Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
2  
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

1 Answer 1

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.
share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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