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### Discrete MathematicsModeling Our World

What is it anyway?

In order to minimize cost to the city, how should weekly garbage collection routes be designed for Detroit’s 350,000 households?Graph TheoryEuler Paths and Circuits

Sears, Roebuck and Company manages a fleet of over 1000 delivery vehicles to bring products they sell to customers’ locations.

How should Sears determine an efficient delivery plan for each day?

Graph TheoryTraveling Salesman ProblemSEARS

SEARS

MatricesComputer Representation of Graphs

How can problems like the Detroit garbage collection or Sear’s delivery service be modeled in order to utilize technology for the solution?

MatricesSolving Systems of Equations

“Problems we solve nowadays have thousands of equations, sometimes a million variables.”Professor George Dantzig, Stanford University

How do telecommunications companies determine how to route millions of long-distance calls using the existing resources of long-distance land lines, repeater amplifiers, and satellite terminals?

MatricesGeometric Transformations

Have you ever wondered how your favorite cartoon characters become animated?

Counting & Arranging

How secure are your passwords?

If your password consists of 3 letters and 3 numerals, how likely is it that someone could successfully guess the configuration?

UR4T82

Coding InformationIdentification Numbers

What mathematics is involved in the design of UPC codes?

Coding InformationError-Detecting Codes

Did you know that many identification codes contain check digits to help catch errors?

VIN 2FTHF26H4RCA06058

Social ChoiceVoting

Sydney Wins! News Clip

2000 Summer Olympics Kansas City Star

go to Australia September 24, 1993

Sydney, Australia, edged out Beijing Thursday for the right to hold the 2000 Summer Olympic Games. Beijing, which was considered the slight favorite, led in each of the first three rounds of voting but could not gain on overall majority. Here’s how the International Olympic Committee voted. A simple majority was required to win.

First Second Third Fourth

roundroundround ** round **

Beijing 32 37 40 43*

Sydney 30 30 37 45

Manchester, England 11 13 11*

Berlin 9 9*

Istanbul, Turkey 7* * Eliminated ** One member did not vote

Social ChoiceApportionment Algorithms

U.S. Constitution: Seats in the House of Representatives “shall be apportioned among the several states within this union according to their respective Numbers …”

1792, First Presidential Veto: George Washington vetoes the apportionment bill

1991 LEGAL CHALLENGES

What’s it all about?

Discrete MathematicsNature of Problems

- Existence of Solutions
- Number of Solutions
- Algorithms for Generating Solutions
- Optimization

Mathematics Curriculum FrameworkProbability and Discrete Mathematics

“Contemporary uses of mathematics demand that students learn to deal with uncertainty, to make informed decisions based on evidence and expectations, to exercise critical judgment about conclusions drawn from data, and to apply mathematical models to real-world phenomena. The technological world in which we live also depends upon information and communication of information and upon applications of systems with separate (discrete) entities. Topics of discrete mathematics such as counting and permutation problems, matrix operations, vertex-edge networks, and relationships among finite sets have significant real-world applications that students will encounter in diverse fields of work and study.”

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