Discussion

Let \(A = \{1, 2, 3, 4\}\) and \(B = \{7, 8, 3, 4\}\) Then the number of elements common to both \(A \times B\) and \(B \times A\) is

Answer: D

The number of common elements is 4.

An ordered pair (x, y) lies in both \(A \times B\) and \(B \times A\) only if x and y each belong to both sets, so

\[ \begin{aligned} &(A \times B) \cap (B \times A) \\ &= (A \cap B) \times (A \cap B) \end{aligned} \]

Here \(A \cap B = \{3, 4\}\), which gives \(2 \times 2 = 4\) pairs: (3, 3), (3, 4), (4, 3), (4, 4).

2 counts only the common elements of the sets, not the pairs, and 8 or 12 count pairs that lie in only one of the products.

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