I'm asked to determine if certain sets are countable or not countable. For the countable sets, I am to exhibit a one-to-one correspondence between the set of natural numbers and that set. One of the sets is:
(A) The real numbers with decimal representations of all 1s or 9s
I said this is uncountable, because it seems that there should be some combination of 1s and 9s such that you have an infinite number of distinct irrational numbers. The set of real numbers is uncountable because it contains an infinite number of irrational numbers, right? (IS that why R is uncountable?) BUT how do I KNOW that this set A contains an infinite number of irrational numbers - and is knowing that enough to conclude that the set is uncountable?
Also, there's another set:
(B) Integers divisible by 5 but not by 7
I know this is countable, because it is a subset of a countable set, but does anyone have any pointers on how to go about writing a one-to-one correspondence between the set of natural numbers and this set?