Question

Consider the following Boolean expression for F: \( F(P, Q, R, S) = PQ + PQR + \overline{P}QR \). The minimum sum of products form of F is
 

• A. \( PQ + QR + QS \)
• B. \( P + Q + R + S \)
• C. \( \overline{P} + \overline{Q} + \overline{R} + \overline{S} \)
• D. \( P \overline{R} + P \overline{R}S + P \)

Hint:

To simplify Boolean expressions, always look for common factors and apply the basic Boolean laws such as \( 1 + R = 1 \) and \( R \cdot 1 = R \).

Answer 1

The Correct Option is A

We are given the Boolean expression \( F(P, Q, R, S) = PQ + PQR + \overline{P}QR \). To simplify the expression, we use Boolean algebra:
1. Combine like terms in the expression \( PQ + PQR + \overline{P}QR \). Notice that \( PQR \) can be written as \( PQ \cdot R \), so: \[ F = PQ + PQ \cdot R + \overline{P}QR \] 2. Factor out the common terms: \[ F = PQ(1 + R) + \overline{P}QR \] 3. Simplifying further, since \( 1 + R = 1 \) in Boolean algebra: \[ F = PQ + \overline{P}QR \] 4. This is the simplest sum of products form of the Boolean expression. Therefore, the minimum sum of products is \( PQ + QR + QS \), matching option (a). Thus, the correct answer is \( \boxed{PQ + QR + QS} \).