Discrete Mathematics Nathan Graf April 23, 2012. Agenda What is Discrete Mathematics? Combinatorics...

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Discrete Mathematics

Nathan Graf

April 23, 2012

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

Agenda

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

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

Discrete Mathematics

• Not Continuous• Not New• Many Mathematical Fields• Key to Computing

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

Combinatorics

• “Pascal’s Triangle” – India (200s BC)– Arabs (600-700s)

• Gambling and Probablility– Cardano (1500s)– Fermat and Pascal

• Leibniz’s De Arte Combinatoria (1666)

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

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

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DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

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

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

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)

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

Sets

• Bolzano (mid 1800s)• Dedekind (1888)• Cantor (1895)

– Provided foundation– Paradoxes of the Infinite

• A Foundation for All Mathematics?

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

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

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

Class Activity

• Markov Chains• Probability/Statistics• Graph Theory to Visualize

AC

DB

{ 1, 2, 3, 4, 5.

P → Q, ⌐Q → ⌐P

(1, 1, 2, 3, 5, 8, 13, ..)Φ

Questions?