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Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Basics of countability
Prove that every collection of disjoint intervals (of positive length) on the real line is countable.
Mathematics
Real Analysis
basics
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High School
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Basics of counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ?
Mathematics
Combinatorics
basics
counting
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High School
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Basics of counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ?
Mathematics
Combinatorics
basics
counting
0
Undergraduate
By
Shiva Kintali
on June 13, 2012 | Updated Dec. 6, 2017
Basics of Pigeonhole Principle
Prove the following : Given \(n\) integers, some nonempty subset of them has sum divisible by \(n\). Let \(A\) be a set of \(n+1\) integers from {\({1, 2,\dots , 2n}\)}. Prove that some element of \…
Mathematics
Discrete Mathematics
basics
counting
pigeonhole principle
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Undergraduate
By
Shiva Kintali
on June 13, 2012 | Updated Dec. 6, 2017
Basics of Pigeonhole Principle
Prove the following : Given \(n\) integers, some nonempty subset of them has sum divisible by \(n\). Let \(A\) be a set of \(n+1\) integers from {\({1, 2,\dots , 2n}\)}. Prove that some element of \…
Mathematics
Discrete Mathematics
basics
counting
pigeonhole principle
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Undergraduate
By
Shiva Kintali
on Aug. 29, 2013 | Updated Jan. 4, 2018
Even Subsets
A set \(T\) is called even if it has even number of elements. Let \(n\) be a positive even integer, and let \(S_1, S_2, \dots, S_n\) be even subsets of the set \(S = \){\(1,2,\dots,n\)}. Prove th…
Mathematics
Combinatorics
counting
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Undergraduate
By
Chandra Chekuri
on July 29, 2012 | Updated Dec. 6, 2017
Diameter and low-degree vertex
Let \(G = (V,E)\) be an undirected connected graph. Suppose \(G\) has a pair of nodes \(s,t\) that are distance \(d\) apart. Show that there is a vertex \(v\in G\) such that the degree of \(v\) is at…
Computer Science
Mathematics
Algorithms
Graph Theory
counting
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Undergraduate
By
Chandra Chekuri
on July 29, 2012 | Updated Dec. 6, 2017
Diameter and low-degree vertex
Let \(G = (V,E)\) be an undirected connected graph. Suppose \(G\) has a pair of nodes \(s,t\) that are distance \(d\) apart. Show that there is a vertex \(v\in G\) such that the degree of \(v\) is at…
Computer Science
Mathematics
Algorithms
Graph Theory
counting
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High School
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Bijective counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ? Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of …
Mathematics
Combinatorics
bijection
counting
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Undergraduate
By
Shiva Kintali
on June 8, 2013 | Updated Dec. 6, 2017
Let \(G\) be a \(k\)-regular simple bipartite graph with vertex partitions \(A\) and \(B\). Then,
Mathematics
Graph Theory
counting
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Graduate
By
Shiva Kintali
on Aug. 1, 2012 | Updated Dec. 6, 2017
Number of Hamiltonian paths
Let \(G\) be an undirected graph on at least five vertices let and \(\overline{G}\) its complement. Let \(h(G)\) be the number of Hamiltonian paths of \(G\). Prove that \(h(G) + h(\overline{G})\) is…
Mathematics
Graph Theory
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