Show that the set $\{x ∈ \Bbb Z|x ≥ 3\}$ is countable.
I'm really stuck on this problem, I have an exam really soon and I'm trying to solve it. So far I know that I have to show that it's onto and one-to-one.
Would I start by saying that $f(n)= 3 + n$ and then showing that it's onto and one-to-one?
Best Answer
Yup, precisely!
Be sure to be clear about your definition for $f$, of course, in that you're using the definition $$f : \Bbb N \to \{x \in \Bbb Z | x \geq 3\}$$
Seems like a minor detail but it makes your argument clear since, usually, I like focusing these sorts of proofs by looking at the integers. Possibly just a personal quirk, but a lack of ambiguity never hurts.
From there, once you have shown $f$ is a bijection - that is, it is both onto (surjective) and one-to-one (injective) - then you can conclude its domain and codomain have the same cardinality. And thus,
$$|\Bbb N | = \aleph_0 = | \{x \in \Bbb Z | x \geq 3\} |$$