If the union of two sets is uncountable, what may we conclude?

Question

I have $2$ questions. Here are these:

1) Assume that the union of sets $A$ and $B$ is uncountable. What exactly can we conclude from here?

I think, at least one of these sets is not countable. Or am I wrong? Or should both sets be uncountable?

2) If the number of sets is more than $2$, will the result change?

Thank you.

Answer 1

The question is designed in such a way so that you think inductively.

If $A_1\cup A_2$ has property $P$, then $Q$ is true.

If $A_1\cup\cdots\cup A_n$ has property $P$, then $Q$ remains true.

It is up to you to figure out what $Q$ is here when $P$ represents the property of being uncountable. I hope your assignment makes more sense now.

Answer 2

Assume a countable infinity of countable sets. You can pick elements from every set in a round-robin fashion, adding one set on every turn. This way, you enumerate all elements of all sets. So the union of countably many countable sets is countable.

Now add a single uncountable set. If that union was countable, the counting process would allow you to count the elements of the new set, a contradiction.