1. ## Sets

How can I prove/disprove that the product of two uncountable sets can be countable?

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2. Originally Posted by Nickolase
How can I prove/disprove that the product of two uncountable sets can be countable?

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One way is if their product is the set of integers such as x as the set of positive real numbers multiplied by 1/x which is the set of 1's. Intergers are countable. That is one way.

3. Originally Posted by Nickolase
How can I prove/disprove that the product of two uncountable sets can be countable?

Tip o the hat
Originally Posted by oldguynewstudent
One way is if their product is the set of integers such as x as the set of positive real numbers multiplied by 1/x which is the set of 1's. Intergers are countable. That is one way.
I don't have an answer, just a comment for clarity. Does the original question refer to the Cartesian product of the two sets, or something different?

4. Start by trying to show whether there is (or is not) a one-to-one correspondence between the cross-product of the two sets and the set of natural numbers.

5. 1. prove NxN is equinumerous to N
2. for uncountable, prove a set (any one of the two sets) is equinumerous to a proper subset of the product of the two sets. thus if product is countable we run into a contradiction

6. First, if B injects into C and B is uncountable, then C is uncountable.
Proof (here 'w', read 'omega', stands for the set of natural numbers):
Toward a contradiction, suppose B injects into C, and B is uncountable, but C is countable.
So C injects into w.
So, since B injects in C, we have B injects into w, so B is countable. Done.

Suppose S and T are uncountable.

Let y be in T.

S X {y} is equinmerous with S.

So S X {y} is uncountable.

S X {y} is a subset of S X T.

So S X T is uncountable.