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• Correspondence
  ...tant in [[set theory]] when dealing with arguments concerning [[infinite]] sets.
  2 KB (289 words) - 17:17, 13 February 2009
• Bijection
  ...tant in [[set theory]] when dealing with arguments concerning [[infinite]] sets or in permutation and probability.
  1,016 bytes (141 words) - 03:39, 29 November 2021
• Cardinality
  '''Cardinality''' is a property of [[set]]s. For [[finite]] sets, the cardinality of is the number of [[element]]s in that set, i.e. the siz ==Infinite==
  2 KB (263 words) - 00:54, 17 November 2019
• 2002 IMO Shortlist Problems/C5
  ...e integer, and let <math> \displaystyle F </math> be an infinite family of sets, each of size <math> \displaystyle r </math>, no two of which are disjoint. ...math> such that <math> A \cup \{x\} </math> is a subset of infinitely many sets in <math> \displaystyle F </math>.
  4 KB (681 words) - 20:10, 19 April 2007
• Countable
  ...divided between those which are [[finite]] and those which are countably [[infinite]]. The name "countable" arises because the countably infinite sets are exactly those which can be put into [[bijection]] with the [[natural nu
  773 bytes (126 words) - 21:35, 26 September 2008
• Permutation
  One can also consider permutations of [[infinite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijecti
  3 KB (422 words) - 11:01, 25 December 2020
• Uncountable
  ...every set that is ''not'' uncountable is either [[finite]] or [[countably infinite]]. The most common example of an uncountable set is the set of [[real numb An alternative proof uses [[Cantor's Theorem]], which says that for all sets <math>S</math>, we have <math>|S|<|\mathcal{P}(S)|</math>, where <math>\mat
  2 KB (403 words) - 20:53, 13 October 2019
• Set
  ...we shall present just a brief discussion of the most common properties of sets and operations related to them. ...he set, in which case it is called a finite set. Otherwise we call it an [[infinite]] set. The objects in a set are called the [[element]]s of the set. A commo
  11 KB (2,021 words) - 00:00, 17 July 2011
• 2005 AIME II Problems
  ...6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> An infinite geometric series has sum 2005. A new series, obtained by squaring each term
  7 KB (1,119 words) - 21:12, 28 February 2020
• Finite
  ...cally) be named, one by one, in a finite amount of time is finite. Finite sets include the [[empty set]], which has zero elements, and every set with a [[ * [[Infinite]]
  532 bytes (92 words) - 10:11, 7 July 2006
• 2002 AIME II Problems
  ...isjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when <math>n</ma Two distinct, real, infinite geometric series each have a sum of <math>1</math> and have the same second
  7 KB (1,177 words) - 15:42, 11 August 2023
• Ring
  ...d therefore also constitute examples of [[field]]s. All these rings are [[infinite]], as well. Among the finite commutative rings are sets of integers mod <math>m</math> (<math>\mathbb{Z}/m\mathbb{Z}</math>), for a
  6 KB (994 words) - 06:16, 8 April 2015
• Power set
  ...The analogous result is also true for [[infinite]] sets (and thus for all sets): for any set <math>S</math>, the cardinality <math>|\mathcal P (S)|</math> ==Size for Finite Sets==
  4 KB (757 words) - 11:44, 8 March 2018
• Maximum
  ...ordered set]] such as the reals has a maximum. However, many [[infinite]] sets do not. The [[integer]]s, <math>\mathbb Z</math> have no maximum, since fo
  1 KB (232 words) - 12:55, 9 February 2007
• Polyhedron
  ...]]. Polyhedra are 3-D analogues of [[polygon]]s. They can be thought of as sets of [[ordered]] triples. ...exist. (In addition, a [[sphere]] could be thought of a polyhedron with an infinite number of faces.)
  1 KB (193 words) - 23:15, 6 December 2016
• 2007 USAMO Problems
  ...<math>n</math> are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family h ...wo classes. Prove that there are at least <math>n</math> pairwise disjoint sets in the same class.
  3 KB (539 words) - 13:42, 4 July 2013
• Zermelo-Fraenkel Axioms
  ...ll's Paradox]], some restrictions must be put on which collections to call sets. This axiom establishes the most basic property of sets - a set is completely characterized by its elements alone. <br/>
  4 KB (732 words) - 20:49, 13 October 2019
• 1951 AHSME Problems
  <math> \textbf{(A) \ } \text{Infinite} \qquad\textbf{(B) \ } 5\frac{1}{4}a \qquad \textbf{(C) \ } 2a \qquad\textb The limit of the sum of an infinite number of terms in a geometric progression is <math> \frac {a}{1- r}</math>
  23 KB (3,641 words) - 22:23, 3 November 2023
• Symmetric group
  ...at in general this theorem is not true on infinite sets without permitting infinite compositions of functions.
  10 KB (1,668 words) - 15:33, 25 May 2008
• Ultrafilter
  ...or ''free'', or ''non-principle''. Evidently, the only filters on finite sets are trivial. ...th>A</math>. Since <math>\mathcal{F}</math> is an ultrafilter, one of the sets <math>\{ a\}</math>, <math>X \setminus \{a\}</math> must be an element of <
  9 KB (1,685 words) - 20:28, 13 October 2019

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