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!

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

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

  • $\begingroup$ 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
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    $\begingroup$ @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
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    $\begingroup$ @DonLarynx: I’m not sure what your $A$ is. $\endgroup$ – Brian M. Scott Dec 17 '13 at 20:20

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