Is the intersection of two countable sets always a countable set or a finite set?
About the intersection of two countable sets
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elementary-set-theory
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3Yes. 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
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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:
- $A=B=\mathbb N$ then $A\cap B=A=B=\mathbb N$ which is infinite;
- $A=\{x\in\mathbb N\mid x\text{ is even}\}$ 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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Yes interaction of any two countable set is countable.
If $A$ & $B$ are any two countable set, then $A \cap B \subseteq A$ , $A \cap B \subseteq B$