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• Correspondence
  ...tant in [[set theory]] when dealing with arguments concerning [[infinite]] sets.
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• 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
• Topological space
  ...plements are elements of <math>\mathcal{T}</math> are then called ''closed sets''. ...of closed sets is closed, and that the intersection of any family of open sets is open.
  6 KB (1,142 words) - 15:38, 21 June 2008
• 2008 iTest Problems
  ...ploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of <math>\dfrac1{r_1^{2008}}+\df ...umber less than <math>2008</math> that is a divisor of some integer in the infinite
  71 KB (11,749 words) - 01:31, 2 November 2023
• Georg Cantor
  ...et of rationals. He first introduced the idea of [[cardinality of infinite sets]] using the symbol <math>\aleph</math> to represent the size. Because his w
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• Tower law
  Furthermore, if either <math>[L:K]</math> or <math>[K:F]</math> is infinite then so is <math>[L:F]</math>. ...ndent sets in <math>L</math> over <math>F</math>, so <math>[L:F]</math> is infinite as well.
  3 KB (567 words) - 08:42, 21 August 2009
• Separation axioms
  ...<math>x,y</math> in a topological space <math>X</math>, there exists open sets <math>U, V</math> such that <math>x \in U, y \in V</math> and <math>U,V</ma ...ath> but not <math>T_2</math> is the [[finite complement topology]] on any infinite space.
  5 KB (672 words) - 13:28, 4 June 2018
• 1967 AHSME Problems
  \textbf{(D)}\ \text{must be infinite}\qquad \textbf{(E)}\ \text{must be zero or infinite}</math>
  20 KB (3,108 words) - 14:14, 20 February 2020
• 2012 USAMO Problems
  ...e which integers <math>n > 1</math> have the property that there exists an infinite sequence <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, <math>\dots< Prove that for any positive number <math>\lambda</math>, the number of sets <math>A</math> satisfying <math>S_A \ge \lambda</math> is at most <math>2^{
  3 KB (507 words) - 03:37, 7 June 2020
• 1962 AHSME Problems
  Which of the following sets of <math>x</math>-values satisfy the inequality <math>2x^2 + x < 6</math>? The limiting sum of the infinite series, <math>\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots</math>
  17 KB (2,459 words) - 22:40, 10 April 2023
• 2014 USAJMO Problems/Problem 5
  ...e integer. Two players <math>A</math> and <math>B</math> play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the ...A</math> cannot play under these circumstances. If we look at the disjoint sets
  4 KB (665 words) - 20:18, 18 April 2016
• Searching the community
  ...posts containing the words "interface", "[[incircle]]", "intermediate", "[[infinite]]", etc. will all match the query "in*e". | Sets the default size of a "unit" to a specified value. Default is very small. N
  21 KB (3,334 words) - 23:53, 23 March 2022
• 2016 AMC 10B Problems
  ...ns as sets of triplets. How many of these <math>1000</math> babies were in sets of quadruplets? The sum of an infinite geometric series is a positive number <math>S</math>, and the second term i
  12 KB (1,868 words) - 17:50, 30 December 2023
• 2016 AMC 12B Problems
  The sum of an infinite geometric series is a positive number <math>S</math>, and the second term i ...</math> games and lost <math>10</math> games; there were no ties. How many sets of three teams <math>\{A, B, C\}</math> were there in which <math>A</math>
  14 KB (2,037 words) - 19:09, 29 July 2023
• 1990 IMO Problems/Problem 4
  ...e non-trivial progress on this we must split all rational numbers into two sets, <math>S_1</math> and <math>S_2</math>, such that if <math>E \in S_1</math> ...into 2 and 1 mod 6 being in one set with 3 and 5 mod 6 in the other. These sets have equal sizes from a strong form of Dirichlet's theorem on arithmetic pr
  5 KB (1,045 words) - 11:01, 30 September 2022
• Axiom of choice
  ...a selection of exactly one object from each box, even if the collection is infinite. ...of choice says, given a set <math>S</math> of non-empty pairwise disjoint sets, there exists a set <math>X</math> with one element from each set in <math>
  1 KB (203 words) - 01:25, 12 January 2024
• 2019 Mock AMC 10B Problems/Problem 23
  ...ncrements per unit (this is because the maximum least value for one of the sets is <math>9</math>) such that no two of them overlap and (2) the number of w Since there are infinite real numbers in a unit of <math>1</math>, there are infinite increments in a unit of <math>1</math>, so we should take the limit as <mat
  2 KB (257 words) - 23:42, 29 November 2019
• Know the Facts: Gmaas
  ...oms, pocket dimensions, and many more fascinating things. The campus spans infinite miles into Gmaasia, all contained in pocket dimensions. Gmaas's room is in ...ft below ground level and filled it with dirt. The university temporarily sets itself up in a mansion in downtown Manhattan, and the university's artifact
  88 KB (14,928 words) - 13:54, 29 April 2024
• 1993 OIM Problems/Problem 6
  ...y of zeros and ones. Let <math>A</math> and <math>B</math> be two infinite sets of non-negative integers, such that <math>B</math> is the set of all number Prove that in one of the sets <math>A</math> or <math>B</math> there are infinitely many pairs of numbers
  686 bytes (118 words) - 14:22, 13 December 2023
• 2023 SSMO Team Round Problems
  ...ath>S = \{1, 2, 3, 4, \cdots, 23\}</math> and let there be randomly chosen sets <math>A, B, C</math> where <math>A, B, C \subseteq S</math>. The probabilit There exists a lane of infinite cars. Each car has a <math>\frac{1}{3}</math> chance of being high quality
  8 KB (1,238 words) - 02:14, 3 January 2024