GATE CS 2010 Q6 — Boolean Algebra

To find the minterm expansion of the Boolean function $f(P, Q, R) = PQ + Q\bar{R} + P\bar{R}$, we can expand each term to include all three variables $P$, $Q$, and $R$ (where $P$ is the MSB and $R$ is the LSB):

1. Term $PQ$:
   $$PQ = PQ(R + \bar{R}) = PQR + PQ\bar{R}$$Converting to binary representation:
   
   • $PQR \implies 111_2 = m_7$
   • $PQ\bar{R} \implies 110_2 = m_6$


2. Term $Q\bar{R}$:
   $$Q\bar{R} = (P + \bar{P})Q\bar{R} = PQ\bar{R} + \bar{P}Q\bar{R}$$Converting to binary representation:
   
   • $PQ\bar{R} \implies 110_2 = m_6$
   • $\bar{P}Q\bar{R} \implies 010_2 = m_2$


3. Term $P\bar{R}$:
   $$P\bar{R} = P(Q + \bar{Q})\bar{R} = PQ\bar{R} + P\bar{Q}\bar{R}$$Converting to binary representation:
   
   • $PQ\bar{R} \implies 110_2 = m_6$
   • $P\bar{Q}\bar{R} \implies 100_2 = m_4$

Combining all the unique minterms obtained, we get:

Therefore, the correct option is A.