Advanced Topics in Mathematics – Logic and Metamathematics
Mr. Weisswange
Assignment #11
Logic
1. Prove that if x  P  x   Q  x   is true, then x P  x   xQ  x  must also be true.
2. Prove that if A and B \ C are disjoint, then A  B  C .
3. Suppose A  P  A . Prove that P  A  P  P  A  .
4. The hypothesis of the theorem proven in the previous exercise is A  P  A .
(a) Can you think of a set A for which this hypothesis is true?
(b) Can you think of another?
5. Suppose x is a real number.
(a) Prove that if x  1 then there is a real number y such that
(b) Prove that if there is a real number y such that
y 1
 x.
y2
y 1
 x , then x  1 .
y2
Metamathematics
Read “Prelude…” and Chapter X, pp. 275-309.
6. Try this experiment in chunking at home. Grab a jar of coins (or something similar), take out a small
number of them (say, 5) and drop them quickly on the table. Get a relative to count them as fast as possible.
Try this with increasing numbers of coins. Afterwards, ask the volunteer their counting strategy and if the
strategy changed as the number of coins increases.
7. Here’s another experiment in chunking. How would you go about memorizing the first 30 digits of pi? Is
the task easier if you group the digits in chunks? What is the optimum chunk size for memorization?
8. How are theorems like chunking?