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CS 2800: Discrete Structures Fall 2014 Sid Chaudhuri Mike George

CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

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Page 1: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

CS 2800: Discrete Structures

Fall 2014

Sid ChaudhuriMike George

Page 2: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Discrete Structures

wallippo.com

Page 3: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Continuous Structures

miriadna.com

Page 4: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

A Discreet Structureindiefix.com

Page 5: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

A Discreet Structureindiefix.com

Page 6: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Things we can count with the integers

ser4kids.com

Page 7: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Prime Numbers

A number with exactly two divisors:1 and itself

2, 3, 5, 7, 11, 13, 17...

Page 8: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

PrimeNumbers

eldhughes.com, pleasureinlearning.com

Page 9: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

PrimeNumbers

pleasureinlearning.com

Page 10: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

How many prime numbers exist?

Page 11: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

How many prime numbers exist?

1,000?

Page 12: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

How many prime numbers exist?

1,000?1,000,000?

Page 13: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

How many prime numbers exist?

1,000?1,000,000?

An infnite number?

Page 14: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

(~300BC)

Page 15: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes

Page 16: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p

Page 17: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p● Consider n = (1 × 2 × 3 × ... × p) + 1

Page 18: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p● Consider n = (1 × 2 × 3 × ... × p) + 1

● n cannot be prime (p is the largest)

Page 19: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p● Consider n = (1 × 2 × 3 × ... × p) + 1

● n cannot be prime (p is the largest)● Therefore it has a (prime) divisor < n

Page 20: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p● Consider n = (1 × 2 × 3 × ... × p) + 1

● n cannot be prime (p is the largest)● Therefore it has a (prime) divisor < n

● But no number from 2 to p divides n

Page 21: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p● Consider n = (1 × 2 × 3 × ... × p) + 1

● n cannot be prime (p is the largest)● Therefore it has a (prime) divisor < n

● But no number from 2 to p divides n● So n has a prime divisor greater than p

Page 22: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Euclid's Proof of Infnitude of Primes

● Suppose there is a fnite number of primes● Then there is a largest prime, p● Consider n = (1 × 2 × 3 × ... × p) + 1

● n cannot be prime (p is the largest)● Therefore it has a (prime) divisor < n

● But no number from 2 to p divides n● So n has a prime divisor greater than p

Contradiction!!!

Page 23: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Discrete Structures

● Number theory● Proof systems● Sets, functions, relations● Counting and probability

Page 24: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 25: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 26: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 27: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 28: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

Is there a city tour that crosseseach bridge exactly once?

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 29: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

Leonhard Euler(1707-1783)

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 30: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

A

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 31: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 32: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

C

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 33: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

C

D

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 34: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

C

D

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 35: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

C

D

Enter by new bridge,Leave by new bridge

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 36: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

C

D

Bridges have toform pairs

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 37: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

B

A

C

D

Odd # of bridgesto each landmass⇒ no solution!

Braun & Hogenberg, “Civitates Orbis Terrarum”, Cologne 1585. Photoshopped to clean up right side and add 7 th bridge.

Page 38: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Bridges of Königsberg

● Cross each bridge once: Euler Path– Easy for a computer to calculate

● Visit each landmass once: Hamiltonian Path– Probably very hard for a computer to calculate– If you can fnd an efcient solution, you will get $1M

and undying fame (answers “P = NP?”)– (Will also break modern crypto, collapse the banking

system, revolutionize automated mathematics and science, bring about world peace...)

Page 39: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

You'll also be terrifc at Minesweeper

baslerweb.com

Page 40: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Discrete Structures

● Number theory● Proof systems● Sets, functions, relations● Counting and probability● Graph theory● Models of computation, automata, complexity

Page 41: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

This sentence is false.

Page 42: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

This sentence is false.If true, it is falseIf false, it is true

Page 43: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

This sentence is false.If true, it is falseIf false, it is true

Page 44: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

Discrete Structures

● Number theory● Proof systems● Sets, functions, relations● Counting and probability● Graph theory● Models of computation, automata, complexity● Logic● Decidability, computability

Page 45: CS 2800: Discrete Structures · Bridges of Königsberg Cross each bridge once: Euler Path – Easy for a computer to calculate Visit each landmass once: Hamiltonian Path – Probably

warrenphotographic.co.uk, auntiedogmasgardenspot.wordpress.com


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