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Recently I thought about concepts for calculating with sets instead of numbers. There you might have axioms like:

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

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

  3. 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 $f(A_1,\ldots, A_n):= \{f(a_1,\ldots, a_n): a_1\in A_1 \land \dots \land a_n\in A_n \}$).

  4. 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.

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    $\begingroup$ Are you sure that $x^2 = 4 \Rightarrow x = \{ 2, -2 \}$? Looks like $x$ could be $\{2\}$ or $\{-2\}$, according to your item 3. $\endgroup$ – user31373 Jun 24 '12 at 18:55
  • $\begingroup$ You are right. I will delete my examples because they might be confusing... $\endgroup$ – Stephan Kulla Jun 24 '12 at 18:59
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    $\begingroup$ You could simply write $x\in \{2,-2\}$ instead of $x=\{2,-2\}$. This holds without changing any axioms or creating any new ones. For the second example $x\in [-1,1]$ will suffice. $\endgroup$ – A.S Jun 24 '12 at 19:01
  • $\begingroup$ Monoid of subsets of a group have been mentioned at MSE a few times, e.g. here. Proofwiki mentions some basic properties. $\endgroup$ – Martin Sleziak Jun 27 '12 at 7:10
  • $\begingroup$ @MartinSleziak thx for the links ;-) $\endgroup$ – Stephan Kulla Jun 27 '12 at 7:12
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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.

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