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Is there a formal definition for a system where you can perform arithmetic operations on sets, for instance, $2 \cdot \mathbb{Z}$ = the set of even integers?

If there is, what is it called?

What happens when I add two sets?

(Note: I have no idea what to tag this.)

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    $\begingroup$ See math.stackexchange.com/questions/659128/… for addition of sets. $\endgroup$ – NickC Nov 6 '14 at 22:31
  • $\begingroup$ What you mean by adding two sets? If you think that $2\mathbb{Z} = \mathbb{Z} + \mathbb{Z}$, you're wrong. $\endgroup$ – brick Nov 6 '14 at 22:32
  • $\begingroup$ @brick I didn't expect $2\mathbb{Z}$ to necessarily be $\mathbb{Z} + \mathbb{Z}$, but was wondering if there was addition where it was. I see that "Minkowski sums" seem to be the standard definition for adding sets? In that case, $\mathbb{R} + \mathbb{R} = \mathbb{R}^2$, right? $\endgroup$ – porglezomp Nov 6 '14 at 23:21
  • $\begingroup$ With the notation suggested in my answer below, $\mathbb Z + \mathbb Z = \mathbb Z$, since every integer is the sum of two integers. $\endgroup$ – Emanuele Paolini Nov 7 '14 at 13:57
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When you have a function $f$, it is customary to abuse the notation to apply $f$ on sets of elements: $$ f(X) = \{ f(x)\colon x \in X\}. $$

So it is not strange to define $$ 2 X = \{2x \colon x \in X\} $$ and $$ X + Y = \{ x+y \colon x \in X,\ y \in Y\}. $$

This is only a notation no special matematical systems are required.

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  • $\begingroup$ This answers perfectly the scenario that prompted my question, and I plan to abuse this notation some myself, thanks! $\endgroup$ – porglezomp Nov 6 '14 at 23:23

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