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Unformatted text preview: Math 100 Quantifiers and Games Martin H. Weissman 1. The ∀∃ game 1.1. The first game. This game relates to the following sentence: ∀ x ∈ R , ∃ y ∈ Z such that y > x. • Player 1: Choose a real number, and call it x . • Player 2: Choose an integer, and call it y . • Game over. Player 2 wins, if y > x . Does Player 2 have a winning strategy? What is it? Try playing multiple times, until you are convinced of your answer. In other words, if Player 2 tries, can he or she always win (no matter which number Player 1 picks)? If so, then the sentence: “ ∀ x ∈ R , ∃ y ∈ Z such that y > x ” is true. 1.2. The second game. This game relates to the following sentence: ∀ M ∈ R , ∃ x ∈ R , such that x 3 x + 1 > M. • Player 1: Choose a real number, and call it M . • Player 2: Choose a real number, and call it x . • Game over. Player 2 wins, if x x + 1 > M . Does Player 2 have a winning strategy? What is it? If so, then the sentence “ ∀ M ∈ R , ∃ x ∈ R , such that x 3 x + 1 > M. ” is true. How is this related to the last homework assignment? 1.3. The third game. This game relates to the following sentence: ∀ z ∈ R , ∃ x ∈ R , ∃ y ∈ R , such that x 2 2 xy...
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This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.
 Fall '08
 Weissman
 Math

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