- #1

Danijel

- 43

- 1

*bijection*from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows:

ℤ is countable, and so iz ℤxℤ. Since ℤ*=ℤ\{0}⊂ℤ, ℤxℤ*⊂ℤxℤ, ℤxℤ* is countable (since every infinite subset of a countable set is also countable). Now, from the construction of ℚ, we know that ℚ=ℤxℤ*|

_{~}, ℚ is the image of τ:ℤxℤ*→ℚ, so ℚ is either finite or countable. Since ℤ⊂ℚ, ℚ is countable.

I understand everything except the part "ℚ is the image of τ:ℤxℤ*→ℚ, so ℚ is either finite or countable"(**), since this is not proven anywere. Maybe it is obvious, but I haven't grasped it. I've found few similar proofs of this statement, but they all consider A being countable if there's an

*injection*from A to ℕ, whereas my book insists on a bijection. So my question is, how do I deduce (**) from (*)?