Discussion
Let U be the universal set and let A and B be any two subsets of U. If \(n(U) = 25, n(A) = 14\) \(n(A \cap B) = 6\) and \(n(A \cup B)\) =20, then \(n(B')\) is equal to
Answer: B
\(n(B') = \) 13.
Find \(n(B)\) from the union formula, then subtract it from \(n(U)\):
\[ \begin{aligned} n(B) &= n(A \cup B) - n(A) + n(A \cap B) \\ &= 20 - 14 + 6 = 12 \\ n(B') &= n(U) - n(B) = 25 - 12 = 13 \end{aligned} \]
12 is \(n(B)\) itself, the usual slip of stopping one step early; 14 is \(n(A)\), and 15 or 16 come from adding \(n(A \cap B)\) with the wrong sign.
Check: \(n(A \cup B) = 14 + 12 - 6 = 20\), as given.
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