So far but you haven't finished. You still have to prove "transitivity": if x R y and y R z then x R z.

So you are saying that all odd numbers are equivalent and all even numbers are equivalent?Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

1.

2. ?

Yes that is correct and your notation is good.