# Size of a union of two sets

We were ask to prove that $|A \cup B| = |A| + |B| - |A \cap B|‫‪$. It was easy to prove it using a Venn diagram, but I think we might be expected to do if more formally. Is there a formal way?

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What does $|A|$ mean? Is $|\cdot|$ a measure? Or maybe the sets are finite? –  Jeff Nov 14 '11 at 16:44
just observe that if you do $|A|+|B|$ you count twice the elements in $A\cap B$. –  Valerio Capraro Nov 14 '11 at 16:48
Suppose that you want to give a dollar to everyone in $A\cup B$. If you give a dollar to everyone in $A$, and then give a dollar to everyone in $B, \dots$. –  André Nicolas Nov 14 '11 at 17:21
I am fairly certain that this question appeared here perhaps infinitely many times before. I cannot find it though... –  Asaf Karagila Nov 14 '11 at 18:22

## 1 Answer

$A\cup B = (A\setminus B) \cup (B\setminus A) \cup (A\cap B)$. These three sets are disjoint, so $$|A\cup B| = |A\setminus B| + |B\setminus A| + |(A\cap B)|$$ But $A\setminus B = A\setminus(A\cap B)$, so $|A\setminus B|=|A|-|A\cap B|$. A similar equality holds for $|B \setminus A|$. Substitution of these into the displayed equation above yields your result.

Of course, one might need to formally show that $|A\setminus (A\cap B)| = |A|-|A\cap B|$. I can't decide if this is any less obvious than the original proposition...

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I have seen $A \setminus B$, $A \smallsetminus B$ and $A - B$ used to denote set difference, but not $A / B$. Do you normally use that notation? =) –  Srivatsan Nov 14 '11 at 16:59
Is using a forward slash for set difference standard notation? –  Henning Makholm Nov 14 '11 at 17:00
bleech... I'm left/right backslash challenged...I got points taken off for doing that on a topology test once. –  David Mitra Nov 14 '11 at 17:00
@David Mitra: The forward slashes were bothersome, so I was changing things to setminus, and in the middle they changed to backslash. Not thinking, I continued. Can turn them to backslash if you wish. –  André Nicolas Nov 14 '11 at 17:40
Thanks. \setminus is better of course. I'm not sure if \backslash is a mathop... –  David Mitra Nov 14 '11 at 17:44