Class 2: Proof Methods

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. cs2102: Discrete Mathematics Class 2: Proof Methods David Evans University of Virginia If you have any questions about syllabus, course organization, etc. ask them in #inclass now
2. Plan Review proof from end of last class How rigorous should proofs be? Aside: “Electronic Signatures” Proof Methods: – “If and only if” What makes a “good” proof?
3. If the product of x and y is even, at least one of x or y must be even.
4. If the product of x and y is even, at least one of x or y must be even.
5. If the product of x and y is even, at least one of x or y must be even. P = “the product of x and y is even” Q = “at least one of x or y must be even” What definitions do we need? Goal: prove P implies Q
6. If the product of x and y is even, at least one of x or y must be even. P = “the product of x and y is even” Q = “at least one of x or y must be even” Goal: prove P implies Q
7. If the product of x and y is even, at least one of x or y must be even. P = “the product of x and y is even” Q = “at least one of x or y must be even” Goal: prove P implies Q Definition: even. An integer, z, is even if there exists an integer k such that z = 2k.
8. If the product of x and y is even, at least one of x or y must be even. P = “the product of x and y is even” Q = “at least one of x or y must be even” Goal: prove P implies Q Definition: even. An integer, z, is even if there exists an integer k such that z = 2k. Definition: odd. An integer, z, is odd if there exists an integer k such that z = 2k + 1.
9. Definition: even. An integer, z, is even if and only if there exists an integer k such that z = 2k. Definition: odd. An integer, z, is odd if and only if there exists an integer k such that z = 2k + 1. Odd-Even Lemma: If an integer is not even, it is odd.
10. Definition: even. An integer, z, is even if and only if there exists an integer k such that z = 2k. Definition: odd. An integer, z, is odd if and only if there exists an integer k such that z = 2k + 1. Odd-Even Lemma: If an integer is not even, it is odd. Prove an implication: assume P, show chain that leads to Q. Assume z is an integer that is not even. Substitute definition: NOT(exists integer k such that z = 2k)
11. Lemma: If an integer is not even, it is odd. Prove an implication: assume P, show chain that leads to Q. Assume z is an integer that is not even. Substitute definition: NOT(exists integer k such that z = 2k) Meaning of NOT: no integer k exists such that z = 2k
12. Lemma: If an integer is not even, it is odd. Prove an implication: assume P, show chain that leads to Q. Assume z is an integer that is not even. Substitute definition: NOT(exists integer k such that z = 2k) Meaning of NOT: no integer k exists such that z = 2k
13. “Electronic Signatures” Digression
14. Ritualistic Signatures • Intended to make signer think seriously and impact signer’s behavior • Varying impact (?): – Quill on parchment – Pen on paper – Scribbling on tablet – Typing
15. Ritualistic Signatures • Intended to make signer think seriously and impact signer’s behavior • Varying impact (?): – Quill on parchment – Pen on paper – Scribbling on tablet – Typing Purposeful Signatures • Verifiable • Non-repudiable • Bound to content
19. Purposeful Signatures Verifiable Non-repudiable Bound to content Real electronic signatures: One-way hard problem Easy to raise to powers, hard to find discrete logs Signature combines message and private key Can be verified by obtaining public key from trusted source and checking signature is valid
20. Signing Pledges
21. If the product of x and y is even, at least one of x or y must be even. P = “the product of x and y is even” Q = “at least one of x or y must be even” Goal: prove P implies Q Definition: even. An integer, z, is even if there exists an integer k such that z = 2k. Lemma: If an integer is not even, it is odd.
22. P = “the product of x and y is even” Q = “at least one of x or y must be even” To prove, P implies Q, we use contrapositive inference rule: NOT(Q) implies NOT(P) P implies Q Observe: this is starting backwards! We are starting the proof from the conclusion we want.
23. P = “the product of x and y is even” Q = “at least one of x or y must be even” To prove, P implies Q, we prove the contrapositive: NOT(Q) implies NOT(P) To prove an implication, assume left side, show right: Assume NOT(at least one of x and y must be even)
24. P = “the product of x and y is even” Q = “at least one of x or y must be even” To prove, P implies Q, we prove the contrapositive: NOT(Q) implies NOT(P) To prove an implication, assume left side, show right: Assume NOT(at least one of x and y must be even) By the meaning of NOT: both x and y are not even
25. P = “the product of x and y is even” Q = “at least one of x or y must be even” To prove, P implies Q, we prove the contrapositive: NOT(Q) implies NOT(P) To prove an implication, assume left side, show right: Assume NOT(at least one of x and y must be even) By the meaning of NOT: both x and y are not even By the Odd-Even Lemma: both x and y are odd
26. P = “the product of x and y is even” Q = “at least one of x or y must be even” To prove, P implies Q, we prove the contrapositive: NOT(Q) implies NOT(P) To prove an implication, assume left side, show right: Assume NOT(at least one of x and y must be even) By the meaning of NOT: both x and y are not even By the Odd-Even Lemma: both x and y are odd By the definition of odd: there exists integers k, m, such that x = 2k + 1 and y = 2m + 1
27. P = “the product of x and y is even” Q = “at least one of x or y must be even” We prove the contrapositive: Assume NOT(at least one of x and y must be even) By the meaning of NOT: both x and y are not even. By the Odd-Even Lemma: both x and y are odd. By the definition of odd: there exists integers k, m, such that x = 2k + 1 and y = 2m + 1 By algebra: xy = (2k + 1)(2m + 1) = 4mk + 2m + 2k + 1 = 2(2mk + m + k) + 1
28. P = “the product of x and y is even” Q = “at least one of x or y must be even” We prove the contrapositive: Assume NOT(at least one of x and y must be even) … By integers closed under multiplication and addition: there exists some integer r where r = 2mk + m + k By definition of odd: So xy = 2r + 1 which means the product of x and y is odd. By the Odd-Even Lemma: the product of x and y is not even.
29. P = “the product of x and y is even” Q = “at least one of x or y must be even” We prove the contrapositive: Assume NOT(at least one of x and y must be even) … By integers closed under multiplication and addition: there exists some integer r where r = 2mk + m + k By definition of odd: So xy = 2r + 1 which means the product of x and y is odd. By the Odd-Even Lemma: the product of x and y is not even. Lemma: If an integer is not even, it is odd.
30. P = “the product of x and y is even” Q = “at least one of x or y must be even” We prove the contrapositive: Assume NOT(at least one of x and y must be even) … By the Even-Odd* Lemma: the product of x and y is not even. So, NOT(product of x and y is even) Thus, we have proven the implication: NOT(at least one of x and y must be even) implies NOT(product of x and y is even) As shown in class, this slide had bugs in it - the version here is fixed
31. P = “the product of x and y is even” Q = “at least one of x or y must be even” We prove the contrapositive: Assume NOT(at least one of x and y must be even) … So, NOT(product of x and y is even) Thus, we have proven the implication: NOT(at least one of x and y must be even) implies NOT(product of x and y is even) By the contrapositive inference rule, this proves: at least one of x and y must be even implies product of x and y is even.
32. Ay, Harambe!
33. Purpose of a Proof Seeking Truth Deepening Understanding
34. In physics, your solution should convince a reasonable person. In math, you have to convince a person who's trying to make trouble. Frank Wilczek (2004 Nobel Prize Physics)
35. “Correct ” Rigorous Proof of P Axioms used are clear and accepted Each step uses a sound inference rule correctly: – Shows antecedents are satisfied – Concludes the conclusion Results in concluding goal proposition: P Do the proofs we do in cs2102 actually do this?
36. “Good” Proofs in cs2102 • Well written and clearly organized: – Should be obvious what you are proving and how • Convincing to a skeptical reader • State assumptions clearly: careful about not assuming non-obvious things • Focus on important steps, not gory details
37. Whitehead and Russell, Principia Mathematica
38. Automated Theorem Proving Coq PVS
39. Example Automated Proofs (thm (implies (and (evenp x) (evenp y)) (evenp (+ x y)))) From https://rosettacode.org/wiki/Proof ACL2
40. Inductive nat : Set := | O : nat | S : nat -> nat. Inductive even : nat -> Set := | even_O : even O | even_SSn : forall n:nat, even n -> even (S (S n)). Theorem even_plus_even : forall n m:nat, even n -> even m -> even (n + m). Proof. … (about 10 lines) Qed. From https://rosettacode.org/wiki/Proof Coq Human-assisted, automatically verified
41. Automatic Theorem Provers in Common Practice?
42. Proposition: input program does not have certain types of errors.
43. Processor Verification FDIV 0x4005FB / 0x2FFFFF
44. “Miracle on the Hudson” ?
46. Charge • Next week: we’ll start proving non-obvious things! • Before Friday (6:29pm): course pledge, registration survey (read “Habits of Highly Mathematical People”) • Read MCS Ch 2, 3; PS1 is due next Friday • TA’s office hours will start Monday