Question about the size of a set.

[cragar]

If I had an uncountable number of sets, and each of these sets had an uncountable number of elements. Then I took the union of all of these sets would the number of elements be uncountable or would it be $2^{\aleph_1}$

 

[micromass]

First of all, $2^{\aleph_1}$ IS uncountable.

Second, you failed to mention the cardinality of your number of sets and the cardinality of the sets in question. The answer depends crucially on that.
Also, the answer depends on whether the sets are disjoint or not.

Right now, the only thing we can say is: if you have an uncountable union of uncountable sets, then this union will be an uncountable set. It might or might not be $2^{\aleph_1}$.

 

[cragar]

I have $2^{\aleph_0}$ sets. And they each have $2^{\aleph_0}$ elements. And all the sets are disjoint. The sets share no common elements.

 

[micromass]

Then you'll end up with $2^{\aleph_0}$ elements.

 

[cragar]

ok thanks for your answer. I am trying to think how you would prove that. Could you give me a hint on how to prove that. I mean if I had a countable number of sets I could just map all the elements in the first set to all the numbers between 0 and 1 and then for the next set map them to 1 to 2. I am not sure how you would do it with an uncountable number of sets.

 

[micromass]

Any good set theory book will prove this.