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, thenis 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
Then assume B is uncountable. Then, somehow show
I would probably do a cardinality argument. You know that
Sincecannot be put into a one-to-one correspondence with the integers, surely
can't as well.
I wouldn't index elements of the set, because that's assuming that there IS a correspondence with the integers, contrary to the uncountable assumption.
Hmm. Well, I think this sort of argument is basically what you have to do, since "uncountable" is defined in terms of one-to-one correspondences (or lack thereof) with the integers. Maybe Plato or Opalg could contribute a more direct argument that doesn't use cardinality.
Alright. Thank you very much for your time and your good explanation.
There is another proof that states... Prove that if A is uncountable and B is countable then A\B is uncountable. I realize this is similar but since card(A\B)<card(A) how would you show this?
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