath.org /encyclopedia/Countable.html (49 words)

	Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)
		Sets A and B have the same cardinality if there is a one-to-one function f with domain A and range B.
		An infinite subset of a countable set is countable.
		The set of all integers Z and the set of all rational numbers Q are countable.
	plato.stanford.edu /entries/set-theory/primer.html (2793 words)

	[No title]
		If A is a countable set, and B is a finite set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Answer.
		If A is a countable set, and B is an uncountable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Comment.
		If A is a countable set, and B is a countable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Answer.
	www.math.ucla.edu /~tao/java/MultipleChoice/countable.txt (804 words)

	Cantor, Georg - Famous mathematicians pictures, posters, gifts items, note cards, greeting cards, and prints
		He distinguished between countable and uncountable sets, and was able to prove that the set of all rational numbers Q is countable, while the set off all real numbers R is uncountable, and therefore, though both were infinite, R was strictly larger.
		The set that remains after continuing this process forever is called the Cantor set.
		The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set, to which color was applied, and then universal operators related to color transition and magnification, ultimately resulting in a unique image whose essence was the Cantor set.
	mathematicianspictures.com /Mathematicians/Cantor.htm (509 words)

	Countable (Site not responding. Last check: )
		If a set can be put in one-to-one correspondence with any countable set, then both sets are countable.
		Construction -- Construction of sets of numbers, starting with the original Peano Axioms, formulated so that zero is included in the set of natural numbers.
		Set Theory -- an introduction to sets, including examples of some standard sets.
	mcraefamily.com /mathhelp/CountingIntro.htm (357 words)

	Perfect Subset Property for co-analytic sets in ZF\P
		A perfect set is the set of all paths through a perfect tree, and can be coded by such perfect tree.
		The only uses of uncountable sets in the standard proof are routine manipulations of projective sets, so the proof goes through (with syntactic modifications) inside ZF\P. Consider a model M of ZF\P. The notions in this paragraph are assumed to be relativized to M.
		The constructible sets under the membership relation form a model L of ZFC\P. Suppose that every set is countable and perfect subset property holds for co-analytic sets.
	web.mit.edu /dmytro/www/other/PerfectSubsetsAndZFC.htm (917 words)

	Countable and Uncountable Sets
		A set is countably infinite if it is countable and infinite, just like the positive integers.
		In fact all finite ordered tuples of the integers, or any other countable set for that matter, are countable.
		As a corollary, the finite sets of integers are countable, as these are all represented, perhaps many times, by various ordered tuples.
	www.mathreference.com /set-card,cable.html (718 words)

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