Thank you for reporting, we will resolve it shortly
Q.
If $A, B$ and $C$ are three sets, then
Relations and Functions
Solution:
Let $(a, b) \subset A \times(B \cap C)$
$\rightarrow a \in A$ 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) }]}$