How to say that there may exist elements in two sets such that their intersection is not null?

Question

I have a very basic understanding of set theory. I am trying to put some English sentences into mathematical notations and I am stuck at this one thing.

I have two sets A and B and they may not be disjoint, i.e., it may be possible that for some cases, their intersection is null and for some cases, they may have overlapping elements. Consider, for example, two courses which may have unique students or there may be a few students who take up both the courses. If $A=\{a_1, a_2, ..., a_n\}$ denotes the first course and $B=\{b_1, b_2, ..., b_n\}$ denotes the second course, then how do I convey this information mathematically?

Is this notation correct?

$\exists{i \in A,j \in B}$ s.t. $A \cap B \ne \emptyset$

This translates to "there exists some $i$ in $A$ and some $j$ in $B$ for which the intersection of $i$ and $j$ is not null". However, does this also put a constraint that there must exist some $i$ and some $j$ for which the intersection is not null? If yes, then what is the right way to denote this information?

Thanks in advance.

Answer 1

The notation is

$$A\cap B\ne \emptyset$$

which means that $A$ and $B$ have at least one element in common (their intersection is not an empty set), or

$$\exists i\in A,B$$

which means there exists (at least) one element which is in $A$ and $B$.

Answer 2

I would probably say something along the lines of: "..where $A\cap B \ne \emptyset$ may hold."

Answer 3

From the discussion in the comments, it sounds like your main purpose is to caution the reader. In that case, you don't need to use terse mathematical symbols. On the contrary, it's probably better to use English so that the reader doesn't skim over the warning. So, something like "Note that $A$ and $B$ might not be disjoint" or "Caution: We cannot assume that $A$ and $B$ are disjoint".