Strategies for Proofs

Transcription

1 Strategies for Proofs Landscape with House and Ploughman Van Gogh Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1

2 Goals of this lecture A bit more logic Reviewing Implication Nested quantifiers Various proof strategies Direct proofs [this week] Proof by cases [this week] Proof by Contrapositive [this week] Proof by Contradiction Induction 2

3 Brief Review Propositional logic AND, OR, T/F, implies, etc. Equivalence and truth tables Manipulating propositions

4 Predicate logic (first order logic) Example: Consider the universe of integers, with constants 0, 1, 2,, functions +,, *, and relations, =,, etc. Every number n added to 0 gives n. There is some number which when multiplied by itself is 0:. There are two numbers whose sum is 1..

5 Quantifiers For some or there exists some x: Some creatures are greyhounds that run fast., 0, 0 For all : For all creatures, if it is fat, then it does not run well., 0 For exactly one :! There is exactly one fat creature than runs well.!, 0

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8 Manipulating quantifiers Negation,,,, Not all dogs are fat is equivalent to At least one dog is not fat. There does not exist one fat dog is equivalent to All dogs are not fat. Contrapositive,,

9 Manipulating quantifiers Negation,,,, Not every number is even is equivalent to There is some number that is not even There is some number that is prime is equivalent to Not all numbers are composite. Contrapositive,, Every number that is a square is a composite number. Every number that is a prime number is not a square.

10 Quantifiers with two variables For all integers and, false,, or,, There are two integers whose sum is 12 (true),. +b = 12) For every real, there exists an integer, such that (true),,

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12 Proving universal statements Claim: For any integers and, if and are odd, then is also odd. Definition: integer is odd iff 2 1 for some integer

13 Note some aspects of the proof 1. Define your variables! You cannot simply start using a variable without telling the reader what it is! 2. Every step of your proof should be a simple verifiable statement. Explain why the argument holds as much as you can. E.g., 1 0 because any squared value is non negative. 3. Make sure you end with the claim you wanted to prove! Tip: work out the proof on scratch paper first, then rewrite it in a clear, logical order with justification for each step.

14 Proving universal statements Claim: For all integers n, 4(n 2 + n + 1) 3n 2 is a perfect square. Definition: is a perfect square iff for some integer

15 Logical equivalences Negating quantified formulas Distributing quantifiers over Bool connectives 15

16 Proving existential statements Claim: There exists a real number x, such that 16

17 Disproving universal statements Claim to disprove: For all real,

18 Disproving existential statements Claim to disprove: There exists a real,

19 Proof by cases 19

20 Proof by cases Claim: For every real x, if 7 8, then 1 20

21 Proof by cases Claim: For every real x, if 7 8, then 1 21

22 Rephrasing claims Claim: There is no integer, such that is odd and is even. 22

23 Proof by contrapositive 23

24 Proof by contrapositive Claim: For all integers and,

25 Proof strategies 1. Does this proof require showing that the claim holds for all cases or just an example? Show all cases: prove universal, disprove existential Example: disprove universal, prove existential 2. Can you figure a straightforward solution? If so, sketch it and then write it out clearly, and you re done 3. If not, try to find an equivalent form that is easier a) Divide into subcases that combine to account for all cases OR in hypothesis is a hint that this may be a good idea b) Try the contrapositive OR in conclusion is a hint that this may be a good idea c) More generally rephrase the claim: convert to propositional logic and manipulate into something easier to solve 25

26 More proof examples Claim: For integers and, if is even or is even, then is even. Definition: integer is even iff 2 for some integer 26

27 More proof examples Claim: For all integers, if 3 5 is even, then is odd. 27

28 More proof examples Claim: For all integers, if is odd, then 4 1 or 4 1 for some integer. (Note, this requires knowing a little about modular arithmetic.) 28

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30 E.g., Clubs and strangers Assume: For any pair of people, either they have met or not. If every pair of people in a group has met, we ll call the group a club. If every pair of people in a group has not met, we ll call it a group of strangers. 30

31 E.g., Clubs and strangers Prove that any group of 6 people includes a club of 3 people or a group of 3 strangers. 31

32 E.g., Clubs and strangers: proof 32

33 E.g., Clubs and strangers: proof 33

34 Ramsey s Theorem For any k, there exists an R such that any group with at least R people will have either k member club, or k mutual strangers. R(3)=6 R(4)=18 R(5) not known! But lies between 43 and 48 34

35 Proof from a recent paper of mine: NEXT CLASS: MORE PROOF STRATEGIES; CONTRAPOSITIVE, CONTRADICTION, INDUCTION 35

36 Reading resources MIT textbook Fleck textbook Madhu s ``succinct notes : Written during this course. Idea: to summarize the contents of lectures, not to describe them from scratch. If you don t understand something here, it s fine. Come back to it later, or ask a staff member. 36

37 Logistics Lectures: Tue/Thu ; Discussions: Friday; Online disc: Piazza Reading quiz: One due Wed (9/6) night (on Moodle) Minihomework: One due Thursday (9/7) night (on Moodle) Longform homework: One will be posted this week (Fri); due Thu next week. Examlet: No examlet this week or next week. Office hours: posted on course website; go to them! Next lecture: More proof strategies. 37

38 Next Week I am traveling! - Tuesday s lecture: unclear; keep an eye on Piazza - Thursday s lecture: Ian Ludden 38