Justin M.

Intern at JP Morgan

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Set Theory

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Question:

Write out the set P(P(P(∅))) where P() represents the power set.

Justin M.

Answer:

∅ = {}, P(∅) = {∅} P(P(∅)) = {∅,{∅}} P(P(P(∅))) = {∅,{∅},{{∅}},{∅,{∅}}}

Discrete Math

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Question:

Consider the sequence of integers a_n defined by a_1=1 and for n>=2, a_n = 2a_{n-1} +1. Find the simplest possible formula for a_n and prove your answer using induction.

Justin M.

Answer:

The sequence starts out 1 3 7 15 31. We can recognize that this sequence is the sequence of Mersenne numbers that are generated by the function a_n = 2^n-1. Now let us prove the base case, P(1) = 2^1-1 = 1 = a_1. Let assume that for all k a_k = 2^k-1 holds. Now let us look at a_{k+1} = 2a_k+1. Since a_k =2^k-1 we see that a_{k+1} = 2(2^k-1)+1 by the inductive hypothesis. 2(2^k-1)+1 = 2^{k+1} -1. Thus by the principle of mathematical induction we have shown that a_n = 2^n -1 for all n.

Computer Science (General)

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Question:

Given a binary search tree what information would be needed on each node to find the kth element in O(log(n))?

Justin M.

Answer:

On each node you would need to have the number of nodes that are under that node. As well as the regular information needed for a binary search tree such as the node to the left and right.

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