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.
Best Answer
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.)