This document provides an overview of discrete mathematics. It discusses key areas like graph theory, counting and probability, social choice, and coding/encryption. Graph theory is used to model problems like finding efficient routes. Counting and probability help answer questions about outcomes and events. Social choice examines voting methods and fair division. Identification numbers, codes, and barcodes are important applications in the digital world. Discrete math provides tools and models for problems in business, industry, and computer science.

1. What is it?
It is an applications driven course that is based
upon the study of events that occur in small, or
discrete, chunks.
Discrete concepts are used extensively in
business, industry, government, and the digital
world.
The major areas of study are counting and
probability, graph theory, the mathematics of
social choice (voting and fair division), and coding
and encryption.

2. Graphing Theory
…is embedded throughout Management Science and Operations
Research.

3. Find the most efficient route for the parking-
control officer, who travels on foot, to check
the meters in an area.
As an example…

4. The previous situation can be represented
by a graph
… in order to simplify the problem…

5. …to optimize the most efficient route.
… and analyze the information…

6. A computer generated street network
Many of the topics in Discrete Math have been
developed since the dawn of the computer.

7. Mail pick up routes
…and the
development
of the
Management
Sciences.
FlightPatterns&Scheduling

8. Graphs are analyzed quantitatively to determine the
optimal solutions to problems…
…and various types of graphs (such as Euler and
Hamiltonian) are introduced as more and more situations
are analyzed.

9. Counting and Probability
What is the probability of
winning the jackpot in the
lottery?
In how many ways can a
lottery participant choose
several numbers from those
on a lottery ticket?
How may different phone
numbers are possible in a
given exchange (same three
digit prefix)?

10. Discrete Math will help to answer some of
these questions.
…as well as explore all possibilities for a given
event such as:
What is the probability of rolling two number
cubes that have a sum of 7?
6 1
36 6
=

11. The Digital World
Identification numbers and codes are associated with
many aspects of our modern world.
Plane Tickets
Check Digits are found on many items for error detection.

12. The Digital World
Identification numbers and codes are associated with
many aspects of our modern world.
Money Orders
This check digit is the remainder when dividing the sum of
the identification numbers by 9.

13. The Digital World
Identification numbers and codes are associated with
many aspects of our modern world.
Euro
The letter preceding the identification number is a code for the
country in which the bill was issued. The check digit is then chosen so
that the the sum of all digits including the number assigned to the
letter is divisible by 9. (X = 7)

14. The Digital World
Identification numbers and codes are associated with
many aspects of our modern world.
Codabar
Add the digits in positions 1, 3, 5, 7, 9, 11, 13, and 15 and double the
result; Next, count the number of digits in those positions that
exceed 4 and add this to the total. Now add in the remaining digits.
The check digit is the number needed to bring this final tally to a
number that ends in 0.
(4+2+0+1+3+5+7+9) x 2 + 3 + (1+8+0+2+4+6+8) + 0 = 94
This credit
card is a fake!

15. The Digital World
A bar code is a series of dark and light spaces that
represent characters
Postnet code
52 vertical bars
2 guard bars
10 sets of 5
Represents the
Zip + 4 Code
Check digit is
such that the sum
of the zip code
digits and the
check digit is
divisible by 10

16. The Digital World
A bar code is a series of dark and light spaces that
represent characters
And, the more complex UPC barcode.

17. The Digital World
A graph can be used to represent the internet…
…where information is presented in binary
code and sent electronically.

18. Social Choice
This area of discrete math involves election theory, voting
and fair division techniques.

19. There are different types of processes by which the
preferences of individuals are combined to produce a group
decision.
Depending upon the Ranking Method used, the results could
be different.
Borda Method: Grape, Runoff Method: Orange, Plurality Method:
Cola and the Arrow Method: Lemon-Lime
What if you were asked to rank your favorite flavor of the
following drink?

20. The electoral college is an example of a weighted voting
system.

21. Social choice also involves strategies for conducting
elections.

22. Fair Division is a process where certain things are divided
among a given number of people so that each person feels
he/she received the best piece (also known as envy free
division).
Other models and applications of fair division will also be
studied.

23. Many of the ideas in Discrete Math will be represented by
Matrices.
In addition, operations with matrices will be studied.