Two set theory questions (isomorphism and countable sets)

[XxLegato666xX]

First

Homework Statement

Let X be a finite non-empty set and Y a countable set. Prove that XxY (X cross Y) is countable

Homework Equations

X isomorphic {1, 2, ..., n} for some n$$\in$$N, N is natural numbers
Y isomorphic to natural numbers

The Attempt at a Solution

I wasn't able to get very far, all attempts to prove 1_1 and onto failed because there's no function rules and I can't think of any that fit.

Second:
Let y denote the set of functions f:{1,2} -> X.
Prove y isomorphic XxX (X cross X)

y = {f:{1,2} -> X}
$$\Phi$$:y->XxX
prove $$\Phi$$ is 1_1 and onto

I got to the same snags as the previous one mentioned. There are no function rules, so I don't know how to approach 1_1 and onto solving.

Help would be very greatly appreciated, even a hint. Thanks.

 

[Tedjn]

We start with the first. You need to explain the function rule for an isomorphism (or bijection) f between X × Y and Y. Consider the special case when X has two elements, which we will call X = {1,2}. How many "more" elements are in X × Y than Y intuitively? Does this parallel a similar situation with the natural numbers and a subset of the naturals?