Question 1: (1)
Which of the following expressions represents the complement of the Boolean expression $$ A' \cdot (B \cdot C' + B' \cdot C) $$
(A) $$A' \cdot (B + C + B' + C)$$
(B) $$A + (B + C') \cdot (B + C')$$
(C) $$A + (B' + C) \cdot (B + C')$$
(D) $$A' \cdot (B' + C' + B \cdot C)$$
(C) $$ A + (B' + C)(B + C') $$
Let $$ F = A' \cdot (B \cdot C' + B' \cdot C) $$
Taking the complement of both sides: $$ F' = \left[A' \cdot (B \cdot C' + B' \cdot C)\right]' $$
Using De Morgan's Theorem: $$ F' = (A')' + (B \cdot C' + B' \cdot C)' $$
$$ F' = A + (B \cdot C')'(B' \cdot C)' $$
Again applying De Morgan's Theorem: $$ (B \cdot C')' = B' + C $$
and $$ (B' \cdot C)' = B + C' $$
Substituting: $$ F' = A + (B' + C)(B + C') $$
Therefore, $$ \boxed{F' = A + (B' + C)(B + C')} $$
Hence, the correct answer is (C).