Discrete Mathematics Nathan Graf April 23, 2012.

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Presentation transcript:

Discrete Mathematics Nathan Graf April 23, 2012

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

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

Combinatorics “Pascal’s Triangle” Gambling and Probablility India (200s BC) Arabs (600-700s) Gambling and Probablility Cardano (1500s) Fermat and Pascal Leibniz’s De Arte Combinatoria (1666)

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

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

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)

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

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

Class Activity Markov Chains Probability/Statistics Graph Theory to Visualize

Questions?

http://www.britannica.com/EBchecked/topic/242012/graph-theory