IST 4Information and Logic

mon tue wed thr fri30 M1 1= todayT

6 M1

13 1 2M2x= hw#x out

oh oh

oh oh

= todayT

M

20 227 M2

x= hw#x due

idt

oh oh

h h T27 M2

4 311 3 4Mx= MQx out

midtermsoh

oh oh

oh = office hours oh T

11 3 418 4 525

Mx= MQx out

Mx= MQx dueoh oh

ohoh

25

1 5oh

oh

oh

oh

So far… True for any Boolean Algebra

T0: duality principle

T1: Single complement per element

L1: self Absorption L1: self Absorption

B l Al bBoolean AlgebraIt is all about syntax...

Boolean ‘editing’

Absorption Theoremp

Th 2Theorem 2:

Absorption Theoremp

Th 2Theorem 2:

Can delete anything!!Can delete anything!!

Absorption Theoremp

Th 2Theorem 2:

Can delete anything!!Can delete anything!!

Absorption Theoremp

Th 2Theorem 2:

Can insert anything!!

Absorption Theoremp

Th 2Theorem 2:

Proof: A1

A4

Absorption Theoremp

Th 2Theorem 2:

Proof: A1

A4A4

??

A1

Q

Simple Absorptionmp p

L 2

A proof of a theorem can help in identifying useful lemmas

Lemma 2:

Simple Absorptionmp p

L 2Lemma 2:

Proof: Q

A1

A2A3

A4

A3 A1A3 A1A2

Absorption Theoremp

Th 2

A proof of a theorem can help in identifying useful lemmas

Theorem 2:

Proof: A1

A4A4

Lemma 2

A1Q

P f FUNProofs are FUN…

N t fNot every proof...

My Ranking of Proofs

Correct and short

Wrong and funWrong and fun

Correct but painful

Wrong and long

PppBoolean wisdom

Mary Boole1832 19161832– 1916

Proofs are fun: B ti t d k it!Be patient and crack it!

B l Al bBoolean AlgebraNow it is your turn

The Positive Win:

The Positive Win:

Proof:

A2

A4

A2

A3

A1Q

B l Al bBoolean AlgebraOne more...

Derive the answer?

A3

A4

A3

A2

A4

A1

B wins:

The positive win:The positive win:

Absorption:

Current state

Next?

B l Al bBoolean Algebraassociativity does not need to be an axiomneed to be an axiom

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) We will prove

the other one follows by duality

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) So far we have seen ‘linear proofs’p

Arrows are axioms, lemmas, theorems....

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) A ‘nonlinear proof’ – has an architecture

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) A ‘nonlinear proof’ – has an architecture

3Prove 1 Prove 2

1 2 4

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) A ‘nonlinear proof’ – has an architecture

3Prove 1 Prove 2

1 2 4

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) A ‘nonlinear proof’ – has an architecture

3Prove 1 Prove 2

1 2 4

multiply

Associativity Theoremy

Th 3Theorem 3:

f ( d ) Proof: (ideas) A ‘nonlinear proof’ – has an architecture

3Prove 1 Prove 2 Prove 3 and 4

1 2 4

multiply

Associativity Theoremy

Th 3Theorem 3:

fProof:B wins:

3Prove 1 Prove 2 Prove 3 and 4

1 2 4

multiply

1 2

3

4

Prove 1

‘text editor’T2

T2

T2

A4

QQ

1 2

3

4

Prove 2

Will appear in HW#3

Associativity Theoremy

Th 3Theorem 3:

fProof:Q

3Proved 1 Prove 2 – HW#3 Proved 3 and 4

1 2 4

multiply

B l Al bBoolean AlgebraDeMorgan Theorem

The complements of the sum and productsum and product

DeMorgan Theoremg

Th 4Theorem 4:

We will prove

the other one follows by duality

DeMorgan Theoremg

P f N d t Proof: Need to prove:

Need to prove that are complementsandNeed to prove that are complementsand

DeMorgan Theoremg

P f N d t Proof: Need to prove:

Need to prove that are complementsand

Idea: need to validate A2Need to prove: HW#3

Need to prove that are complementsand

next

DeMorgan TheoremgQ

HW#3

A4

HW#3

A4

T3A3

A2

L2

Q

L2

L1Q

T3.

DeMorgan Theoremg

Th 4Theorem 4:Q: Simple proof of DeMorgan without Associativity??

We will prove y

Brian LawrenceIST4 2009

Michael GottliebIST4 2009

DeMorgan TheoremgProof with Associativity: Q

i 0 HW#3

A4

is 0 HW#3

A4

T3A3Identify a lemmaA2

L2

Q

L2

L1Q

T3.

DeMorgan Theoremg

Proof f th lemma:

Proof without Associativity:

Proof of the lemma:is 0

A1

A2

A3 A4

A3 T2

A3A2

So Far so Good...

Syllogism to Algebra G B l 1847George Boole, 1847

George Boole1815 –1864

George Boole Early Days

George Boole1815-1864 Early Days5 6

Born in Lincoln England Born in Lincoln, England an industrial town

When his was 15 he had to go to His father was a shoemaker with a passion for mathematics and science

When his was 15 he had to go to work to support his family, he became a math teacher in the Wesleyan Methodist academy in

When George was 8 he surpassed his father’s knowledge in mathematics

y yDoncaster (40 miles…)

Lost his job after two years….knowledge in mathematics

By age 14 he was fluent in Latin German French Italian

Lost two more teaching jobs…

When he was 20 he opened his Latin, German, French, Italian and English… and algebra…

pown school in his hometown -Lincoln

George Boole1815-1864

George BooleEarly Career5 6

Born in Lincoln, England

y

Born in Lincoln, England an industrial town

In 1841 (26) he published three papers in the newly establishedCambridge Mathematical Journal (edited by DG).

1844 (29) h bli h d “O G l M h d f l i ” h In 1844 (29) he published “On a General Method of Analysis”; he considered it to be his best paper. This paper won the first (newly established) Gold medal for Mathematics awarded by Royal Society

In 1846 (31) he applied for a professor position in the newly established Queen’s College – 3 campuses in Ireland

source: wikipedia

In 1847 (32) ‘while waiting to hear from Ireland’, he published “The Mathematical Analysis of Logic”

George Boole1815-1864

8848 m 29,029 ft George BooleIreland 5 6

Born in Lincoln, England Born in Lincoln, England an industrial town

I 1847 (32) ‘ hil i i h f I l d’ In 1847 (32) ‘while waiting to hear from Ireland’, he published “The Mathematical Analysis of Logic”

In 1849 (34) his was offered a position of the first professor In 1849 (34), his was offered a position of the first professor of mathematics at Queen’s college at Cork

He married Mary Everest (1832- 1916) in 1855 (23,40)d h h d f d hand they had five daughters

Niece of George Everest (Mt. Everest…) led the expedition to map the Himalayasled the expedition to map the Himalayas

source: wikipediaIn 1864, died of pneumonia (49)

1849-1864 taught at George BooleGeography

Born in Lincoln, England an industrial town

1849 1864, taught at at Queen’s college in Cork

g p y

an industrial town

source: wikipedia

1849-1855: lived here until got married

Cork, Ireland Source: www.flickr.com

http://georgeboole.com/