0
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ please use latex :) $\endgroup$ – Albanian_EAGLE Dec 17 '13 at 3:16
7
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – user2933041 Dec 17 '13 at 2:19
  • $\begingroup$ Read it again... $\endgroup$ – Asaf Karagila Dec 17 '13 at 2:20
  • $\begingroup$ aii broski, thanks I gotta wait 3 mins to accept $\endgroup$ – user2933041 Dec 17 '13 at 2:23
  • 1
    $\begingroup$ @user2933041: Now that you’ve worked it out, here’s a moderately natural example: $A=\Bbb R\times\Bbb Q$ and $B=\Bbb Q\times\Bbb R$. $\endgroup$ – Brian M. Scott Dec 17 '13 at 5:48
  • 1
    $\begingroup$ @DonLarynx: I’m not sure what your $A$ is. $\endgroup$ – Brian M. Scott Dec 17 '13 at 20:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.