Intersection of countable and uncountable sets

Question

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

1. Empty.
2. Finite, countable, or uncountable.
3. Countable.
4. Uncountable.
5. Countable or uncountable.
6. Finite.
7. 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?

Answer 1

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: $A\cap B\subset A\implies \lvert A\cap B\rvert\le\lvert A\rvert$.

Answer 2

Option 7 is the right one, the others are all false:

1. empty is possible, not necessary (take $\mathbb{N}$ and $\mathbb{R}$).
2. uncountable is not possible as $A \cap B \subseteq A$ so the intersection is at most countable. As "finite, countable or uncountable" essentially means "all sets", I discount it as an option, also because it suggests uncountable is even possible.
3. countable, not, as it can also be finite or empty. ($\mathbb{N}$ and the irrationals, say).
4. uncountable, not, as $A \cap B \subseteq A$ so this is out.
5. countable or uncountable, not as finite is also possible, see 3. or for even suggesting uncountable is possible which it is not (see 2.)
6. finite, not, because infinite is an option too, as we saw.
7. Clearly true, as $A \cap B \subseteq A$ and a subset of a countable set is at most countable. Also, the last option remaining...

Answer 3

There are many text that define countable and finite as mutually incompatible. A finite set is not countable and a countable set must be infinite. Many others, and my person preference, define countable as including finite as well and countably infinite.

BOTH conventions will define "at most countable" as "not uncountable" to include the options of being finite or empty.

Although I do prefer the latter, I am sympathetic to this question. To ask what is "the most" we can say, I figure we must specify that although the set could be countably infinite (but never uncountable), that it need not be infinite.

And "at most countable" is the most accurate answer.