Thread: Sets

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

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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.

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?

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.

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

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.