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.