Question

Three variable Boolean expression $PQ+PQR+\bar{P}Q+\bar{P}\bar{Q}R$ can be written as

• A. $\bar{Q} + \bar{P}R$
• B. $\bar{P} + \bar{Q}R$
• C. $Q + \bar{P}R$
• D. $Q + \bar{P}R$
• E. $P + QR$

Hint:

Whenever you see a variable ($Q$) and its complement ($\bar{Q}$) multiplied by something else, use the rule $Q + \bar{Q}X = Q + X$ to drop the complemented term.

Answer 1

The Correct Option is C

Concept:
We use the laws of Boolean algebra, specifically the Absorption Law ($A + AB = A$) and the Distributive Law ($A + \bar{A}B = A + B$).

Step 1: Simplify the expression step by step.
Original: $PQ + PQR + \bar{P}Q + \bar{P}\bar{Q}R$ [label=\alph*), itemsep=8pt]
• Simplify first two terms: $PQ + PQR = PQ(1 + R) = PQ$.
• New expression: $PQ + \bar{P}Q + \bar{P}\bar{Q}R$.
• Combine $PQ$ and $\bar{PQ$:} $Q(P + \bar{P}) = Q(1) = Q$.
• Final result: $Q + \bar{P}\bar{Q}R$.
• Apply $A + \bar{AB = A + B$:} Here $A = Q$ and $\bar{A} = \bar{Q}$. So, $Q + \bar{Q}(\bar{P}R) = Q + \bar{P}R$.