IST 4 Information and Logic

Transcription

1 IST 4 Information and Logic

2 T = today x= hw#x out x= hw#x due mon tue wed thr fri 30 M1 1 6 oh M1 oh 13 oh 1 oh 2M2M 20 oh oh 2 Mx= MQx out 27 oh M2 h T oh = office hours oh T oh 3 4 oh oh midterms oh Mx= MQx due 18 oh oh oh 5 oh oh oh

3 So far True for any Boolean Algebra T0: duality principle T1: Single complement per element L1: self Absorption

4 Boolean Algebra It is all about syntax... Boolean editing

5 Absorption Theorem Theorem 2:

6 Absorption Theorem Theorem 2: Can delete anything!!

7 Absorption Theorem Theorem 2: Can delete anything!!

8 Absorption Theorem Theorem 2: Can insert anything!!

9 Absorption Theorem Theorem 2: Proof: A1 A4

10 Absorption Theorem Theorem 2: Proof: A1 A4?? A1 Q

11 Lemma 2: Simple Absorption A proof of a theorem can help in identifying useful lemmas

12 Simple Absorption Lemma 2: Proof: Q A1 A3 A2 A4 A3 A1 A2

13 Absorption Theorem A proof of a theorem can help in identifying useful lemmas Theorem 2: Proof: A1 A4 Lemma 2 Q A1

14 Proofs are FUN Not every proof...

15 My Ranking of Proofs Correct and short Wrong and fun Correct but painful Wrong and long

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24 Boolean wisdom Ppp Proofs are fun: Be patient t and crack it! Mary Boole

25 Boolean Algebra Now it is your turn

26 The Positive Win:

27 The Positive Win: Proof: A4 A2 A3 A1 Q

28 Boolean Algebra One more...

29 Derive the answer? A3 A4 A2 A1

30 B wins: The positive win: Absorption:

31 Next? Current state

32 Boolean Algebra associativity does not need to be an axiom

33 Associativity Theorem Theorem 3: Proof: (d (ideas) We will prove the other one follows by duality

34 Associativity Theorem Theorem 3: Proof: (d (ideas) So far we have seen linear proofs Arrows are axioms, lemmas, theorems...

35 Associativity Theorem Theorem 3: Proof: (d (ideas) A nonlinear proof has an architecture

36 Associativity Theorem Theorem 3: Proof: (d (ideas) A nonlinear proof has an architecture Prove 1 Prove

37 Associativity Theorem Theorem 3: Proof: (d (ideas) A nonlinear proof has an architecture Prove 1 Prove

38 Associativity Theorem Theorem 3: Proof: (d (ideas) A nonlinear proof has an architecture Prove 1 Prove multiply

39 Associativity Theorem Theorem 3: Proof: (d (ideas) A nonlinear proof has an architecture Prove 1 Prove 2 Prove 3 and multiply

40 Associativity Theorem Theorem 3: Proof: B wins: Prove 1 Prove 2 Prove 3 and multiply

41 Prove 1 T2 T2 T2 A4 text editor Q

42 Prove 2 Will appear in HW#3

43 Associativity Theorem Theorem 3: Proof: Q Proved 1 Prove 2 HW#3 Proved 3 and multiply

44 Boolean Algebra DeMorgan Theorem The complements of the sum and product

45 DeMorgan Theorem Theorem 4: We will prove the other one follows by duality

46 DeMorgan Theorem Proof: Need to prove: Need to prove that and are complements

47 DeMorgan Theorem Proof: Need to prove: Need to prove that Need to prove: HW#3 and are complements Idea: need to validate A2 next

48 DeMorgan Theorem A4 Q HW#3 A3 T3 A2 L2 L1 Q T3.

49 DeMorgan Theorem Theorem 4: Q: Simple proof of DeMorgan without Associativity?? Michael Gottlieb IST Brian Lawrence IST We will prove

50 DeMorgan Theorem Proof with Associativity: Q is 0 HW#3 A4 Identify a lemma A2 L2 A3 T3 L1 Q T3.

51 Proof of the lemma: DeMorgan Theorem Proof without Associativity: is 0 A1 A2 A3 A3 A2 A4 T2 A3

52 So Far so Good...

53 Syllogism to Algebra George Boole, 1847 George Boole

54 George Boole George Boole Early Days Born in Lincoln, England an industrial town His father was a shoemaker with a passion for mathematics and science When George was 8 he surpassed his father s knowledge in mathematics By age 14 he was fluent in Latin, German, French, Italian and English and algebra 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 Doncaster (40 miles ) Lost his job after two years. Lost two more teaching jobs When he was 20 he opened his own school in his hometown - Lincoln

55 George Boole George Boole Early Career Born in Lincoln, England an industrial town In 1841 (26) he published three papers in the newly established Cambridge Mathematical Journal (edited by DG). 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 In 1847 (32) while waiting to hear from Ireland, he published The Mathematical Analysis of Logic source: wikipedia

56 George Boole George Boole Ireland 8848 m 29,029 ft Born in Lincoln, England an industrial town In 1847 (32) while waiting i to hear from Ireland, he published The Mathematical Analysis of Logic In 1849 (34), his was offered a position of the first professor of mathematics at Queen s college at Cork He married Mary Everest ( ) in 1855 (23,40) and they had five daughters Niece of George Everest (Mt. Everest ) led the expedition to map the Himalayas In 1864, died of pneumonia (49) source: wikipedia

57 , taught at at Queen s college in Cork George Boole Geography Born in Lincoln, England an industrial town source: wikipedia

58 : lived here until got married Cork, Ireland Source:

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