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) If A is a subset of U is uncountable, the complement of A is countable.
b) If A is a subset of U is countable, the complement of A is uncountable.
5. Let f: X -> Y be 1-1. Show that if Y is countable, then X is countable.
Thanks A Million