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Discrete Mathematics

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.

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Discrete Mathematics

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  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” • India (200s BC) • Arabs (600-700s) • 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?

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