Is the Range of a Countable Function also Countable?

[Noxide]

If the domain of a function is countable, then is its range also countable?

also

if A is countable and B is countable is A(cartersian product)B countable?

 

[disregardthat]

If the domain of a function is countable, then is its range also countable?

also

if A is countable and B is countable is A(cartersian product)B countable?

Both are true. A function is a set of pairs (x,y) where x belongs to the domain, and y to the range. For each element x in the domain the associated pair (x,y) occurs only once in the function, so by counting the elements of the domain, you are counting all elements of the range (possible more than once).

To show that AxB is countable, you can define a way of counting each pair. By labeling the elements of A and B, do you see how this amounts to showing that $$\mathbb{Z} \times \mathbb{Z}$$ is countable? In other words, the integer gridpoints of the plane must be counted. Can you find an intuitive way of doing so?