Advanced Topics in Mathematics – Logic and Metamathematics
Mr. Weisswange
Assignment #10
Logic
1. Prove without using truth tables:
(a) Suppose P  Q and Q  R are both true. Prove that P  R is true.
(b) Suppose ~ R   P ~ Q  is true. Prove that P   Q  R  is true.
2. Suppose A  C , and B and C are disjoint. Prove that if x  A then x  B .
3. Prove indirectly: Suppose A  C  B and a  C . Prove that a  A \ B .
4. Consider the following incorrect theorem: Suppose x and y are real numbers and x  y  10 . Then x  3 and
y  8.
(a) What’s wrong with the following proof?
Suppose the conclusion of the theorem is false. Then x = 3 and y = 8. But then x + y = 11, which
contradicts the given information that x + y = 10. Therefore the conclusion must be true.
(b) Disprove the theorem by finding a counterexample.
5. Can we modify the proof in class to prove that if x 2  y  13 and x  3 then y  4 ? Explain.
Metamathematics
Read “A Mu Offering” and Chapter IX, pp. 231-272.
6. Consider the following proof that the saying “every rule has an exception” is false:
1. Suppose that every rule has an exception.
2. Line (1) is a rule.
3. Therefore, by (1), line (1) has an exception.
4. Since (1) has an exception, there exists a rule which has no exception.
5. Since (1) and (4) contradict, our assumption must be false.
6. Therefore, there must exist a rule which has no exception.
What do you think? Is this proof valid? If so, is it possible that a false premise can lead to a true conclusion?
If not, why not?