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 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 =
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] [15] 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
relation.
(a) [5 pts] Rn := cfw_(x, y) Z Z s.t. x y
Fallacies with Innity
David Shin
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
n
i2 x i
i=0
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:
n
22k1 ?
k=1
(b) [5 pts] Find a closed expression for
n
m
3i+j
i=0 j=0
(c) [5
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