symbol denoting the empty set
Factsheet
This is sufficiently rare, and handled in sufficiently many different ways, that you should always state explicitly how you're treating it. In my experience, the most common symbols are:
$\mathsf{null}$, $\mathsf{nil}$, $\mathsf{NaN}$ or similar on the more computer-sciencey side, and
$\perp$ or $\uparrow$ on the more logicy side.
- Note that "$\perp$" is also used to denote contradiction, and "$\uparrow$" is also used as a predicate to denote "is undefined" or "doesn't halt" with "$\downarrow$" denoting "is defined"/"does halt."
But again, I'd explicitly state which you're using - although admittedly multiple of these would almost certainly make it obvious from context.
You can use some kind of Many-valued logic, but you said you want to put it simply. In SQL there is 3-valued logic with "null"/"unknown", for example.
The distinction between the empty set $\emptyset$ and the number $0$ is similar to that between NULL and ZERO. For example, the set of real solutions (or informally "the solution") to $x^2=-1$ is $\emptyset$, but the solution to $x^2=0$ is $0$.
In my mind there is no need for a concept like NULL in mathematics if you think of NULL as in NULL-pointers.
NULL in this sense is a technical necessity because you cannot un-define a variable: Once a variable has been assigned a value, a certain bit of memory is reserved for this variable and this memory is marked as re-usable only if the variable goes out of scope (simplified speaking).
You cannot say "The variable with this name doesn't exist anymore." without letting it go out of scope, because that would make language interpretation much more complicated without many benefits. Therefore, to indicate that the value of the variable has no meaning, one uses NULL.
What NULL stands for in the end depends upon the programming language: In some it is a special keyword, but in some it is also just a different name for the integer $0$.
You can assign an arbitrary value to NULL in mathematics as mentioned in the other replies ($\emptyset$, $0$, etc.) but as mathematics has nothing to do with memory allocation there is really no need for such a thing as NULL.