IST 4 Information and Logic

Transcription

1 IST 4 Information and Logic

2 T = today mon tue wed thr 3 M1 oh 1 fri sun x= hw#x out 10 oh M1 17 oh oh 1 2 M2 oh oh x= hw#x due 24 oh oh 2 Mx= MQx out 1 oh M2 oh = office hours oh T oh 3 4 oh oh midterms oh oh Mx= MQx due 22 oh oh oh oh oh 5 oh 5 oh

3 HW2 will be returned today average is B

4 Boolean Algebra helps us we did about paradigms that T0: duality principle T1: Unique complement per element L1: self Absorption

5 More on the Complement T1: one complement per element We also proved that an element is not self-complement Q: Is the complement of the complement the original element? True for two elements? Four elements?

6 Saying No Twice? uniqueness of complement (T1) NONO Property: Proof: Need to prove that is the complement of Definition of complement: A2 A3 Q

7 Boolean Algebra How many elements?

8 Is there a Boolean algebra with 3 elements? Idea: use the properties of the complement NO!! Must have an even number of elements!? Question resolved only in

9 Quiz #5 review

10 0-1 Boolean Algebra: set of elements B={0,1} two binary operations OR and AND xy OR(x,y) xy Prove that the following statement is true for a 0-1 Boolean algebra: AND(x,y) a =

11 0-1 Boolean Algebra: set of elements B={0,1} two binary operations OR and AND xy OR(x,y) xy Prove that the following statement is true for a 0-1 Boolean algebra: AND(x,y) a =

12 0-1 Boolean Algebra: set of elements B={0,1} two binary operations OR and AND xy OR(x,y) xy Prove that the following statement is true for a 0-1 Boolean algebra: Generalization: AND(x,y)

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

14 Absorption Theorem Theorem 2:

15 Absorption Theorem Theorem 2: Can delete anything!!

16 Absorption Theorem Theorem 2: Can delete anything!!

17 Absorption Theorem Theorem 2: Can insert anything!!

18 Absorption Theorem Theorem 2: Proof: options for the first step? A1 A4

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

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

21 Lemma 2: Simple Absorption A proof of a theorem can help in identifying useful lemmas You can use associativity = T* You can use self absorption = L1 A2 A3 T* L1 A3 A2 Q

22 Simple Absorption Lemma 2: without associativity Proof: Q A1 A3 A2 A4 A3 A1 A2

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

24 Proofs are FUN Not every proof...

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

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34 Boolean wisdom Ppp Mary Boole Do not trust first impressions Proofs are fun: Be patient and crack it!

35 Boolean Algebra Now it is your turn

36 The Positive Wins:

37 The Positive Wins: Proof: A4 A2 A3 A1 Q

38 Boolean Algebra One more...

39 Derive the answer? A3 A4 A2 A1

40 B wins: The positive wins: Absorption:

41 Next? Current state

42 Boolean Algebra associativity does not need to be an axiom

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

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

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

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

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

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

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

50 Associativity Theorem Theorem 3: Proof: B wins: Prove 1 Prove 2 Prove 3 and 4?? multiply

51 Prove 1 T2 T2 T2 A4 text editor Q

52 Prove 2 Will appear in HW#3

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

54 A question about complements: What is the complement of:

55 Pythagoras BC Boolean Algebra DeMorgan DeMorgan Theorem The complements of the sum and product

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

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

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

59 DeMorgan Theorem Q HW#3 A4 A3 T3 A2 L2 A1 Q T3.

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

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

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

63 So Far so Good...

64 Pre-quiz: What is the complement of Use the following DeMorgan Laws: And the axioms:

65 Pre-quiz: What is the complement of: DM DM A3 A4

66 Quiz time

67 Quiz #6 10min What is the complement of Use the following DeMorgan Laws: And the axioms: