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???