Which theories and concepts exist where one calculates with sets? Recently I thought about concepts for calculating with sets instead of numbers. There you might have axioms like:


*

*For every $a\in\mathbb{R}$ (or $a\in\mathbb{C}$) we identify the term $a$ with $\{a\}$.

*For any operator $\circ$ we define $A\circ B := \{a\circ b : a\in A\land b\in B\}$.

*For any function $f$ we define $f(A) := \{ f(a) : a\in A \}$. (More general: For a function $f(x_1,\ldots, x_n)$ we define $f(A_1,\ldots, A_n):= \{f(a_1,\ldots, a_n): a_1\in A_1 \land \dots \land a_n\in A_n \}$).

*One has to find a good definition for $f^{-1}(A)$ which might be the inverse image of $A$.
((3.) is just the normal definition of the image and (2.) is a special case of (3.))
Now I am interested to learn about theories and concepts where one actually calculates with sets (similar to the above axioms).
After a while I found interval arithmetic. What theories or approaches do you know?
Because there will not be just one answer to my question, I will accept the answer with the most upvotes.
Update: The theories do not have to follow the above axioms. It's okay when they make there own definitions how a function shall act on sets. It is just important that you calculate with sets in the theory, concept or approach.

I like Minkowski addition, aka vector addition. It is a basic operation in the geometry of convex sets. See: zonotopes & zonoids, Brunn-Minkowski inequality, polar sets... and here's a neat inequality for an arbitrary convex set $A\subset\mathbb R^n$: 
$$
\mathrm{volume}\,(A-A)\le \binom{2n}{n}\mathrm{volume}\,(A)
$$ 
with equality when $A$ is a simplex. (Due to Rogers and Shepard, see here)
The case $n=1$ isn't nearly as exciting. 
