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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
Undergraduate
By
Shiva Kintali
on Sept. 29, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and primes
The Fibonacci numbers are defined by \(F_1 = F_2 = 1\) and \(F_n = F_{n−1} + F_{n−2}\) for \(n \geq 3\). If \(p\) is a prime number, prove that at least one of the first \(p + 1\) Fibonacci numbers mu…
Mathematics
Combinatorics
fibonacci
pigeonhole principle
primes
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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 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
1
Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and Induction
The Fibonacci numbers, \(F_0, F_1, F_2, \dots\) , are defined recursively by the equations \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2},\) for \(n > 1\). Prove that …
Mathematics
Discrete Mathematics
fibonacci
induction
tiling
0
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
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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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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
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
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