# 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.

## 3 Answers

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 $A.$