1
$\begingroup$

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

  • 3
    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.2012-04-10

2 Answers 2

8

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.
1

Yes interaction of any two countable set is countable.

If $A$ & $B$ are any two countable set, then $A \cap B \subseteq A$ , $A \cap B \subseteq B$