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) What is the converse of this statement?
ANSWER: If B then
Discrete Mathematics (550.171)
Homework 9 (Due Friday, April 24, 2015)
Objectives: The student will
explore properties of prime numbers
encrypt/decrypt simple messages using the RSA method
learn about the fundamentals of graph theory: vertices, edges,
Discrete Mathematics (550.171) Final Exam
Solutions to Practice Problems
Reminder: All graphs are simple (no multi-edges, no self-loops) and undirected. A cycle in
a simple undirected graph must contain three or more distinct vertices.
1. Let p be a prime
Discrete Mathematics (550.171)
Homework 1 (Due Friday, February 06, 2015)
Objectives: The student will be exposed to
applications of discrete mathematics
the concepts of denition and conjecture
the foundations of mathematical proof
General Directions:
Discrete Mathematics (550.171)
Homework 8 (Due Friday, April 17, 2015)
Objectives: The student will
use congruences to encrypt and decrypt simple messages
learn the concepts of modular arithmetic
solve linear and nonlinear congruences.
General Directio
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 proofs do exist for this problem.
Combinatorial Proof:
Discrete Mathematics (550.171)
Homework 6 (Due Friday, March 27, 2015)
Objectives: The student will
learn how to prove theorems using induction
identify when a relation is a function
understand the terms one-to-one and onto
General Directions: You must
Discrete Mathematics (550.171)
Homework 4 (Due Friday, February 27, 2015)
Objectives: The student will
work with set operations
understand and use properties of relations
General Directions: You must show all work and document any assumptions to receive
Discrete Mathematics (550.171)
Homework 2 (Due Friday, February 13, 2015)
Objectives: The student will
prove concepts using direct proof techniques
use list counting to answer real-world questions
gain familiarity with set operations
General Directions
Discrete Mathematics
Objectives: The student will be exposed to
applications of discrete mathematics
the concepts of denition and conjecture
the foundations of mathematical proof
General Directions: You must show all work and document any assumptions.
Discrete Mathematics (550.171) Exam II
Practice Problem Solutions
1. Use contradiction to show that the sum of any four consecutive integers is not divisible
by 4.
ANSWER:. Let x Z. Then x, x 1, x 2, x 3 are four consecutive integers.
Let s x px 1q px 2q
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
Number Theory (Sections 3639)
Cryptography (Sections 44
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 set of all possible rst elements of the ordered pairs in
Discrete Mathematics (550.171)
Homework 5 (Due Friday, March 13, 2015)
Objectives: The student will
count the number of partitions of a set with given properties
prove identities involving binomial coecients using algebra/arithmetic and combinatorial pr
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:
Suppose x A . . . Then x B.
Suppose x B . . . Then x A.
Relations
Intuitively, a relation is a test. Formally, a relation is a set of ordered pairs. It is somewhat
confusing that the word relation can be used in these two dierent (yet related!) contexts:
R = cfw_(x, y) : xRy.
If A and B are sets, we say the re
Quantiers
Existential Quantier
Symbol:
Meaning: There exists, There is
Example:
There is an integer less than 5 that is even.
This can also be written:
There exists an x, a member of Z, such that x < 5 and x is even.
More formally:
x Z, x < 5 and x is ev
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m visn evarg gomlr If n if oddi-
K=§ mud} fl. mi?)
V / \ /
5* =93)
V
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3 r ,7 L/Lm divmbié EU. 3';
d
Fov KEZ ,K?27 we vltmnj mm d MO?- n have (10
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Discrete Mathematics (550.171) Exam I
Practice Problems / Study Guide
General Information: The exam covers Sections 3-12, 14-17 from the course text. You
are responsible for all the material covered up to 02/27/2015. This includes material from
lecture, s
Discrete Mathematics (550.171) Exam II
Practice Problems / Study Guide
General Information
The exam covers all of sections 17, 20, 22, 24, 35, and parts of sections 15 (congruence
mod n), 26 (denition of a composition of two or more functions), 36 (using
550.171 - D ISCRETE M ATH
JHU S UMMER 2017
C OURSE S YLLABUS
Course Description
Discrete math is a fun and exciting branch of mathematics that is often overlooked
in high school curricula, which tend to focus primarily on continuous math building
up to ca