GATE CS 2014 Set 1 Q7 — Minimization (K-Maps etc.)

We are given the Boolean expression:

$$F(P,Q,R,S) = PQ + \bar{P}QR + \bar{P}Q\bar{R}S$$

To find the minimal sum-of-products (SOP) form, we can simplify the expression algebraically by factoring out $Q$:

$$F = Q \left( P + \bar{P}R + \bar{P}\bar{R}S \right)$$

Now, let's simplify the expression inside the parentheses using the Boolean identity $A + \bar{A}B = A + B$:

1. First, group the terms with $\bar{P}$:
   $$P + \bar{P}(R + \bar{R}S)$$
2. Apply the identity $R + \bar{R}S = R + S$:
   $$P + \bar{P}(R + S)$$
3. Apply the identity again with $P$ and $\bar{P}$:
   $$P + (R + S) = P + R + S$$

Substituting this back into the factored expression:

$$F = Q(P + R + S) = PQ + QR + QS$$

Thus, the minimal sum-of-products form is $PQ + QR + QS$, which corresponds to option A.