Please help me with these countable, uncountable problems I am having trouble with.

1. Determine whether each of these sets is countable or uncountable. For those that are countable, exhibit a one-to-one correspondence between the set of natural numbers and that set.

a) integers not divisible by 3

d) the real numbers with decimal representations of all 1s or 9s

2. Show that if A and B are sets with the same cardinality, then the power set of A and the power set of B have the same cardinality.

3. Show that the set of real numbers that are solutions of quadratic equations ax^2 + bx + c = 0, where a,b, and c are integers, is countable.

4. Let U be an uncountable universal set. Prove or disprove the following statements:

a) IfAis a subset ofUis uncountable, the complement ofAis countable.

b) IfAis a subset ofUis countable, the complement ofAis uncountable.

5. Let f: X -> Y be 1-1. Show that if Y is countable, then X is countable.

Thanks A Million

Anu