1. Prove: If S is denumerable, then S is equinumerous with a proper subset of itself. 2. Prove: Every infinite set has a denumerable subset 3. Prove: Every infinite set is equinumerous with a proper...

1. Prove: If S is denumerable, then S is equinumerous with a proper subset of itself.

2. Prove: Every infinite set has a denumerable subset

3. Prove: Every infinite set is equinumerous with a proper subset of itself.

4. Prove that our ordering of cardinal numbers is antisymmetric: If | S | ≤ | T | and | T | ≤ | S |, then | S | = | T |. This result is known as the Schröder−Bernstein theorem and is very useful in proving sets equinumerous. (The proof is hard, but the hint in the back of the chapter will help.)