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 fou
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.
Discrete Mathematics (550.171) Exam II
Practice Problem Solutions
1. Let A be a set of nonzero integers and let be the relation on A A defined by
(a, b) (c, d) whenever a + d = b + c.
Prove that is an
Discrete Math HW 1 Solutions
Problem 1
Convex Set: A set S is convex if, for any x, y S the line segment connecting x
and y is contained in S.
Convex Function: A function f is convex if for any x, y
Discrete Math 550.171
Homework 10 Solutions
1. (Scheinerman 50.7) Prove the following converse to Proposition 50.8:
Let T be a tree with at least two vertices and let v V (T ). If T v is a tree, then
Discrete Mathematics (550.171)
Homework 1 (Due Friday, February 10, 2017)
Objectives: The student will be exposed to
the concepts of definition, theorem, and conjecture
the process of using given in
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 Math Homework 4 Solutions
1. Prove: a relation R on a set A is antisymmetric if and only if
R R1 cfw_(a, a) : a A.
Suppose R is antisymmetric. We need to prove that the set R R1 is a subset o
Discrete Mathematics (550.171) Exam II
Practice Problems / Study Guide
General Information
You are responsible for all the material covered up to 3/31/2017 with special emphasis on
the material (lectu
Discrete Mathematics (550.171)
Homework 3 (Due Friday, February 24, 2017)
Objectives: The student will be exposed to
proofs involving sets
more counting (including use of the inclusion/exclusion pri
Discrete Mathematics (550.171)
Homework 5 (Due Friday, March 17, 2017)
Objectives: The student will
practice counting unordered collections
use indirect proof to show mathematical statements are tru
Discrete Mathematics (550.171) Exam I
Practice Problems / Study Guide
General Information: The exam covers Sections 3-6, 8-12, 14 (through example 14.3)
from the course text. You are responsible for a
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.
Then x, x 1, x 2, x 3 are four conse
Discrete Mathematics (550.171) Exam II
Practice Problem Solutions
1. Let n be a positive integer. Use induction to prove that
n
X
n2 (n + 1)2
4
j3 =
j=1
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Chinese Postmen and
Traveling Salesmen
Introduction to Applied Math and Statistics
Beryl Castello
Problem 1
A postman is seeking the cheapest (cost, time, distance,
etc.) way to deliver mail to the st
Discrete Math Exam 1 Review
Sections 10 - 12, and beginning of 14
February 28, 2017
This document has brief notes and a few examples for each of the sections 10 through 12
and the beginning of section
Discrete Math 550.171
Homework 7 Solutions
1. (Scheinerman 15.1) For each of the following congruences, find all integers N , with N > 1, that make the
congruence true.
a. 23 13 (mod N )
We are lookin
SOLUTION MANUAL FOR HW 6 (Discrete Spring 2017)
Question 1 (22.12 The Tower of Hanoi)
Prove: For every positive integer n, the Tower of Hanoi puzzle (with n disks) can be solved in 2n
1
Proof by ind
Discrete Mathematics (550.171) Final Exam
Solutions to Practice Problems
Reminder: All graphs are simple (no multi-edges, no self-loops) and undirected unless otherwise stated. A cycle in a simple und
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 ca
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
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
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
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