Is there any symbol for intersects?

Question

Given two polygons $A$ and $B$ the intersection of them is represented by the notation $A \cap B$ that returns a geometry (more precisely the set of intersecting points) resulting from the intersection operation.

Is there any symbol for intersects (a boolean function that returns 0 (false) or 1 (true) if $A$ intersects $B$) ?

Otherwise, what would be a good notation for that?

Thanks!

Answer 1

I do not know of any specific notation for this.

But what about

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

This statement is true if $A$ and $B$ have any overlap, false otherwise.

This returns true for any overlap, including edges and vertices. If you want to exclude those, use the notation for the interior of the polygons, $\operatorname{int}(A)\cap\operatorname{int}(B)\ne\emptyset$.

Answer 2

The characteristic function of a set $S$ is $$\chi_S(x) = \begin{cases}1,& x\in S\\0,& x\notin S\end{cases}.$$ So what you're looking for is $$\sup_{x\in\mathcal U}\chi_{A\cap B}(x),$$ where $\mathcal U$ is our universal set (i.e. $A,B\subset\mathcal U$).

Answer 3

It's a little awkward, but perhaps you could use $\not\perp$, because, according to this Wikipedia page:

the notation $a \perp b$ is sometimes used to indicate that $a$ and $b$ are relatively prime

a usage which could be extended to a general lattice (unless the symbol $\bot$ was already in use to denote the bottom element, which would not be a problem in this case).

The notation (for the case of coprime integers) seems to have been invented by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics (2nd edition, 1989), section 4.5.