Let $A=\{1,2\}, B=\{1,2,3,4\}, C=\{5,6\}$ and $D=\{5,6,7,8\}$. Verify that $A \times (B \cap C) = (A \times B) \cap (A \times C)$.

(N/A) To verify: $A \times (B \cap C) = (A \times B) \cap (A \times C)$.
First,find the intersection of sets $B$ and $C$:
$B \cap C = \{1, 2, 3, 4\} \cap \{5, 6\} = \varnothing$.
Now,calculate the Left Hand Side $(L.H.S.)$:
$L.H.S. = A \times (B \cap C) = \{1, 2\} \times \varnothing = \varnothing$.
Next,calculate the Right Hand Side $(R.H.S.)$:
$A \times B = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)\}$.
$A \times C = \{(1, 5), (1, 6), (2, 5), (2, 6)\}$.
$R.H.S. = (A \times B) \cap (A \times C) = \varnothing$.
Since $L.H.S. = R.H.S. = \varnothing$,the identity $A \times (B \cap C) = (A \times B) \cap (A \times C)$ is verified.