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• Probability
  ...oss. In this case, <math>\Omega = \{H, T\}</math>, <math>\mathfrak{a} = \{\emptyset, \{H\}, \{T\}, \{H, T\}\}</math>, and <math>\mathit{P}</math> assigns the f <math>\mathit{P}(\emptyset)=0</math>,
  4 KB (588 words) - 12:47, 2 October 2022
• Subset
  <math>\emptyset \subset \{1, 2\} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \
  1 KB (217 words) - 09:32, 13 August 2011
• LaTeX:Symbols
  |<math>\prime</math>||\prime||<math>\emptyset</math>||\emptyset||<math>\nabla</math>||\nabla
  16 KB (2,324 words) - 16:50, 19 February 2024
• Set
  ...nt}S</math>. For example, <math>\bigcap_{S\,is\,P(A)\,for\,some\,set\,A}S=\emptyset</math>, or the empty set defined next. An [[empty set]], denoted <math>\emptyset</math> is a set with no elements.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Empty set
  The '''Empty Set''' (generally denoted <math>\emptyset</math> or <math>\varnothing</math>) is the (unique) [[set]] containing no e
  489 bytes (84 words) - 21:33, 27 February 2020
• 2004 AIME I Problems/Problem 15
  ...not in any of the other sets. Also, we note that <math>\mathcal{A}_{1,k}=\emptyset</math> for <math>k=0,2,3,4,5,6,7,8,9</math>.
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2002 AIME II Problems/Problem 9
  ...to organize the elements of <math>\mathcal{S}</math> such that <math>B = \emptyset</math> and <math>\mathcal{S} = A+C</math>. But, the combination such that <math>A = B = \emptyset</math> and <math>\mathcal{S} = C</math> is counted twice.
  3 KB (404 words) - 23:07, 4 May 2024
• Power set
  ...set]] has only one subset, itself. Thus <math>\mathcal{P}(\emptyset) = \{\emptyset\}</math>. ...ets, the empty set and the entire set. Thus <math>\mathcal{P}(\{a\}) = \{\emptyset, \{a\}\}</math>.
  4 KB (757 words) - 11:44, 8 March 2018
• Partially ordered set
  :* <math>\emptyset</math> is a minimal element of the poset; every other element is larger.
  4 KB (717 words) - 20:01, 25 April 2009
• Boolean lattice
  ...\{1, 3\}, \{2, 3\}</math> and <math>\{1, 2, 3\}</math>, among which <math>\emptyset</math> is smaller than all others, <math>S = \{1, 2, 3\}</math> is larger t
  1 KB (168 words) - 22:46, 20 April 2008
• 2007 USAMO Problems/Problem 4
  ...ack) be two complementary animals in <math>D</math>, i.e. <math>R\cap B = \emptyset</math> and <math>R\cup B = D</math>. Suppose <math>|R|\leq s - 1</math>. Th
  10 KB (1,878 words) - 14:56, 30 June 2021
• Zermelo-Fraenkel Axioms
  ...tement:''' There exists a set (called the [[empty set]] and denoted <math>\emptyset</math>) which contains no elements.
  4 KB (732 words) - 20:49, 13 October 2019
• Graph (graph theory)
  ...G = (V, E_1 \cup E_2)</math> is a complete graph and <math>E_1 \cap E_2 = \emptyset</math> then <math>G_2</math> is said to be the ''complement'' of <math>G_1<
  8 KB (1,428 words) - 10:26, 27 August 2020
• Filter
  * <math>F\neq\emptyset</math>.
  2 KB (368 words) - 21:14, 13 October 2019
• 2001 USAMO Problems/Problem 5
  Since <math>S\neq \emptyset</math> by (A), it suffices to prove that <math>m = 1</math>. For the sake o
  3 KB (599 words) - 07:44, 23 October 2022
• 2009 USAMO Problems/Problem 2
  .../math> for all <math>a,b,c\in S</math> implies that <math>-S\cap(S_++S_-)=\emptyset</math>. Since any element of <math>S_++S_-</math> has absolute value at mos
  8 KB (1,444 words) - 03:44, 2 October 2015
• 2010 AIME I Problems
  ...\cap B| = |B \cap C| = |C \cap A| = 1</math> and <math>A \cap B \cap C = \emptyset</math>. For example, <math>(\{1,2\},\{2,3\},\{1,3,4\})</math> is a minimall ...a pair of sets <math>A</math>, <math>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>.
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 7
  ...\cap B| = |B \cap C| = |C \cap A| = 1</math> and <math>A \cap B \cap C = \emptyset</math>. For example, <math>(\{1,2\},\{2,3\},\{1,3,4\})</math> is a minimall
  2 KB (255 words) - 17:03, 9 August 2018
• 2010 AIME I Problems/Problem 12
  ...a pair of sets <math>A</math>, <math>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>.
  2 KB (284 words) - 21:56, 25 November 2023
• 2010 AIME II Problems
  <LI> <math>\mathcal{A} \cap \mathcal{B} = \emptyset</math>, </LI>
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AIME II Problems/Problem 8
  <LI> <math>\mathcal{A} \cap \mathcal{B} = \emptyset</math>, </LI>
  4 KB (699 words) - 20:57, 20 July 2023
• Bézout's Identity
  ...a_iu_i: u_1, \dots, u_m \in \mathbb{Z}\}</math>. Obviously, <math>P \neq \emptyset</math>. Thus, because all the elements of <math>P</math> are positive, by t
  4 KB (768 words) - 16:50, 6 September 2023
• 1997 USAMO Problems/Problem 6
  ...th> and <math>k</math> are positive integers, then <math>I_n\cap I_{k}\ne \emptyset</math>. To prove this, we need the following lemma. ...his contradicts the statement of Lemma 1. Therefore, <math>I_k\cap I_n\ne \emptyset</math>.
  6 KB (1,107 words) - 14:12, 12 April 2023
• 2006 Romanian NMO Problems/Grade 10/Problem 1
  ...ve <math>f(\emptyset) = f(\emptyset) + f(\emptyset)</math>, hence <math>f(\emptyset) = 0</math>. Also, if <math>A \subset B \subset M</math>, then <math>f(B) = Let <cmath>A_{0} = \emptyset \subsetneq A_{1} \subsetneq \dotsb \subsetneq A_{n-1} \subsetneq A_{n} = M<
  4 KB (796 words) - 10:53, 8 May 2012
• 2014 AMC 12B Problems/Problem 22
  <cmath>a=0.1\emptyset+0.9b,b=0.2a+0.8c,c=0.3b+0.7d,d=0.4c+0.6e</cmath> <cmath>10a=\emptyset+9b,10b=2a+8c,10c=3b+7d,10d=4c+6e</cmath>
  7 KB (1,082 words) - 22:35, 3 April 2024
• 2014 AIME I Problems/Problem 13
  <cmath>51=\emptyset</cmath> <cmath>68=\emptyset\text{ others}</cmath>
  9 KB (1,404 words) - 21:07, 13 October 2023
• 2006 iTest Problems
  ...by the rule <cmath>P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}</cmath> Let <math>d</math> be the least up
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2016 USAMO Problems
  ...union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldot
  4 KB (608 words) - 13:49, 22 November 2023
• 2016 USAJMO Problems
  ...union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldot
  3 KB (414 words) - 16:43, 5 August 2023
• 2016 USAMO Problems/Problem 1
  ...union is not the whole set <math>S</math>, that is, <math>X_i\cap X_{i+1}=\emptyset</math> and <math>X_i\cup X_{i+1}\neq S</math>, for all <math>i\in\{1, \ldot \emptyset & i\equiv 0 \text{ (mod } 9\text{)} \\
  7 KB (985 words) - 18:11, 11 June 2017
• 1994 USAMO Problems/Problem 5
  ...can use PIE to show that <math>\left|\bigcup_{k \in U}A_k\right| = \sum_{\emptyset \neq U \subseteq S} (-1)^{|U|+1}\left|\bigcap_{k \in U}A_k\right|</math>. <math>\binom{m}{|S|}-\pi(S) = \sum_{\emptyset \neq U \subseteq S} (-1)^{|U|+1}\binom{m- \sigma(U)}{|S|} \iff \pi(S) = \su
  3 KB (628 words) - 10:42, 4 August 2023
• 2017 AMC 10B Problems/Problem 17
  ...ce it will always be a leading digit and that is not allowed. Also, <math>\emptyset</math> (the empty set) isn't included because it doesn't generate a number. ...s included since we are allowed to end numbers with zeros. However, <math>\emptyset</math> (the empty set) still isn't included because it doesn't generate a n
  7 KB (1,158 words) - 19:34, 27 March 2024
• 2018 AIME I Problems
  ...th>s(T)</math> be the sum of the elements of <math>T</math>, with <math>s(\emptyset)</math> defined to be <math>0</math>. If <math>T</math> is chosen at random
  8 KB (1,284 words) - 14:35, 9 August 2021
• 2018 AIME I Problems/Problem 12
  ...th>s(T)</math> be the sum of the elements of <math>T</math>, with <math>s(\emptyset)</math> defined to be <math>0</math>. If <math>T</math> is chosen at random ...^3=8</math> total subsets, and <math>w(3)=4</math> (the subsets are <math>\emptyset, \{0\}, \{1,2\},</math> and <math>\{1,2,0\}</math>). Now consider the first
  26 KB (4,044 words) - 13:58, 24 January 2024
• 2019 USAMO Problems/Problem 4
  ...h S_{ij}</math>. Begin by considering <math>C_0</math> and <math>S_{00} = \emptyset</math>. Then given <math>S_{i0}</math> we can create <math>S_{(i+1)0}</mat
  7 KB (1,288 words) - 19:17, 26 April 2023
• 2006 iTest Problems/Problem 34
  ...by the rule <cmath>P(n):=\begin{cases}\min(s)_{s\in S_n}&\text{if }S_n\neq\emptyset,\\0&\text{otherwise}.\end{cases}</cmath> Let <math>d</math> be the least up
  2 KB (346 words) - 18:04, 13 September 2020
• 2023 AIME II Problems
  ...<math>X</math> and <math>Y</math> in the collection, <math>X \cap Y \not= \emptyset.</math>
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2022 AIME I Problems/Problem 12
  (A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2
  9 KB (1,471 words) - 16:41, 1 February 2024
• 2022 AIME I Problems
  ...sum is taken over the pairs of subsets <cmath>(A,B) \in \left\{(\emptyset,\emptyset),(\{1\},\{1\}),(\{1\},\{2\}),(\{2\},\{1\}),(\{2\},\{2\}),(\{1,2\},\{1,2\})\
  9 KB (1,520 words) - 19:06, 2 January 2023
• 2023 AIME II Problems/Problem 11
  ...<math>X</math> and <math>Y</math> in the collection, <math>X \cap Y \not= \emptyset.</math> This entails <math>\emptyset \in \mathcal C</math>.
  12 KB (2,014 words) - 00:58, 12 February 2024
• 2023 AMC 10B Problems/Problem 16
  ...<math>(1, 2, 3, 4, 5, 6, 7, 8, 9)</math>, excluding the empty set (<math>\emptyset</math>) and the one-digit integers. So, the number of upno integers is <mat
  8 KB (1,195 words) - 12:03, 26 February 2024
• 2024 AIME I Problems/Problem 3
  <math>\text{heap}(0) = \{\} = *\text{mex}(\emptyset) = 0</math>
  10 KB (1,628 words) - 18:50, 7 February 2024
• 2024 AIME II Problems/Problem 14
  We can easily see that all ordered partitions (except <math>A = \emptyset</math>) guarantee feasible solutions of <math>z</math>.
  4 KB (682 words) - 17:07, 2 May 2024