# Two uncountable sets A and B are countably infinite

I'm having trouble finding the solution to the following problem...

Give an example of an uncountable set $A$ and an uncountable set $B$ such that $A$ intersect $B$ is countable infinite.

Answer - I know that $(-\infty, 1]$ and $[1, \infty)$ has an intersection of $\{1\}$. $\{1\}$ is countable but $\{1\}$ is also finite. I need to find two sets that have a result of countably infinite. Thanks guys!

• please use latex :) – Albanian_EAGLE Dec 17 '13 at 3:16

## 1 Answer

HINT: Can you find two uncountable sets which are disjoint? Now find a countable set and add it to both of them.

• The intersection of the two disjoint sets would be an empty set, and wouldn't that be not uncountable? So I wouldn't have two uncountable sets anymore right? – user2933041 Dec 17 '13 at 2:19
• Read it again... – Asaf Karagila Dec 17 '13 at 2:20
• aii broski, thanks I gotta wait 3 mins to accept – user2933041 Dec 17 '13 at 2:23
• @user2933041: Now that you’ve worked it out, here’s a moderately natural example: $A=\Bbb R\times\Bbb Q$ and $B=\Bbb Q\times\Bbb R$. – Brian M. Scott Dec 17 '13 at 5:48
• @DonLarynx: I’m not sure what your $A$ is. – Brian M. Scott Dec 17 '13 at 20:20