Question

If $A,B$ and $C$ are three sets, then

• A. $A\times (B\cap C)=(A\times B)\cap (A\times C)$
• B. $A\times \left(B^{\prime }\cup C^{\prime }\right)=(A\times B)\cap (A\times C)$
• C. Both (a) and (b)
• D. None of the above

Answer 1

Answer: (C)

Let $(a,b)\subset A\times (B\cap C)$
$\to a\in A$ and $b\in (B\cap C)$
$\Rightarrow a\in A$ and $(b\in B$ and $b\in C)$
$\Rightarrow (a\in A$ and $b\in B)$ and $(a\in A$ and $b\in C)$
$\Rightarrow (a,b)\in A\times B$ and $(a,b)\in (A\times C)$
$(a,b)\in (A\times B)\cap (A\times C)$
$\Rightarrow A\times (B\cap C)\subset (A\times B)\cap (A\times C)$....(i)
Again, let $(x,y)\in (A\times B)\cap (A\times C)$
$(x,y)\in A\times B$ and $(x,y)\in A\times C$
$\Rightarrow (x\in A\text{ and }y\in B)\text{ and }(x\in A\text{ and }y\in C)$
$\Rightarrow x\in A\text{ and }(y\in B\text{ and }y\in C)$
$\Rightarrow x\in A\text{ and }y\in (B\cap C)$
$\Rightarrow (x,y)\in A\times (B\cap C)$
$\Rightarrow (A\times B)\cap (A\times C)\subset A\times (B\cap C)$....(ii)
From Eqs. (i) and (ii), we get
$A\times (B\cap C)=(A\times B)\cap (A\times C)$....(ii)
Now, $A\times \left(B^{\prime }\cup C^{\prime }\right)$
$=A\times \left[{\left(B^{\prime }\right)}^{\prime }\cap {\left(C^{\prime }\right)}^{\prime }\right]\text{ (by De-Morgan’s law) }$
$=A\times (B\cap C)\left[\because \left(A^{\prime }\right)=A\right]$
$=(A\times B)\cap (A\times C)[\text{ by Eq. (iii) }]$