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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
0
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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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
counting
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Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Basics of decidability
State whether each of the following statements are TRUE or FALSE. Your answers should be accompanied by a proof. Every recognizable set has a decidable subset. Any subset of a recognizable language …
Computer Science
Complexity Theory
basics
true or false
undecidability
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Graduate
By
Shiva Kintali
on June 9, 2012 | Updated Dec. 6, 2017
Basics of Pfaffians
Let \(G(V,E)\) be an undirected graph. An orientation of \(G\) is Pfaffian if every even cycle \(C\) such that \(G \setminus V(C)\) has a perfect matching, has an odd number of edges directed in eithe…
Mathematics
Graph Theory
basics
pfaffian
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Basics of Expectation and Variance
If the random variable \(X\) takes values in non-negative integers, prove that: \(E[X] = \sum_{t=0}^\infty \Pr(X > t)\) Prove that if \(X_1\) and \(X_2\) are independent random variables then …
Mathematics
Probability
basics
expectation
variance
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Graduate
By
Shiva Kintali
on Nov. 15, 2012 | Updated Dec. 6, 2017
Basics of Perfect Graphs
A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Prove that bipartite graphs are perfect. Prove that the lin…
Mathematics
Graph Theory
basics
bipartite graph
chordal graph
graph coloring
interval graph
perfect graphs
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Graduate
By
Shiva Kintali
on June 10, 2012 | Updated Dec. 6, 2017
Basics of Treewidth
Treewidth of an undirected graph \(G\) measures how close \(G\) is to a tree. A tree decomposition of a graph \(G(V, E)\) is a pair \(\mathcal{D} = ({X_i\ |\ i \in I}, T(I, F))\) where …
Mathematics
Graph Theory
basics
treewidth
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Basics of Connectivity
Prove that a connected graph \(G\) with \(2k\) odd vertices (odd degree vertices) decomposes into \(k\) paths (not necessarily simple paths) if \(k > 0\). Does this remain true if \(G\) is not connec…
Mathematics
Graph Theory
basics
connectivity
0
Undergraduate
By
Shiva Kintali
on May 22, 2013 | Updated Dec. 6, 2017
Basics of probability
Prove that, if \(E_1, E_2, \dots, E_n\) are mutually independent, then so are \(\overline{E_1}, \overline{E_2}, \dots, \overline{E_n}\) Give an example of three random events \(X,Y,Z\) for which any …
Mathematics
Probability
basics
conditional probability
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