Discrete Math Homework #9
Theodore D. Drivas
Question #1: Scheinerman Problem 47.4
Answer #1: If the countries were allowed to be in multiple pieces, then we can create
maps that require more than four colors, such as the following example: Notice that al
Discrete Math Homework #5
Theodore D. Drivas
Question #1: Scheinerman Problem 17.16
Algebraic Proof: Note
n
n1
k1
(n 1)!
n!
=
(k 1)!(n 1) (k 1)!
(k 1)!(n k)!
n!
n
=k
=k
k!(n k)!
k
=n
and we are done.
Combinatorial Proof: Question: in how many ways can we
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 3537; 39)
Cryptography (Sections
1. Let A and B be sets such that
| A|=8|B|=6|A B|=3
Evaluate
A
B
a.
2 2
b. ( A B ) ( B A )
( A A ) ( B B)
c.
2. Prove or disprove
( A B ) ( B A )=( A A ) (B B)
3. Prove or disprove
x R , y R , x= y
a.
b.
x R , y R , x= y
c.
x R , y , z R , xz= yz
d.
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1. Let R be any equivalence relation on an arbitrary non-empty set of integers S
S 2
a. Show that
|R|
S2
b. Find an equivalence relation R such that
|R|=
c. Show that |R|S
d. Find an equivalence relation R such that |R|=|S|
2. Let P be any partial order
Some Simple Cryptosystems
In cryptography, a Caesar cipher, also known as the shift cipher, is one of the simplest and
most widely known encryption techniques. It is a type of substitution cipher in which each
letter in the plaintext is replaced by a lett
More Counting
1. A mathematical modeling class consists of 6 seniors, 5 juniors, and 4 sophomores. A
team of four students must created using only students in this class. In how many
ways may the team be created if
(a) There are no additional restrictions
Functions
Definition: A relation f is called a function if (a, b) f and (a, c) f implies b = c.
In other words, for every input the function generates a UNIQUE output.
(Informal) Definition: Let f be a function. The set of all possible first elements of t
Induction
Example 1: Let n N. If a R and a 6= 1, a 6= 0 then
n
X
an+1 1
.
a1
aj =
j=0
Proof: First we establish the base case which is for the smallest number in the set under
consideration. Since the set under consideration is the natural numbers, our ba
Urban Services
he nnderlying theme of management science, also called operations research,
@- is nding the best method for solving some problemwhat mathematicians
cail the optimal solution. In some cases, the goal may he to nish a job as quickly
as possib
Discrete Math Homework #6
Theodore D. Drivas
Question #1: Use contradiction to prove that log3 5 is irrational. You may use the fact
that very positive integer has a unique prime factorization (up to the ordering of the primes
involved).
Proof. Suppose fo
Discrete Math Homework #8
Theodore D. Drivas
Question #1: Solve using Euclid?s Algortihm and the Chinese remainder theorem
x5
x4
mod 6
mod 11
Answer #1:
Proof. First we note that 6 and 11 are relatively prime, i.e. gcd(6, 11) = 1. Let n1 = 6
and n2 = 11 D
Discrete Math Homework #3
Theodore D. Drivas
Question #1:
a) Scheinerman Problem 10.1 (c), (e), (g)
b) Scheinerman Problem 10.2 (b), (d)
c) Scheinerman Problem 10.3 (d), (e), (f), (g)
d) Scheinerman Problem 10.4 (d), (e), (f), (g)
Answer #1:
a) (c) cfw_2,
PROOF TEMPLATES
Proving two sets are equal:
Let A and be be sets. To show A=B,
-Spse xA.Therefore xB.
-Spse xB.Therefore xA
Therefore A=B
Proving one set is a subset of another:
To show AB:
-Let xA.Therefore xB. Therefore AB
Proving existential statements
TREES
Cycles: A cycle, Cn, is a walk of length at least three in which the first and last vertex
are the same, but no other vertices are repeated.
V=cfw_v0,v1,.,vn and E=cfw_v1v2, v2v3,., vn-1vn, vnv1
Forest: Let G be a graph. If G contains no cycles, the
MODULAR ARITHMETIC
Modular addition
ab = (a+b) mod n
Modular multiplication
ab = (ab) mod n
*They are commutative, associative, distributive and identity elements
Modular subtraction
Let n be a positive integer and let a, b Zn. We define ab to be the uniq
INDUCTION
Principle of Mathematical Induction: Let A be a set of natural numbers. If 0A and
kN, kA k+1A, then A=N
RECURRENCE RELATIONS
First Order, an expressed in terms of an-1
All solutions to the recurrence relation an =s an-1+t, where s 1 have form:
a
SETS
A set is a repetition-free, unordered collection of objects
Let A be a finite set. The number of subsets of A is 2|A|.
DeMorgans Laws
Let A, B and C be sets. Then
A-(B C)=(A-B)(B-C) and A-(BC)=(A-B)(A-C)
PROPERTIES OF RELATIONS
-If for all xA we have
Definitions
Counting numbers: cfw_1,2,3,4,.
Natural numbers ( ): cfw_0,1,2,3.
Integers ( ): cfw_.,-2,-1,0,1,2,.
Rational numbers ( ): can be written in form p/q where p and q are integers and q!=0
Irrational numbers: set of numbers that can be written as
Discrete Math Homework #1
Theodore D. Drivas
Question #1: Scheinerman 3.2: Here is a possible alternative to Defnition 3.2: We say
that a is divisible by b provided a/b is an integer. Explain why this alternative denition
is dierent from Denition 3.2. Her
Discrete Math Homework #10
Theodore D. Drivas
Question #1: Scheinerman Problem 50.6
Answer #1: Let G be a graph in which every pair of vertices is joined by a unique path.
We want to show that G is a tree. By denition 50.3, we must show that G is connecte
Discrete Math Homework #7
Theodore D. Drivas
Question #1:
1. q = 14 and r = 2 and therefore a div b = 14 and a mod b = 2.
2. q = 15 and r = 5 and therefore a div b = 15 and a mod b = 5.
3. q = 12 and r = 0 and therefore a div b = 12 and a mod b = 0.
4. q
Proving a function is one-to-one
Direct: Spse f(x)=f(y).Therefore x=y
Contrapositive: Spse xy.Therefore f(x)f(y)
Contradiction, Spse f(x)=f(y), but xy.
Proving a function is onto
Direct: Let b be an arbitrary element of B. Explain how to construct an ele