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AC
DB
{ 1, 2, 3, 4, 5.
P → Q, ⌐Q → ⌐P
(1, 1, 2, 3, 5, 8, 13, ..)Φ
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
AC
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?