1. Discrete Mathematics
Nathan Graf
April 23, 2012

2. Agenda What is Discrete Mathematics? Combinatorics Number Theory
Mathematical Logic
Sets
Graphs
Class Activity

3. Discrete Mathematics Not Continuous Not New Many Mathematical Fields
Key to Computing

4. Combinatorics “Pascal’s Triangle” Gambling and Probablility
India (200s BC)
Arabs ( s)
Gambling and Probablility
Cardano (1500s)
Fermat and Pascal
Leibniz’s De Arte Combinatoria (1666)

5. Greek Number Theory Pythagoreans (beginning 6th Century BC)
Number mysteries
Figurative Numbers
Euclid (350 BC)
Divisibility
Primes
Diophantus - (ca. AD 250)
Rational Solutions to Indeterminant Polynomials

6. Number Theory Resurgence
"Presurgence" - Fibonacci (early 1200s)
Fermat - divisibility, perfect numbers (mid 1600s)
Marsenne - primes
Euler - proofs of Fermat's theorems (mid 1700s)
Gauss 
Disquisitiones Arithmeticae (1801)
Congruence
Prime Numbers

7. Mathematical Logic Informal Logic - Euclid Calculating Machines
Pascal - Pascaline (1642)
Leibniz - Stepped Reckoner (1694)
Babbage - Difference/Analytical Engines (1800s)
Mathematical Logic
Boole, De Morgan (mid 1800s)
C.S. Pierce (late 1800s)

8. Sets Bolzano (mid 1800s) Dedekind (1888) Cantor (1895)
Provided foundation
Paradoxes of the Infinite
A Foundation for All Mathematics?

9. Graph Theory Euler – Konigsberg Bridge Problem (1735)
Hamilton – Circuits on Polyhedra (1857)
Four Color Problem
Asked in 1850
Proven in 1976 by computer
Modeling Chemical Compounds
Modern Usage
Computer Programming

10. Class Activity Markov Chains Probability/Statistics
Graph Theory to Visualize

11. Questions?