What is Discrete Math?

1. What is Discrete Math? Fall 2011 Day 1 Sarah Spence Adams Professor of Mathematics and Electrical and Computer Engineering Some slides and graphics adapted from Denise Troxell

2. Discrete Means… • Discrete: • consisting of distinct or unconnected elements • taking on or having a finite or countably infinite number of values • Not Continuous: • Real numbers are no longer the base • Integers are the primary tool

3. Why Discrete? • DM provides models and tools for real world phenomena that change abruptly or have distinct states • DM has become increasingly important in the digital/computer age

4. DM intersects other areas • Computer Science • Electrical Engineering • Operations Research • Probability • Statistics • Number Theory • Cryptology • Group Theory • Graph Theory • Coding Theory • Set Theory • Logic, and more

5. Applications • algorithms • network flows • telephone routing • delivery routes • computer networks • airplane schedules • personnel assignments • genetics • election procedures • secure and reliable wireless communications • design of statistical experiments • bin packing, and more…

6. More on Applications • Software engineering – uses sets, graphs, trees, and other structures • Analysis of algorithms – requires ability to count number of operations, proofs of correctness • Recursive algorithms – require solution to recurrence relations, proofs of correctness through induction • Cryptology – requires number theory • AI – requires logic • Theory of computation and compiler design – requires proofs including proofs by induction

7. What’s in store this semester? • Learn how to count! • You may be surprised that counting certain things can be really, really hard! • But you may also be surprised at how good you’ll get at counting!

8. Count things like.. • Number of ways to buy a dozen donuts from a choice of 32 different varieties • Number of ways to triangulate an n-gon • Number of ways to configure a network so that certain connectivity requirements are met • Number of ways to assign students to groups, considering certain constraints on student preferences

9. The Pigeon-hole Principle • Learn how to use pigeons to “unlock the common sense in your head”

10. Find out how many colors it takes to color any map such that no “neighbor states” have the same color

11. Learn about the Königsberg Bridge Problem • Is it possible? • Start at locations a, b, c, or d • Cross each bridge exactly once • Return to the starting location c Euler - 1736 River Pregel d a b

12. Study how the NASA Mariner Mission sent pictures back to Earth

13. Unlock the secrets of ISBN and UPC

14. Discover why this is perhaps the coolest figure in mathematics

15. RSA Cryptosystem • Learn how the famous RSA algorithm actually works

16. Learn how to prove things like: • Every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps. • For all positive integers n, a 2nx 2nchessboard with one square removed can be tiled using L-shaped pieces, where these pieces cover 3 squares at a time, as shown

17. What Else Can You Expect? • Work lots of hard but fun problems • Learn to argue clearly, convincingly, and flawlessly • Improve technical writing and presentation skills • Investigate topics in small groups • Participate actively in class • Get help early and often • Work closely with classmates and professor

18. Number 1 Piece of Advice from Previous Students Do Practice Problems