Qn 9. Simplify the following Boolean function a) \( F(A, B, C, D)=\sum(0,1,2.3,4,6,7,8,9,10,15) \) b) \( F(A, B, C, D)=\sum(0,4,6,8,9,11,11,14,15) \mathrm{m} \)

To simplify the given Boolean functions, we can use the Karnaugh map (K-map) method. Let's go through each function step by step. ### a) \( F(A, B, C, D) = \sum(0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 15) \) 1. **Construct the K-map**: We will fill in the K-map with the minterms. The K-map for 4 variables (A, B, C, D) is structured as follows: ``` CD AB | 00 | 01 | 11 | 10 ----------------------- 00 | 1 | 1 | 1 | 1 (0, 1, 3, 2) 01 | 1 | 1 | 0 | 1 (4, 5, 7, 6) 11 | 0 | 0 | 1 | 1 (12, 13, 15, 14) 10 | 1 | 1 | 0 | 0 (8, 9, 11, 10) ``` Filling in the K-map with the given minterms: ``` CD AB | 00 | 01 | 11 | 10 ----------------------- 00 | 1 | 1 | 1 | 1 (0, 1, 3, 2) 01 | 1 | 1 | 0 | 1 (4, 5, 7, 6) 11 | 0 | 0 | 1 | 1 (12, 13, 15, 14) 10 | 1 | 1 | 0 | 0 (8, 9, 11, 10) ``` 2. **Group the 1s**: We can form groups of 1s in the K-map. - Group 1: (0, 1, 2, 3) -> Covers \( A'B' \) - Group 2: (4, 5, 6, 7) -> Covers \( A'B \) - Group 3: (8, 9) -> Covers \( AB' \) - Group 4: (10, 15) -> Covers \( ACD \) 3. **Write the simplified expression**: The simplified expression is: \[ F(A, B, C, D) = A' + B'C + ACD \] ### b) \( F(A, B, C, D) = \sum(0, 4, 6, 8, 9, 11, 14, 15) \) 1. **Construct the K-map**: We will fill in the K-map with the minterms. Filling in the K-map with the given minterms: ``` CD AB | 00 | 01 | 11 | 10 ----------------------- 00 | 1 | 0 | 0 | 1 (0, 1, 3, 2) 01 | 1 | 0 | 1 | 0 (4, 5, 7, 6) 11 | 0 | 1 | 1 | 0 (12, 13, 15, 14) 10 | 1 | 1 | 0 | 0 (8, 9, 11, 10) ``` 2. **Group the 1s**: We can form groups of 1s in the K-map. - Group 1: (0, 4) -> Covers \( A'B' + A'B \) - Group 2: (8, 9) -> Covers \( AB' \) - Group 3: (14, 15) -> Covers \( AB \) 3. **Write the simplified expression**: The simplified expression is: \[ F(A, B, C, D) = A'B' + A'B + AB' \] ### Final Results a) \( F(A, B, C, D) = A' + B'C + ACD \) b) \( F(A, B, C, D) = A'B' + A'