Proving equality of cardinality two sets

Question

Problem : Recall that the closed interval [a, b] = {x ∈ R : a ≤ x ≤ b}. Prove that |[−4, 2]| = |[5, 7]|.

From my understanding, |[−4, 2]| and |[5, 7]| are both uncountable sets (since they are in the reals) and thus it proves that the equation is correct. However, I have no idea how I would go about with a formal proof or that my idea is even correct in the first place.

Any help please?

Answer 1

Hint: Try to find $f(x)=ax+b\,$ so that $f(-4)=5$ and $f(2)=7$.

And this function is bijective. Can you verify it?

Answer 2

Your idea is on the right track (toward one way to do it, at least), but not quite enough. It's true that they are both uncountable sets. But what if they have different uncountable cardinalities?

You can try to make your line of thinking more precise, but the easier way to show that these two sets have the same cardinality is to construct an explicit bijection between them.