Problem 1. Let A be a finite set and B be an infinite set and let A B. Prove that B A is infinite. Problem 2. Let A be a countably infinite set and x...

Problem 1. Let A be a finite set and B be an infinite set and let A ⊂ B. Prove that B − A is infinite.

Problem 2. Let A be a countably infinite set and x A be an element of the same universe. Prove that

A ∪ {x} is countably infinite.

Problem 3. You own a restaurant which has seats numbered 1, 2, 3, 4, . . . .

One night all of the seats are occupied. Someone shows up at the restaurant and asks for a seat. How can you

accommodate that person? (Hint: problems 2 and 3 are more-or-less equivalent.)

Problem 4. Prove that N×N is countably infinite in two different ways (one, by repeating the "diagonalization

argument" as we did for Q, you do not need to write a explicit bijection, but you should draw a convincing

picture)