Problem Set 3
Due: February 26
Reading: Chapter 6, Induction; Chapter 7, Partial Orders, 13.
Problem 1.
For any sets, A, and B, let [A B] be the set of total functions from A to B. Prove that if A is not
empty and B has more than one element, then NOT(A s
Problem Set 2
Due: February 19
Reading: Chapter 4, Mathematical Data Types; Chapter 5, First-Order Logic. (Assigned readings do
not include the Problem sections.)
Reminder: Comments on the reading using the NB online annotation system are due at times ind
Problem Set 8
Due: April 9
Reading: Notes Ch. 14; Ch. 15
Problem 1.
Suppose m, n are relatively prime. In the problem you will prove the key property of Eulers
function that (mn) = (m)(n).
(a) Prove that for any a, b, there is an x such that
xa
(mod m),
(
Problem Set 10
Due: April 23
Reading: Notes Ch. 16.1016.12
Problem 1.
Lets develop a proof of the Inclusion-Exclusion formula using high school algebra.
(a) Most high school students will get freaked by the following formula, even though they actu
ally kn
Problem Set 11
Due: April 30
Reading: Redene the \reading command to get this weeks reading assignment here!
Problem 1.
Let x0 := 0, x1 := 1 and for n 2, let xn be dened by the linear recurrence:
xn = 3xn1 2xn2 + n.
Find a closed form expression for xn .
Problem Set 9
Due: April 16
Reading: Notes Ch.16.116.9
Problem 1.
Show that for any set of 201 positive integers less than 300, there must be two whose quotient is a
power of three (with no remainder).
Problem 2.
Answer the following questions with a numb
Problem Set 12
Due: May 7
Reading: This pset covers Notes Ch. 18, skipping 18.3.518.3.7 and Ch.?. Additional reading
for class May 712 is specied in online Tutor Problems 13.
Problem 1.
Outside of their hum-drum duties as 6.042 LAs, Oscar is trying to lea
Problem Set 6
Due: March 19
Reading: Notes Ch. 10.3 10.6; Ch. 11
Problem 1.
An edge is said to leave a set of vertices if one end of the edge is in the set and the other end is not.
(a) An n-node graph is said to be mangled if there is an edge leaving eve
Problem Set 7
Due: April 2
Reading: Notes Ch. 12; Ch. 14
Problem 1.
A simple graph is sevenish when it has no simple cycles of length less than seven. A map is a planar
graph whose faces are all simple cycles (no dongles or bridges). Let M be a sevenish,
Problem Set 5
Due: March 12
Reading: Ch. 9.1.4, Derived Variables; Ch. 9.2, Stable Marriage; Ch. 10.1, Graph Isomorphism
Problem 1.
The following procedure can be applied to any digraph, G:
1. Delete an edge that is traversed by a directed cycle.
2. Delet
Problem Set 4
Due: March 5
Reading: Chapter 7, Partial Orders, 46; Ch. 8, Digraphs; Ch. 9, State Machines, 9.19.1.3
Problem 1.
Let be a partial order on a set, A, and let
Ak := cfw_a | depth (a) = k
where k N.
(a) Prove that A0 , A1 , . . . is a parallel
Problem Set 1
Due: February 12
Reading: Chapters 1, What is a Proof?; 2, The Well Ordering Principle; 3, Propositional Formulas.
These assigned readings do not include the Problem sections. (Many of the problems in the text
will appear as class or homewor