Math 308 Discrete Mathematics

1. Math 308Discrete Mathematics Discrete Mathematics deals with • “Separated” or discrete sets of objects (rather than continuous sets) • Processes with a sequence of individual steps (rather than continuously changing processes)

2. Kind of problems solved by discrete mathematics • How many ways are there to choose a computer password? • What is the probability of winning a lottery? • Is there a link between two computers in a network? • What is the shortest path between two cities using a transportation system? • How can a list of integers sorted in increasing order? How many steps are required to do such a sorting?

3. Importance of Discrete Mathematics • Information is stored and manipulated by computers in a discrete fashion • Applications in many different areas • Discrete mathematics is a gateway to more advanced courses • Develops mathematical reasoning skills • Emphasizes the new role of mathematics

4. The new role of Mathematics • Make the computer to solve the problem for you • Modeling (vs. calculations) • Using logic • to choose the right model • to write a correct computer program • to justify answers • Efficiency • make the computer to solve the problem fast • choose the more efficient model

5. Goals of Math 308 • Study of standard facts of discrete mathematics • Development of mathematical reasoning skills (emphasis on modeling, logic, efficiency) • Discussion of applications

6. Review of auxiliary topics Mathematical Logic Proofs and Induction Main topics Graph Theory Counting Techniques Application areas Computer Science Discrete Optimization Discrete Probability Game Theory and Voting Systems Outline of Topics

7. Situations where counting techniques are used • You toss a pair of dice in a casino game. You win if the numbers showing face up have a sum of 7. • Question: What are your chances of winning the game?

8. Situations where counting techniques are used • To satisfy a certain degree requirement, you are supposed to take 3 courses from the following group of courses: Math300, Math301, Math302, Math304, Math305, Math306, Math308. • Question: In how many different ways the requirement can be satisfied?

9. Job 3 Job 1 Job 4 Job 2 Situations where counting techniques are used • There are 4 jobs that should be processed on the same machine. (Can’t be processed simultaneously). Here is an example of a possible schedule: • Counting question: What is the number of all possible schedules? • Optimization question: Find a schedule that minimizes the average completion time of the four jobs.

10. Examples of Graph Models:Shortest Path Problem • In a directed network, we have distances on arcs ; source node s and sink node t . • Goal: Find a shortest path from the source to the sink. 3 a d 4 1 1 1 4 7 2 s c t 2 2 1 2 5 b e

11. Examples of Graph Models:Traveling Salesman Problem There are n cities. The salesman  starts his tour from City 1,  visits each of the cities exactly once,  and returns to City 1. For each pair of cities i, j there is a cost cijassociated with traveling from City i to City j . • Goal: Find a minimum-cost tour. 1

12. B C E A D people v jobs z u x y Examples of Graph Models:AssignmentProblem • Given: n people and n jobs Each person can be assigned to exactly one job Each job should be assigned to exactly one person Person-job compatibility is given by a directed network (e.g., having a link A  x means “person A can do job x ”) • Goal: Find an assignment of n jobs to n people (if such an assignment exists). • Example: