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I know the Cartesian product of any two countable sets is countable. Also, it is safe to say that the Cartesian product of uncountable sets is uncountable.

I am totally lost on how to prove at least one of A or B is uncountable. I need some suggestions.

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    $\begingroup$ You’ve already proved it. Your first sentence is the contrapositive of the required statement. $\endgroup$
    – ziggurism
    Commented Dec 5, 2017 at 18:55

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You just need to restate the claim as its own contrapositive, given a fact you said you already know: if they're both countable, then they have a countable Cartesian product.

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