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If $A, B$ and $C$ are non-empty sets such that
$A \cap C=\phi$, then what is $(A \times B) \cap(C \times B)$ equal to?
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$A \cap C=\phi$, then what is $(A \times B) \cap(C \times B)$ equal to?
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The correct answer is:
$\phi$
Let $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are non-empty sets such that $\mathrm{A} \cap \mathrm{C}=\phi$
Consider, $(A \times B) \cap(C \times B)=(A \cap C) \times(B \cap B)$
$=(\mathrm{A} \cap \mathrm{C}) \times \mathrm{B}=\phi \times \mathrm{B}=\phi$
Consider, $(A \times B) \cap(C \times B)=(A \cap C) \times(B \cap B)$
$=(\mathrm{A} \cap \mathrm{C}) \times \mathrm{B}=\phi \times \mathrm{B}=\phi$
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