Discussion
Let A, B, C be any three finite sets.
If \(n(A \times B) = 160, n(B \times C) = 80\) and \(n(C \times A) = 200\), then \(n(A) =\)
Answer: E
\(n(A) = \) 20.
The number of elements in a Cartesian product is the product of the sizes: \(n(A \times B) = n(A)\,n(B)\). Write \(a, b, c\) for the three sizes:
\[ \begin{aligned} ab &= 160,\quad bc = 80,\quad ca = 200 \\ (abc)^{2} &= 160 \times 80 \times 200 = 2560000 \\ abc &= 1600 \\ a &= \frac{abc}{bc} = \frac{1600}{80} = 20 \end{aligned} \]
Check: then \(b = 8\) and \(c = 10\), and \(20 \times 8 = 160\), \(8 \times 10 = 80\), \(10 \times 20 = 200\). The other values (10, 12, 16, 18) do not divide the three products consistently.
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