Math 3355
Spring 2014
Some Review Problems
for Exam 1
1. Consider the proposition (p q) r (p r) (q r).
(a) Use a truth table to show that the proposition is true.
(b) Show that the proposition is true some other way.
(c) Find a statement logically equival
Math 3355
Spring 2014
Combinatorial
proofs
These are some notes on combinatorial proofs. First, recall that a combinatorial proof is
a proof of something based on counting. We sometimes refer to a combinatorial model,
which describes the types of objects
Math 3355
Spring 2014
Final Exam Review
All questions below are from past nal exams except for problem 5d.
1. One of the following statements is a tautology, the other is not. Which is which? You
MUST give reasons, but you need not use a truth table.
(i)
Math 3355
Spring 2014
Final Exam Review
The Solutions
1. One of the following statements is a tautology, the other is not. Which is which? You
MUST give reasons, but you need not use a truth table.
(i) (p q) (p q)
(ii) (p q) (p q).
Solution: (ii) is a tau
Math 3355
Spring 2014
Exam 3
Solutions
1. (20 points) A digit is an integer between 0 and 9. How many strings of 5 digits are
possible if:
(a) There are no repeated digits?
Solution: P (10, 5) = 10 9 8 7 6 = 30, 440
(b) Repeated digits are allowed?
Soluti
Math 3355
Spring 2014
Some Review Problems
for Exam 3: Solutions
I thought Id start by reviewing some counting formulas.
Counting the Complement: Given a set U (the universe for the problem), if you
want to know how many things in U have some property, an
Math 3355
Spring 2014
Some Review Problems
for Exam 3
1. The English alphabet contains 21 consonants and ve vowels. How many strings of four
lowercase letters of the English alphabet contain:
(a) A vowel in position 2?
(b) Vowels in positions 2 and 3?
(c)
Math 3355
Spring 2014
Some Review Problems
for Exam 2: Solutions
1. Prove that for sets A, B, C, if A B = A C, then B = C.
Solution: We assume that A B = A C and try to show that B = C. We do
this by trying to show both B C and C B. For B C, suppose that
Math 3355
Spring 2014
Exam 2
solutions
1. (25 points) Dene a function f : N N by f (x) = x/2 . Let A = cfw_0, 1, 2, 4, B =
cfw_2, 3, 5, 9, and C = cfw_5, 6, 7. In each answer, list elements of a set in increasing order,
and list an element of a set only o
Math 3355
Spring 2014
Exam 1
Solutions
1. (15 points) Let P (m, n) be the proposition that m is divisible by n. So, for example,
P (20, 5) is true but P (20, 6) is false. Let E(n) be the proposition n is even and
O(n) the proposition n is odd. One could w
Math 3355
Spring 2014
Some Review Problems
for Exam 2
1. Prove that for sets A, B, C, if A B = A C, then B = C.
2. (10 points) What can be said about sets A and B if
(a) A B = A?
(b) A B = ?
3. Let S = cfw_2, 1, 0, 1, 2, 3,
f (x) = x2 and A = cfw_1, 0, 1,
Math 3355
Spring 2014
Some Review Problems
for Exam 1: Solutions
Here is my quick review of proof techniques. I will focus exclusively on propositions of
the form p q, or more properly, x P (x) Q(x) or x y P (x, y) Q(x, y). The basic
proof techniques:
Dir
Math 3355
Spring 2014
Undertermined
Coecients
I thought I would write up some examples of undetermined coecients since todays class
was shortened. First, the theorem, as I mentioned in class. We are dealing with recurrence
relations of the form
an = c1 an