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Topic: Cartesian product of sets

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	Cartesian product - Wikipedia, the free encyclopedia
		The Cartesian product is named after René Descartes whose formulation of analytic geometry gave rise to this concept.
		The assertion that the Cartesian product of arbitrary non-empty collection of non-empty sets is non-empty is equivalent to the axiom of choice.
		Categorically, the cartesian product is the product in the Category of sets.
	en.wikipedia.org /wiki/Cartesian_product (546 words)

	PlanetMath: generalized cartesian product
		The generalized Cartesian product is the product in the category of sets.
		The axiom of choice is the statement that the generalized Cartesian product of nonempty sets is nonempty.
		This is version 7 of generalized cartesian product, born on 2001-10-19, modified 2006-02-01.
	planetmath.org /encyclopedia/GeneralizedCartesianProduct.html (160 words)

	Product
		In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
		When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative.
		The dot product and cross product are forms of multiplication of vectors.
	www.ebroadcast.com.au /lookup/encyclopedia/pr/Product.html (142 words)

	CmSc 365 Theory of Computation (Site not responding. Last check: 2007-10-31)
		All sets are subsets of the universal set E.
		The Cartesian product of A and B is defined as the set
		A partition of a nonempty set A is a nonempty set
	storm.simpson.edu /~sinapova/cmsc365-02/L01-Sets.htm (930 words)

	Cartesian product - Wikipedia (Site not responding. Last check: 2007-10-31)
		In mathematics, given two sets X and Y, the Cartesian product (or direct product) of the two sets, written as X × Y is the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.
		For example, if set X is the 13-element set {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} and set Y is the 4-element set {spades, hearts, diamonds, clubs}, then the Cartesian product of those two sets is the 52-element set { , ,...
		Another example is the 2-dimensional plane R × R where R is the set of real numbers.
	wikipedia.findthelinks.com /ca/Cartesian_product.html (186 words)

	Logic, Sets and how they F.I.T.
		Sets have subsets, which are themselves sets comprising some of the elements of the original set.
		The Cartesian Product of two sets is the set of all pairs of elements taken one from each set.
		The powerset of a set is the set of all subsets of the set.
	www.sfu.ca /~robw/FoC/sets.html (685 words)

	[No title]
		The first argument should be the set which contains the elements of the set we wish to represent, the second argument should be a predicate, that is a function from the set to the booleans which describes if an element is to be in the set returned.
		The set {3, 6, 9} 3 6 9 intersect application This symbol is used to denote the n-ary intersection of sets.
		It is used to denote that the first set is a proper subset of the second, that is a subset of the second set but not actually equal to it.
	www.win.tue.nl /~amc/oz/om/cds/set1.html (1084 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}.
		The transitive closure of {1} under + is the set of positive integers.
		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 (1387 words)

	Immutable Cartesian Products
		The Cartesian product of two sets is the set of all possible pairs that can be formed by choosing one element from the first set and one element from the second set.
		For example, the Cartesian product of {1,2,3} and {a,b} is the set of pairs {(1,a), (2,a), (3,a), (1,b), (2,b), (3,b)}.
		We can think of the set of all Person objects as being the Cartesian product of the set of all names, the set of all ages, and the set of all heights.
	www.cs.utah.edu /classes/cs2010-zachary/book/product.html (1034 words)

	Cartesian Product (Site not responding. Last check: 2007-10-31)
		Cartesian product is an operator on sets (tuples, bags,...).
		The result from the cartesian product of n sets, is a set of all possible ordered tuples, containing on i-th place an element from i-th set.
		For example, the cartesian product of {a,b,c} and {c,d,e} is {(a,c),(a,d),(a,e),(b,c),(b,d),(b,e),(c,c),(c,d),(c,e)}.
	c2.com /cgi/wiki?CartesianProduct (263 words)

	PlanetMath: product topology
		The product topology is generally more useful than the box topology.
		In particular, any product of closed sets is closed.
		This is version 28 of product topology, born on 2002-06-12, modified 2006-01-09.
	planetmath.org /encyclopedia/ProductTopology.html (184 words)

	Untitled Document (Site not responding. Last check: 2007-10-31)
		A Cartesian product of two sets A and B is the set of all ordered pairs (a,b) where a ∈ A and b ∈ B. That is that if A={a
		The odds of winning becomes the number of different permutations you buy divided by the number of elements in the cartiesian product which is called the cardinality.
		The cardinality of this cartesian product is 60
	cse.unl.edu /~zharris/cartesian.html (153 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		The set of natural numbers or counting numbers is the set of positive integers and is denoted by N = Z + = {1, 2, 3, }.
		The set of integers is closed under the operations of addition, multiplication, and subtraction; but not under the operation of division.
		The set of counting numbers is closed under the operations of addition and multiplication but not under the operations of subtraction and division.
	www.upd.edu.ph /~stat/faculty/tgc/S117Ch1_2.doc (2327 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		If there are a small fixed number of sets the usual way to iterate over the Cartesian product is to write nested loops, one for each set.
		This module is useful when the numbers of sets is not known at the time the program is being written, or when the number of sets is "large", e.g.
		For efficiency this simply uses the references to the sets that are passed in, rather than making a "deep copy" of each one.
	home.comcast.net /~tolkin.family/CartesianProduct.pm (3115 words)

	Set Operations
		Sets can be combined in a number of different ways to produce another set.
		Then the Cartesian product of multiple sets is defined using the concept of n-tuple.
		The concept of Cartesian product can be extended to that of more than two sets.
	www.cs.odu.edu /~toida/nerzic/content/set/set_operations.html (524 words)

	Unit: Functions and Graphs (Site not responding. Last check: 2007-10-31)
		The sibling names being placed in set A while their corresponding brothers and sisters are placed in set B. These ideas will placed on the board and the concept of mapping diagrams will be illustrated as well as the definitions of Cartesian product and ordered pairs.
		The class will asked if the product of sets A x B would be equal in result to the Cartesian product of sets B x A. Then working individually, students will be assigned seatwork in which they will asked to construct five more examples of relations in their own lives.
		There will be five different sets of equipment where a group will either set up a pendulum and determine the effect of mass of the bob on the length of the swing, or will determine the effect of the string length on the length of the swing.
	plato.acadiau.ca /courses/educ/reid/up/Graphing-unit.htm (2472 words)

	Encyclopedia :: encyclopedia : Cartesian product (Site not responding. Last check: 2007-10-31)
		An n-tuple can be viewed as a function on {1, 2,..., n} that takes its value at i to be the ith element of the tuple.
		The assertion that the Cartesian product of a non-empty collection of non-empty sets is non-empty is equivalent to the axiom of choice.
		Categorically, the cartesian product is the direct product in the Category of sets.
	www.hallencyclopedia.com /Cartesian_product (566 words)

	SparkNotes: Functions: Sets and Relations
		In a given problem, two sets might be the scores of a class on one test, and the scores of the same students on another test.
		Given two sets, A and B, the set of all the possible ordered pairs in which the first element comes from A and the second element comes from B is called the Cartesian product A×B.
		The domain of the relation is the set D = {2, 3}, and the range is the set R = {14, 21}.
	www.sparknotes.com /math/precalc/functions/section1.html (453 words)

	Power Sets (Site not responding. Last check: 2007-10-31)
		True if enumerated set S is in the power set P, that is, if all elements of the set S are contained in or coercible into R, where P is the power set of R; false otherwise.
		True if indexed set S is in the power set P, that is, if all elements of the set S are contained in or coercible into R, where P is the power set of R; false otherwise.
		True if multiset S is in the power set P, that is, if all elements of the set S are contained in or coercible into R, where P is the power set of R; false otherwise.
	www.math.ufl.edu /help/magma/text140.html (451 words)

	Untitled
		We write the Cartesian product of, for example, sets A and B as A x B. You may be thinking, that looks like A times B. Well, in a way that's what it is. After all, it is called the Cartesian product.
		The intersection of sets A and B is a set contaning the elements that are in both A and B. This is written as A
		The union of sets A and B is a set contaning the elements that are either in A or B. This is written as A
	cse.unl.edu /~aebbeka/sets/p5.html (677 words)

	polyhedra.mathmos.net - Cartesian Product (Site not responding. Last check: 2007-10-31)
		This new set, the cartesian product, is usually denoted
		This definition may be extended in the obvious way to form the cartesian product of more than two sets.
		Its elements are those of the cartesian product of the sets of elements of the two original groups.
	polyhedra.mathmos.net /entry/cartesianproduct.html (63 words)

	Math 3000 Sample Exam II
		The Cartesian Product of sets A and B is...
		A partial function is a relation between sets A and B such that no two ordered pairs of the relation have the same first coordinate.
		The Cartesian Product of sets A and B is the set of all ordered pairs whose first coordinate is in A and whose second coordinate is in B. partition of a set S is a collection of non-empty subsets of S whose union is all of S and which are mutually disjoint.
	www-math.cudenver.edu /~wcherowi/courses/m3000/abexs2.html (870 words)

	Recursive Definitions of Sets (Site not responding. Last check: 2007-10-31)
		of the sets A and B is the union of two non-intersecting sets one of which is in 1-1 correspondence with A and the other with B.
		We shall denote the set of sequences of elements of A by seq(A).
		When sets are formed by this kind of recursive definition, the canonical mappings associated with the direct sum and Cartesian product operations have significance.
	www-formal.stanford.edu /jmc/basis1/node8.html (573 words)

	Untitled Document
		Each element of this cartesian product is an n-term sequence of real numbers called an n-tuple (or a vector).
		is the familiar cartesian plane from basic algebra, and the cartesian product results in a set of ordered pairs (2-tuples).
		This states that f maps the nth cartesian product of the set of real numbers into the oth cartesian product of the set of real numbers.
	www.sas.org /E-Bulletin/2003-04-25/mathCorner/body.html (658 words)

	Sets, Venn diagrams, Ghostbusters problems (Site not responding. Last check: 2007-10-31)
		VIP: Sets HW clarification and hints, animated GIF presentation, etc. The animation might be very useful.
		Sets, Cartesian Product (Cross Product) and subsets of...
		Sets, FRIENDS who went to same high school, principle of inclusion/exclusion and counting overlapping sets.
	www.cs.uni.edu /~jacobson/080/sets.html (239 words)

	Power Sets
		Return a set with universe R consisting of the elements of the set S, where P is the power set of R. An error results if not all elements of S can be coerced into R. S : PowSetIndx, SetIndx -> SetIndx
		Return an indexed set with universe R consisting of the elements of the set S, where P is the power set of R. An error results if not all elements of S can be coerced into R. S : PowSetMulti, SetMulti -> SetMulti
		, it is possible to create the Cartesian product of sets (or, in fact, of any combination of structures), but the result will be of type `Cartesian product' rather than set, and the elements are tuples -- we refer the reader to the Chapter on tuples for details.
	www.umich.edu /~gpcc/scs/magma/text169.htm (444 words)

	[No title]
		Aleksidze st., 1 380093 Tbilisi, Georgia sane@rmi.acnet.ge The paper introduces the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets.
		The chain complex of this twisted Cartesian product in fact is a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre.
		This construction is formalized as a theory of twisted tensor products for Hirsch algebras.
	www.lehigh.edu /~dmd1/h114.txt (844 words)

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