WHAT IS DISCRETE MATH?Fall 2015Day 1

Sarah Spence Adams

Associate Dean for Faculty Affairs and Development

Professor of Mathematics and Electrical and Computer Engineering

Some slides and graphics adapted from Denise Troxell

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

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

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

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…

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

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!

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

THE PIGEON-HOLE PRINCIPLE

Learn how to use pigeons to “unlock the common sense in your head”

FIND OUT HOW MANY COLORS IT TAKES TO COLOR ANY MAP SUCH THAT NO “NEIGHBOR STATES” HAVE THE SAME COLOR

LEARN ABOUT THE KÖNIGSBERG BRIDGE PROBLEM

Is it possible? Start at locations a, b, c, or dCross each bridge exactly onceReturn to the starting location

a

c

d

b

River Pregel

Euler - 1736

STUDY HOW THE NASA MARINER MISSION SENT PICTURES BACK TO EARTH

UNLOCK THE SECRETS OF ISBN AND UPC

DISCOVER WHY THIS IS PERHAPS THE COOLEST FIGURE IN MATHEMATICS

RSA CRYPTOSYSTEM

Learn how the famous RSA algorithm actually works

LEARN HOW TO PROVE THINGS LIKE:

Every positive integer has a unique prime factorization

For all positive integers n, a 2n x 2n

chessboard with one square removed can be tiled using L-shaped pieces, where these pieces cover 3 squares at a time, as shown

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

NUMBER 1 PIECE OF ADVICE FROM PREVIOUS STUDENTS

Do more problems