Countable and Uncountable sets

Here is a proof that was in my notes. My professor did not do a great job of explaining how to prove a set is countable or uncountable so any explanation would be appreciated.

Prove that if A is uncountable and B is any set, then is uncountable.

I'm thinking I start by saying that A is some set .

Then, I'm thinking maybe you do a proof by cases? Let B be a set defined by . Then assume B is countable. Then, somehow show is uncountable.

Then assume B is uncountable. Then, somehow show is uncountable???