Addition of Sets which isn't union today a student asked me to prove
$${A} \cup B \cup C = A+ B+ C- A\cap B - A\cap C$$
I really had no idea what precisely the "+" sign meant, they insisted, "You know you just add the sets together"; of course, they also had no textbook.  
I assumed then that perhaps
$$A\cup B = A+B - A\cap B$$
But I'm not really sure what "-" means here, normally I'd interchange that with set subtraction \ and I occasionally use + to mean $\cup$.  In any case, I think + sort of gets extra copies of the parts they share.  But I really have no idea, any clues.
 A: It looks like the student was talking about cardinalities of sets rather than the sets themselves. For instance, it holds true that
$$|A \cup B| = |A| + |B| - |A \cap B|$$
But in this scenario the first equation still doesn't hold, because
$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$
But if a student is prepared to say "you know, just add the sets" then it's not entirely infeasible that they just omitted/forgot/incorrectly copied the last two terms.
(There's no obvious definition of $+$ that would make your first equation hold, where by 'obvious definition' I mean either with cardinalities as above, or where $+$ means disjoint union.)
A: Before looking it up just now, I didn't realize you couldn't have duplicate elements in a set. 
Intuitively "$A\cup B\cup C = A \cup  B - A\cap B - A\cap C$" made a lot of sense to me because of this.
If you were under the misconception that sets could have multiple copies of the same element (as I was just a couple of minutes ago), then it's not much of a stretch to also see the union of A and B ($A\cup B$) as just grouping all of the elements into one set (A+B) and filtering out the duplicates ($-A\cap B$).
I assumed that A + B is (informally) defined as simply conjoining two sets without regard to duplicity of elements.
First off:
The operation "+" maps two sets or "setts" to a "sett", where a "sett" is essentially a set with duplicity allowed.
So $\{1,2,3\} + \{2,3,4\} = \{\{1,2,2,3,3,4\}\}$.
And subtraction is simply your regular set subtraction expanded to operate on "setts".  
From this definition you can see that $A\cup B + A\cap B = A+B$. 
So yes, essentially $A\cup B = A+B - A\cap B$.
You were exactly right with your guess that $A+B$ "sort of gets extra copies of the parts they share". 
I could work on proving the student's original statement, but I'm sure people smarter than me can figure it out. Not to mention I didn't give very clear formal definitions for '+'.
Also, I apologize for my terrible formatting.
