Discrete Mathematics (550.171) Exam I
Friday, October 04, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calculator. To receive credit for a problem you must SHOW A
ANSWERS, HINTS, and TYPOS for HOMEWORK 5
1. I am not sure how to approach problem 3(c). Can you help?
ANSWER: I suggest you do a few examples (like 7 choose 4 and 8 choose 3 which were
given in the as
Discrete Mathematics (550.171) Exam II
Practice Problems / Study Guide
General Information
The exam covers Sections 1517, 2022, 24, and 26 from the course text. You are responsible
for all the materia
Discrete Mathematics (550.171) Exam I
Friday, October 04, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calculator. To receive credit for a problem you must SHOW A
Discrete Mathematics (550.171) Exam I
Solutions to Practice Problems
1. Consider the true statement: If A then B.
(a) What is the contrapositive of this statement?
ANSWER: If not B then not A.
(b) Wha
Discrete Mathematics (550.171) Exam I
Practice Problems / Study Guide
General Information
The exam covers Sections 212, and 14, from the course text. You are responsible for all the
material covered f
Discrete Mathematics (550.171) Exam II
Friday, November 08, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calcu-
lator. To receive credit for a problem you must SH
Discrete Mathematics (550.171) Exam II
Friday, November 08, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calcu-
lator. To receive credit for a problem you must SH
Discrete Mathematics (550.171) Exam III
Practice Problem Solutions
1. Let t be a positive integer. Prove that if p, q1 , q2 , . . . , qt are prime numbers and
p|q1 q2 qt
then p = qi for some 1 i t.
AN
Discrete Mathematics (550.171) Final Exam
Practice Problems / Study Guide
General Information
The exam is CUMULATIVE; however, the focus will be on the material covered since Exam
2. This includes Sec
Fundamentals of Discrete Mathematics
It is said that if we think of mathematics as a dramatic production there are three main
characters:
Denition
Theorem
Proof
Denitions. Mathematical objects come
Contradiction, Smallest Counterexample
Suppose we are trying to prove that a statement is true. One way to approach this is to suppose
that it is not true. If supposing that it is not true leads us to
Working With Sets
Proving two sets are equal Let A and B be sets. To show A = B we must show every
element of A is also in B AND that every element of B is also in A. The template is as
follows:
Supp
Discrete Mathematics (550.171) Exam II
Practice Problem Solutions
1. Let A = cfw_1, 2, 3 and let B = cfw_4, 5. Let R = (A A) (B B).
(a) Prove that R is an equivalence relation on A B.
ANSWER: Observe
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Functions
Denition: A relation f is called a function if (a, b) f and (a, c) f imply b = c.
In other words, for every input the function generates a UNIQUE output.
Denition: Let f be a function. The s
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Proof by Induction
Example: Let n N. If a R and a = 1, a = 0 then
n
aj =
j=0
an+1 1
.
a1
Proof: First we establish the base case which is for the smallest number in the set under
consideration. Since
A Binomial Identity
Proposition: Let n N. Then
n
k=0
n
k = n2n1 .
k
We will give two proofs, one combinatorial and the other algebraic. It should be noted that
alternative combinatorial and algebraic
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Discrete Mathematics (550.171)
Homework 5 (Due Friday, October 18, 2013)
Objectives: The student will be able to
prove identities involving binomial coecients
apply various counting techniques in a
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