I came across following problem:
If $A$ is a countable set, and $B$ is an uncountable set, then the most we can say about $A\cap B$ is that it is
- Empty.
- Finite, countable, or uncountable.
- Countable.
- Uncountable.
- Countable or uncountable.
- Finite.
- At most countable.
I understand it has to be countable, that is option 3. But the answer given was option 7 – "at most countable". I am confused what it means to say by prefix "at most". The explanation given was "The intersection of A and B could be smaller than countable." What does it means "smaller than countable"? Empty and/or finite? Or is there something more to it?
Best Answer
I'm going to opt for $7$. The reason is that there are three possibilities: $$i)\,\emptyset \\ ii)\text { finite }\\iii)\text { countably infinite }$$.
Note: $A\cap B\subset A\implies \lvert A\cap B\rvert\le\lvert A\rvert$.