Chapter 3.3, Problem 11E

1. Prove that the notion of same cardinality is an equivalence relation on the family of all sets. Explain, with reference to Definition 3.3.2.

2. Use the result of (a) and Problem 27 to show that $\left(1,\infty \right)\text{ and }\left(3,\infty \right)$ have the same cardinality.

3. Try to find an explicit one-to-one correspondence between $\left(1,\infty \right)\text{ and }\left(3,\infty \right)$.

4. Find a one-to-one correspondence between $\left(a,\infty \right)\text{ and }\left(b,\infty \right)$.