Problem Set 5
Readings: Section 5.4 to 5.7 and 6.1-6.2.
Problem 1. [20 points] Recall that a tree is a connected acyclic graph. In particular, a
single vertex is a tree. We dene a Splitting Binary Tree, or SBTree for short, as either the
lone vertex, or a
Problem Set 2
Problem 1. [12 points] Dene a 3-chain to be a (not necessarily contiguous) subsequence
of three integers, which is either monotonically increasing or monotonically decreasing. We
will show here that any sequence of ve distinct integers will
Problem Set 6
Problem 1. [20 points]  For each of the following, either prove that it is an equivalence
relation and state its equivalence classes, or give an example of why it is not an equivalence
(a) [5 pts] Rn := cfw_(x, y) Z Z s.t. x y
Problem Set 3
Problem 1. [16 points] Warmup Exercises
For the following parts, a correct numerical answer will only earn credit if accompanied by
its derivation. Show your work.
(a) [4 pts] Use the Pulverizer to nd integers s and t such that 135s + 59t =
Fallacies with Innity
Consider the following claim:
Claim 1. There exists an innite decreasing sequence of natural numbers.
Proof. Assume for sake of contradiction that the longest decreasing sequence of natural numbers is nite.
Let S = cfw_a1
Problem Set 7
Problem 1. [15 points] Express
i2 x i
as a closed-form function of n.
Problem 2. [20 points]
(a) [5 pts] What is the product of the rst n odd powers of two:
(b) [5 pts] Find a closed expression for
Problem Set 8
Problem 1. [25 points] Find bounds for the following divide-and-conquer recurrences.
Assume T (1) = 1 in all cases. Show your work.
(a) [5 pts] T (n) = 8T (n/2) + n
(b) [5 pts] T (n) = 2T (n/8 + 1/n) + n
(c) [5 pts] T (n) = 7T (n/20) + 2T (n
Problem Set 10
Problem 1. [15 points] Suppose Pr cfw_ : S [0, 1] is a probability function on a sample
space, S, and let B be an event such that Pr cfw_B > 0. Dene a function PrB cfw_ on outcomes
w S by the rule:
Pr cfw_w / Pr cfw_B if w B,
PrB cfw_w =
Problem Set 11
Problem 1. [20 points] You are organizing a neighborhood census and instruct your census
takers to knock on doors and note the sex of any child that answers the knock. Assume that
there are two children in a household and that girls and boy
Problem Set 1
Problem 1. [24 points]
Translate the following sentences from English to predicate logic. The domain that you are
working over is X, the set of people. You may use the functions S(x), meaning that x has
been a student of 6.042, A(x), meaning
Problem Set 12
Problem 1. [15 points]
In this problem, we will (hopefully) be making tons of money! Use your knowledge of
probability and statistics to keep from going broke!
Suppose the stock market contains N types of stocks, which can be modelled by in
Problem Set 9
Problem 1. [10 points]
(a) [5 pts] Show that of any n + 1 distinct numbers chosen from the set cfw_1, 2, . . . , 2n, at
least 2 must be relatively prime. (Hint: gcd(k, k + 1) = 1.)
(b) [5 pts] Show that any nite connected undirected graph wi
Problem Set 4
Problem 1. [15 points] Let G = (V, E) be a graph. A matching in G is a set M E
such that no two edges in M are incident on a common vertex.
Let M1 , M2 be two matchings of G. Consider the new graph G = (V, M1 M2 ) (i.e. on the
same vertex se