Post on 25-May-2018
mon tue wed thr fri30 M1 1= todayT
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13 1 2M2x= hw#x out
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So far… True for any Boolean Algebra
T0: duality principle
T1: Single complement per element
L1: self Absorption L1: self Absorption
Absorption Theoremp
Th 2
A proof of a theorem can help in identifying useful lemmas
Theorem 2:
Proof: A1
A4A4
Lemma 2
A1Q
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!
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:Q
3Proved 1 Prove 2 – HW#3 Proved 3 and 4
1 2 4
multiply
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 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
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