- #1
Danijel
- 43
- 1
I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a 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 (*)?
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 (*)?