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Advanced Topics in Mathematics – Logic and Metamathematics Mr. Weisswange Assignment #14 Logic 1. Prove A B \ C A B \ C 2. Prove that for any sets A and B, P A P B P A B 3. Prove that for every real number x, if x 3 3 then x 2 6 x . 4. Prove that for every integer x, x 2 x is even. 5. Consider the following (possible) theorem: For every real number x, if x 3 3 then 0 x 6 . Is the following proof correct? If so, what proof strategies does it use. If not, can it be fixed? Is the theorem correct? Proof(?) Let x be an arbitrary real number, and suppose x 3 3 . We consider two cases. Case 1. x 3 0 . Then x 3 x 3 . Plugging this into the given, we get x 3 3 , so clearly x 6 . Case 2. x 3 0 . Them x 3 3 x . Similarly plugging in, we get 3 x 3 , so 0 x . Since we have proven both 0 x and x 6 , we can conclude that 0 x 6 . Metamathematics Read Chapter XII (pp. 369-390) and “Aria with Diverse Variations” (pp. 391-405). I especially recommend that you read the “Aria” dialogue—it is full of many interesting mathematical facts and factoids for you to delve into.