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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.
Graphing Theory
…is embedded throughout Management Science and Operations Research.
Find the most efficient route for the parking-control officer, who
travels on foot, to check the meters in an area.
As an example…
The previous situation can be represented by a graph
… in order to simplify the problem…
…to optimize the most efficient route.
… and analyze the information…
A computer generated street network
Many of the topics in Discrete Math have been developed since the dawn of the
computer.
Mail pick up routes
…and the development
of the Management Sciences.
Flight Patterns &
Scheduling
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.
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)?
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=
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.
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.
The Digital World
Identification numbers and codes are associated with many aspects of our modern
world.
EuroThe 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)
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!
The Digital World
A bar code is a series of dark and light spaces that represent characters
Postnet code
52 vertical bars2 guard bars10 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
The Digital World
A bar code is a series of dark and light spaces that represent characters
And, the more complex UPC barcode.
The Digital World
A graph can be used to represent the internet…
…where information is presented in binary code and sent
electronically.
Social Choice
This area of discrete math involves election theory, voting and fair division techniques.
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?
The electoral college is an example of a weighted voting system.
Social choice also involves strategies for conducting elections.
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.
Many of the ideas in Discrete Math will be represented by Matrices.
In addition, operations with matrices will be studied.