Math 99 (Assignments) encoding

Problem Set 3:

1. Prove that (AÈB)´C = (A´C)È(B´C).

2. Prove that AÇ(B - A) = Ø for all A and B.

3. Prove that Ö3 is irrational.

   (Hint: You will need to prove a theorem like the one we did in class for even and odd numbers, but this time for divisibility by 3. You can assume that every integer n can be written in one of the forms 3k, 3k+1 or 3k+2 for some integer k. That is, ("nÎZ)($kÎZ)((n=3k)Ú(n=3k+1)Ú(3k+2)).)

4. Prove: AÍB if, and only if, AÈ(BÇC) = (AÈC)ÇB for all sets C.

   (Hint: The "for all C" goes with the right-hand part. You will need to handle the converse very carefully. You will need to use the fact that you know the "for all" part is true by picking an appropriate set C that will get you the result you need. This one is subtle.)

5. Prove that A´B = Ø if, and only if, A = Ø or B = Ø.

   (Hint: We already did most of one direction in class. Use the contrapositive for the other direction.)

6. Suppose A¹Ø and B¹Ø. Prove: A´CÍB´D if, and only if, AÍB and CÍD.

   (Hint: One direction is easy. For the other direction, you will need to use the fact that A and B are non-empty. This one is also subtle.)

Math 99 (Fall 2000) web pages Created: 21 Sep 2000 Last modified: Sep 24, 2000 9:21:06 PM Comments to: dpvc@union.edu

Assignments