Prove \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} is Countable

[The Captain]

Homework Statement

Prove that $$\mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+}$$ is countable, where X is the Cartesian product.

Homework Equations

The Attempt at a Solution

I'm lost as to where to start proving this.

 

[annoymage]

Homework Statement

Prove that $$\mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+}$$ is countable, where X is the Cartesian product.

Homework Equations

The Attempt at a Solution

I'm lost as to where to start proving this.

do you know how to prove this

$$\mathbb Z^{+} X \: \mathbb Z^{+}$$

is countable?

 

[╔(σ_σ)╝]

There are many ways to do this you show that there it bijection from N to ZxZ ,that is , ZxZ is countable. Then show that there is bijection from ZxZ to ZxZxZ.

EDIT
someone beat me to it.

 

[The Captain]

I had to prove that $$\mathbb Z^{+} \: X \: Z^{+} \: \rightarrow \: Z^{+}$$ was one-one and onto using $$f(a,b)=2^{a-1}(2b-1)$$, does that count for proving it's countable, and if it's not, no I don't know how to prove it's countable.

The class I'm taking is a giant leap from Calc 4, and Abstract Algebra isn't even pre-req though it probably should be because the professor keeps asking who's taking it before.

 

[╔(σ_σ)╝]

Yes, that is a bijection. So you have already shown the first part all you need to do provide a bijection from ZxZxZ to ZxZ .