Q63 / 180 · KCET 2012 · Mathematics

If $A$ and $B$ have $n$ elements in common, then the number of elements common to $A \times B$ and $B \times A$ is

  • A
    $\mathrm{n}$
  • B
    $2 \mathrm{n}$
  • Correct answer
    C
    $\mathrm{n}^{2}$
  • D
    0

Solution

Let $\quad \mathrm{C}=\mathrm{A} \cap \mathrm{B}$, then $C \times C=(A \cap B) \times(A \cap B)$ $=(A \cap B) \times(B \cap A)$ $$ \begin{gathered} =(A \times B) \cap(B \times A) \\ {[\because(A \cap C) \times(B \cap D)=(A \times B) \cap(C \times D)]} \end{gathered} $$ Since, $A \cap B$ has $\mathrm{n}$ elements, so $C$ has n elements. Hence, $\mathrm{C} \times \mathrm{C}$ has $\mathrm{n}^{2}$ elements. $\therefore \quad(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{B} \times \mathrm{A})$ has $\mathrm{n}^{2}$ elements. Hence, $\mathrm{A} \times \mathrm{B}$ and $\mathrm{B} \times \mathrm{A}$ has $\mathrm{n}^{2}$ elements in common.

Hence, the correct answer is option: 'C'.

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