Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 5 Solutions
Due: Monday, March 7 at 9 PM
Problem 1. An undirected graph G has width w if the vertices can be arranged in a se
quence
v 1 , v 2 , v3 , . . . , v n
such that each vertex vi is joined by an edge to at most w preceding vertices. (V
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 8 Solutions
Due: Monday, April 11 at 9 PM
Problem 1. An electronic toy displays a 44 grid of colored squares. At all times, four are
red, four are green, four are blue, and four are yellow. For example, here is one possible
conguration:
'
R
Y
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 3 Solutions
Due: Tuesday, February 22 at 9 PM
Problem 1. An urn contains 75 white balls and 150 black balls. While there are at least
2 balls remaining in the urn, you repeat the following operation. You remove 2 balls
selected arbitrarily and
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 4 Solutions
Due: Monday, February 28 at 9 PM
Problem 1. Prove all of the following statements except for the two that are false; for
those, provide counterexamples. Assume n > 1. When proving each statement, you may
assume all its predecessors
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 2 Solutions
Due: Monday, February 14 at 9 PM
Problem 1. Use induction to prove that
1
1
1
1
1
1
1
1
1
=
2
3
4
n
n
for all n 2.
Solution. The proof is by induction on n. Let P (n) be the proposition that the equation
above holds.
Base case. P
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 6 Solutions
Due: Monday, March 28 at 9 PM
Problem 1. Sammy the Shark is a nancial service provider who offers loans on the fol
lowing terms.
Sammy loans a client m dollars in the morning. This puts the client m dollars in
debt to Sammy.
Ea
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 7 Solutions
Due: Monday, April 4 at 9 PM
Problem 1. Every function has some subset of these properties:
injective
surjective
bijective
Determine the properties of the functions below, and briey explain your reasoning.
(a) The function f : R R
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 9 Solutions
Due: Monday, April 25 at 9 PM
Problem 1. There are three coins: a penny, a nickel, and a quarter. When these coins are
ipped:
The penny comes up heads with probability 1/3 and tails with probability 2/3.
The nickel comes up heads
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 10 Solutions
Due: Monday, May 2 at 9 PM
Problem 1. Justify your answers to the following questions about independence.
(a) Suppose that you roll a fair die that has six sides, numbered 1, 2, . . ., 6. Is the
event that the number on top is a
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 11 Solutions
Due: 5PM on Friday, May 6
This is a miniproblem set. The rst problem reviews basic facts about expectation.
The second and third are typical nal exam questions.
Problem 1. Answer the following questions about expectation.
(a) The
Mathematics Integrated with Computers and Applications I
MATH 1p40

Summer 2014
Problem Set 1 Solutions
Due: Monday, February 7 at 9 PM
Problem 1. The connectives (and), (or), and (implies) come often not only in com
puter programs, but also everyday speech. But devices that compute the nand operation
are preferable in computer chip