lectures9-10

lectures9-10 - Lecture 9 Proofs, Intro to sets...

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Lecture 9 Proofs, Intro to sets
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Announcements Hw1 on GRIT Hw2 due Friday Study groups
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Recap Basic proof techniques Direct proof Proof by contraposition Proof by contradiction Proof by cases Existence proofs Constructive vs. non-constructive Backward reasoning
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Backward reasoning example Prove that (x+y)/2 > (xy) whenever x y and x and y are positive
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Backward reasoning example Consider a game where two players remove 1,2 or 3 stones from a pile of 15 stones. The person who removes the last stone wins the game. Claim: there exists a winning strategy for player 1
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Backward reasoning Player 1 wins: Last (Player 1) move: 1,2 or 3 stones on the pile. Previous (Player 2) move: 4 stones Previous (Player 1) move: 5,6,7 stones Previous (Player 2) move: 8 stones Previous (Player 1) move: 9, 10, 11 stones Previous (Player 2) move: 12 stones Player 1 move: 13, 14,15 stones We can now construct the winning strategy
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Vacuous and Trivial Proofs When proving p q If p is false, the statement is true Vacuous proof If q is (independently) true, the statement is true Trivial Proof These often show up as special cases in proof by cases
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Recap Basic proof techniques Direct proof Proof by contraposition Proof by contradiction Proof by cases Existence proofs Backward reasoning
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Discussion Proofs Reasoning techniques Rest of the course
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Exercise: A theorem about friendship Suppose two people are either friends or strangers Three people a, b and c are mutual friends iff (a and b are friends) and (b and c are friends) and (a and c are friends) Three people a, b and c are mutual strangers iff (a and b are strangers) and (b and c are strangers) and (a and c are strangers) Prove or disprove: In a group of six people, there are either three mutual friends or three mutual strangers.
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Proof Let A be an arbitrary person in the group. Claim: A has the same type of relationship with at least three people. Proof: Case 1: A has three or more friends. The claim holds. Case 2: A has less than three friends. In this case, there are at least three people who are strangers to A.
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Proof (cont) Suppose A has three friends. (The proof for the case when there are three strangers to A is symmetric) Let B,C and D be A’s friends.
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lectures9-10 - Lecture 9 Proofs, Intro to sets...

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