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Is the intersection of two countable sets always a countable set or a finite set?

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Yes. If $A$ and $B$ are countable, $A\cap B$ is a subset of the countable set $A$ and therefore is countably infinite or finite. – Brian M. Scott Apr 10 '12 at 11:40
up vote 6 down vote accepted

Recall that $A\cap B\subseteq A$ as well $A\cap B\subseteq B$.

Now using the fact that a subset of a countable set is either finite or countably infinite we have that $A\cap B$ is either finite or countably infinite.

It can, of course, be both:

  1. $A=B=\mathbb N$ then $A\cap B=A=B=\mathbb N$ which is infinite;
  2. $A=\{x\in\mathbb N\mid x\text{ is even}\}$ and $B=\{x\in\mathbb N\mid x\text{ is odd}\}$, now $A\cap B=\varnothing$ which is empty and finite.
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