Here's my question: Let A ⊆ ℝ be a countable set. Prove that ℝ \ A is uncountable.
In class, we just learned about cardinality, power sets, and injections, surjections, and bijections.
So I know that a set is countable if it has the same cardinality as the natural numbers N. I also know that the set R is uncountable. Usually if wanting to prove that R is uncountable, it is sufficient to prove that (0,1) is uncountable and does not have the same cardinality as the natural number N. So, I know I have to proceed by contradiction.
I feel like I have the information, but I do not know where to start. Do I prove by contradiction that a bijection from N to R\A cannot exist. If yes, how do I do so?
My attempt was:
Let a ∈ N and b ∈ R\A. So, my plan is to prove surjectiviy and injectivity of a function f: N → R\A. Now, proving surjectivity, I know that if b ∈ R\A, there must be a ∈ N.
Am I going on the right track? Where do I go from here? How do I use the information given to me in the question? Is there perhaps bigger concepts I am missing?
Can someone please help me with this or show me how they would prove this?