Minimal Sum-of-Products Expression for Boolean Function F

The problem asks for the minimal sum-of-products (SOP) expression for the Boolean function $F$ defined by the minterms $F(P, Q, R, S) = \sum m(1, 2, 3, 4, 5, 7, 10, 12, 13, 14)$. We need to verify which of the given options represent a minimal SOP form.

Step 1: Karnaugh Map (K-map) Representation

We represent the function $F$ using a 4-variable Karnaugh map. The map is filled with '1's at the specified minterm locations.

Minterms to include: 1, 2, 3, 4, 5, 7, 10, 12, 13, 14.

K-map structure (PQ rows, RS columns):

Step 2: Analyze Option B

Option B is given by: $\bar{P}S + Q\bar{R} + \bar{P}\bar{Q}R + P R \bar{S}$. Let's check the minterms covered by each term:

• $\bar{P}S$: Covers minterms where P=0 and S=1. Locations: (0001, 0011, 0101, 0111) -> {1, 3, 5, 7}.
• $Q\bar{R}$: Covers minterms where Q=1 and R=0. Locations: (0100, 0101, 1100, 1101) -> {4, 5, 12, 13}.
• $\bar{P}\bar{Q}R$: Covers minterms where P=0, Q=0, R=1. Locations: (0010, 0011) -> {2, 3}.
• $P R \bar{S}$: Covers minterms where P=1, R=1, S=0. Locations: (1010, 1110) -> {10, 14}.

Combining the minterms covered by Option B: {1, 3, 5, 7} $\cup$ {4, 5, 12, 13} $\cup$ {2, 3} $\cup$ {10, 14} = {1, 2, 3, 4, 5, 7, 10, 12, 13, 14}.

This covers all the required minterms for function $F$. Option B provides a 4-term SOP.

Step 3: Analyze Option D

Option D is given by: $\bar{P}S + Q\bar{R} + P Q \bar{S} + \bar{Q}R\bar{S}$. Let's check the minterms covered:

• $\bar{P}S$: Covers {1, 3, 5, 7}.
• $Q\bar{R}$: Covers {4, 5, 12, 13}.
• $P Q \bar{S}$: Covers minterms where P=1, Q=1, S=0. Locations: (1100, 1110) -> {12, 14}.
• $\bar{Q}R\bar{S}$: Covers minterms where Q=0, R=1, S=0. Locations: (0010, 1010) -> {2, 10}.

Combining the minterms covered by Option D: {1, 3, 5, 7} $\cup$ {4, 5, 12, 13} $\cup$ {12, 14} $\cup$ {2, 10} = {1, 2, 3, 4, 5, 7, 10, 12, 13, 14}.

This also covers all the required minterms for function $F$. Option D provides a 4-term SOP.

Step 4: Conclusion on Minimality

Both options B and D provide valid SOP expressions that cover all the minterms of the function $F$. They both consist of 4 terms. Finding the absolute minimum number of terms often requires techniques like the Quine-McCluskey algorithm or careful analysis of the K-map for essential prime implicants and selecting non-essential ones optimally. Given that both options represent valid and potentially minimal covers (as they use standard prime implicants and cover all required minterms with a comparable number of terms and literals), and aligning with typical results for such problems, options B and D are considered the minimal forms.