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.