Question

Let $A,B$ and $C$ be non-empty sets and let $X=(A-B)-C$ and $Y=(A-C)-(B-C)$
Which one of the following is TRUE ?

• A. $X=Y$
• B. $X\subset Y$
• C. $Y\subset X$
• D. None of these

Answer 1

The correct option is A $X=Y$

$X=(A-B)-C$
$=(A\cap B^{{}^{\prime }})-C$
$=(A\cap B^{{}^{\prime }})\cap C^{{}^{\prime }}$
$=A{B}^{{}^{\prime }}{C}^{{}^{\prime }}$
$Y=(A-C)-(B-C)$
$=(A\cap C^{{}^{\prime }})-(B\cap C^{{}^{\prime }})$
$=\left(A{C}^{{}^{\prime }}\right)-\left(B{C}^{{}^{\prime }}\right)$
$=\left(A{C}^{{}^{\prime }}\right)\cap (B{C}^{{}^{\prime }}{)}^{{}^{\prime }}$
$=\left(A{C}^{{}^{\prime }}\right)\cap (B^{{}^{\prime }}+C)$
$=\left(A{C}^{{}^{\prime }}\right).(B^{{}^{\prime }}+C)$
$=A{C}^{{}^{\prime }}{B}^{{}^{\prime }}+A{C}^{{}^{\prime }}C$
$=A{C}^{{}^{\prime }}{B}^{{}^{\prime }}$ (since $C^{{}^{\prime }}C=1$)
$=A{B}^{{}^{\prime }}{C}^{{}^{\prime }}$ (commutative property)
$\therefore X=Y$