Question 48

On the 4-variable Karnaugh map for $F(P, Q, R, S) = \sum m(1, 2, 3, 4, 5, 7, 10, 12, 13, 14)$:

1. Essential prime implicants:
   • $\overline{P}S$ covers $\{1, 3, 5, 7\}$ (essential for $m_1$ and $m_7$).
   • $Q\overline{R}$ covers $\{4, 5, 12, 13\}$ (essential for $m_4$ and $m_{13}$).
2. Remaining uncovered minterms: $\{2, 10, 14\}$.
   • The prime implicants that cover elements of $\{2, 10, 14\}$ are:
     • $\overline{P}\,\overline{Q}R = \{2, 3\}$ (covers $2$)
     • $\overline{Q}R\overline{S} = \{2, 10\}$ (covers $2, 10$)
     • $PR\overline{S} = \{10, 14\}$ (covers $10, 14$)
     • $PQ\overline{S} = \{12, 14\}$ (covers $14$)
   • Choosing $\overline{P}\,\overline{Q}R$ (covers $2$) and $PR\overline{S}$ (covers $10, 14$) yields Option (B): $\overline{P}S + Q\overline{R} + \overline{P}\,\overline{Q}R + PR\overline{S}$.
   • Choosing $PQ\overline{S}$ (covers $14$) and $\overline{Q}R\overline{S}$ (covers $2, 10$) yields Option (D): $\overline{P}S + Q\overline{R} + PQ\overline{S} + \overline{Q}R\overline{S}$.
   • Option (A) fails to cover $m_{14}$, and Option (C) fails to cover $m_2$.