Set theory II

Probability Level 2

Which of the following sequence of set operations counts the number of elements that are in exactly one of $A$ , $B$ or $C$ ?"

Options:

$|A\cap B|+|B\cap C|+|C\cap A|-3(|A\cap B\cap C|)$
$|A|+|B|+|C|-2(|A\cap B|+|B\cap C|+|C\cap A|)+3|A\cap B\cap C|$
$|A\cap B|+|B\cap C|+|C\cap A|-2|A\cap B\cap C|$
$|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|$
$|(A\cap B\cap C)\cup(A\cap B^{C}\cap C^{C})\cup(A^{C}\cap B\cap C^{C})\cup(A^{C}\cap B^{C}\cap C)|$
| $A\cup B\cup C$ |

1 4 6 3 2 5

1 solution

Zandra Vinegar Staff
Oct 12, 2015

Imagine painting a color chart by overlapping the 3 circles primary colors of pigment: magenta (A), yellow (B), and cyan (C). The very center gets covered 3 times and the three regions bordering the center each get covered twice. Therefore, if we want to count only the three outermost areas (aka, the things in exactly one of A, B, or C but not shared between both or all 3 sets), if we start by counting $|A|+|B|+|C|$ , we then need to remove $2(|A\cap B|+|B\cap C|+|C\cap A|)$ . But in doing so, we "over-remove" $|A\cap B\cap C|$ (the very center) by three extra times, so we need to add that back in ( $+ 3|A\cap B\cap C|$ ). In other words, the answer is option $\fbox{2}$ .

Wording of the problem is confusing. Better phrasing would be "...elements that are only in A, only in B and only in C.

David Johnson - 3 years, 3 months ago

What David said

Frank Siran - 2 years, 6 months ago

Wording is not exact so agree with David, Brilliant should remove this question

Darko Doko - 11 months ago

Set theory II

1 solution

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