Proving the Countability of Nx{0}

[aaaa202]

Homework Statement

Proove that Nx{0} is countable. x stand for a product i.e. like the cartesian product NxN

Homework Equations

N is countable.

The Attempt at a Solution

This is so obvious since Nx{0} is just (1,0),(2,0) etc. But how do you write a proof formally?

 

[Dick]

Homework Statement

Proove that Nx{0} is countable. x stand for a product i.e. like the cartesian product NxN

Homework Equations

N is countable.

The Attempt at a Solution

This is so obvious since Nx{0} is just (1,0),(2,0) etc. But how do you write a proof formally?

Show there is a 1-1 correspondence between Nx{0} and N. Write an explicit bijection. Yes, it is easy.

 

[aaaa202]

So I can define the map:
f: N->Nx{0}
f(x)=(x,0)
and state that this is clearly bijective as a proof?

 

[jbunniii]

So I can define the map:
f: N->Nx{0}
f(x)=(x,0)
and state that this is clearly bijective as a proof?

I suggest writing the extra few lines to prove injectivity and surjectivity instead of writing "clearly." The problem statement itself seems so obvious that it's tempting to write "clearly" as a one-word proof, but clearly that isn't the intent.