Discrete Mathematics Modeling Our World

1
Discrete MathematicsModeling Our World

• What is it anyway?

2
Graph TheoryEuler Paths and Circuits

• In order to minimize cost to the city, how
  should weekly garbage collection routes be
  designed for Detroits 350,000 households?

3
Graph TheoryTraveling Salesman Problem

• 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?

SEARS
SEARS
4
MatricesComputer Representation of Graphs
How can problems like the Detroit garbage
collection or Sears delivery service be modeled
in order to utilize technology for the solution?
5
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?
6
MatricesGeometric Transformations
Have you ever wondered how your favorite cartoon
characters become animated?
7
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
8
Coding InformationIdentification Numbers
What mathematics is involved in the design of
UPC codes?
9
Coding InformationError-Detecting Codes
Did you know that many identification codes
contain check digits to help catch errors?
VIN 2FTHF26H4RCA06058
10
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.
Heres how the International Olympic Committee
voted. A simple majority was required to win.
First Second
Third Fourth
round round
round 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
11
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
Whats it all about?
12
Discrete MathematicsNature of Problems

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

13
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.