# Definition of overlapping sets

Is this definition of overlapping sets correct?

If the intersection of two sets is non-empty set but neither is a subset of the other, the sets are called overlapping sets

It is written in a book. I couldn't make out the last part (i.e neither is a subset of the other) how is it possible? If intersection of two sets has some common elements, then this resulting set will be subset of any of two sets.

The "neither is a subset of the other" is not referring to the intersection, but to the two sets themselves. So it is saying that the first set is not a subset of the second, and the second is not a subset of the first.

So, a rewritten definition would be:

$A$ and $B$ are overlapping if $A\cap B\neq \emptyset$ and it is not true that $A\subseteq B$ or $B\subseteq A$.

Under this definition, $\{1,2\}$ and $\{2,3\}$ are overlapping, but $\{1,2\}$ and $\{2\}$ are not, because $\{2\}$ is a subset of $\{2,3\}$.

This is a good example of how spoken language can be vague and confusing, and a great case for using mathematical symbols for clarity.

Let $A$ and $B$ two sets. If $A \cap B \ne \emptyset$, $A$ is not a subset of $B$ and if $B$ is not a subset of $A$ , then the sets are called overlapping .

Let's call the two sets $A$ and $B.$ What is meant is that the two sets have elements in common ($A\cap B\neq \emptyset$), but $A$ contains elements which are not in $B,$ and $B$ contains elements which are not in $A.$