Question

Solve the following SOP expressions with the help of K-map: (i) F(A, B, C, D) = Σm(0, 1, 3, 4, 5, 7) (ii) F(A, B, C) = Σm(0, 1, 4, 5, 6, 7).

Answer 1

To solve the given Sum of Products (SOP) expressions using Karnaugh Maps (K-maps), follow these steps:

### (i) \( F(A, B, C, D) = \Sigma m(0, 1, 3, 4, 5, 7) \)

This is a 4-variable function. To simplify it, we first construct a 4-variable K-map and then group the 1s to find the simplified expression.

#### Construct the K-map

The 4-variable K-map is a 4x4 grid, where each cell corresponds to a minterm. The arrangement of variables in the K-map is:

| AB\CD | 00 | 01 | 11 | 10 |
|-------|----|----|----|----|
| 00    | m0 | m1 | m3 | m2 |
| 01    | m4 | m5 | m7 | m6 |
| 11    | m12| m13| m15| m14|
| 10    | m8 | m9 | m11| m10|

Fill in the K-map with the given minterms (0, 1, 3, 4, 5, 7):

| AB\CD | 00 | 01 | 11 | 10 |
|-------|----|----|----|----|
| 00    | 1  | 1  | 1  | 0  |
| 01    | 1  | 1  | 1  | 0  |
| 11    | 0  | 0  | 0  | 0  |
| 10    | 0  | 0  | 0  | 0  |

#### Group the 1s

- **Group 1:** A group of four 1s in the first two rows (minterms 0, 1, 4, 5). This group covers all columns where \(CD\) is 00 or 01. This group can be simplified to \(A'B'\) because it covers cells where \(A\) and \(B\) are both 0, regardless of \(C\) and \(D\).

- **Group 2:** A single 1 left in cell (1,1) (minterm 7). This 1 is not part of a larger group. So, this contributes \(A'BD\) (where \(A = 0\), \(B = 1\), and \(D = 1\)).

The final simplified expression is the sum of these groups:

\[ F(A, B, C, D) = A'B' + A'BD \]

### (ii) \( F(A, B, C) = \Sigma m(0, 1, 4, 5, 6, 7) \)

This is a 3-variable function. Construct a 2x4 K-map and group the 1s.

#### Construct the K-map

The 3-variable K-map is a 2x4 grid. The arrangement of variables is:

| AB\C | 0  | 1  |
|------|----|----|
| 00   | m0 | m1 |
| 01   | m4 | m5 |
| 11   | m6 | m7 |
| 10   | 0  | 0  |

Fill in the K-map with the given minterms (0, 1, 4, 5, 6, 7):

| AB\C | 0  | 1  |
|------|----|----|
| 00   | 1  | 1  |
| 01   | 1  | 1  |
| 11   | 1  | 1  |
| 10   | 0  | 0  |

#### Group the 1s

- **Group 1:** A group of six 1s (all except cell (10,0) and (10,1)). This covers all cases where \(A = 0\) or \(B = 0\). This group can be simplified to \(B'C + A'C'\).

The final simplified expression is:

\[ F(A, B, C) = B'C + A'C' \]

### Summary

1. For \(F(A, B, C, D) = \Sigma m(0, 1, 3, 4, 5, 7)\), the simplified expression is \(A'B' + A'BD\).

2. For \(F(A, B, C) = \Sigma m(0, 1, 4, 5, 6, 7)\), the simplified expression is \(B'C + A'C'\).

Answer 2

To simplify the given Sum of Products (SOP) expressions using Karnaugh Maps (K-Maps), we follow these steps:

### 1. F(A, B, C, D) = Σm(0, 1, 3, 4, 5, 7)

For a 4-variable function (A, B, C, D), we will use a 4x4 K-Map. Each cell in the K-Map corresponds to a minterm of the function. Here's how to fill in and simplify the expression:

#### Step-by-Step Solution:

**1. Draw the K-Map:**

| \AB\CD | 00 | 01 | 11 | 10 |
|--------|----|----|----|----|
| 00     |    |    |    |    |
| 01     |    |    |    |    |
| 11     |    |    |    |    |
| 10     |    |    |    |    |

**2. Fill in the K-Map:**

- Minterm 0: A'B'C'D'
- Minterm 1: A'B'C'D
- Minterm 3: A'B'CD
- Minterm 4: A'BC'D'
- Minterm 5: A'BC'D
- Minterm 7: A'BCD

Fill the map with 1s at the positions corresponding to these minterms:

| \AB\CD | 00 | 01 | 11 | 10 |
|--------|----|----|----|----|
| 00     |  1 |  1 |  0 |  0 |
| 01     |  1 |  1 |  1 |  0 |
| 11     |  0 |  0 |  1 |  0 |
| 10     |  0 |  0 |  0 |  0 |

**3. Group the 1s into the largest possible groups (1, 2, 4, 8 cells):**

- **Group 1:** The four 1s in the bottom-left 2x2 submap covering minterms 0, 1, 4, 5. This simplifies to \(A' \text{ and } D' \text{ (since } B \text{ and } C \text{ are changing)}\).

- **Group 2:** The four 1s in the bottom-right 2x2 submap covering minterms 4, 5, 7 (and extending to include minterms 6, which is not listed but completes the 2x2 map). This simplifies to \(A'B \text{ and } C\).

**4. Write the simplified expression:**

Combining these groups, we get:

\[ F(A, B, C, D) = A'D' + A'BC \]

### 2. F(A, B, C) = Σm(0, 1, 4, 5, 6, 7)

For a 3-variable function (A, B, C), we will use a 2x4 K-Map. Here's how to fill in and simplify the expression:

#### Step-by-Step Solution:

**1. Draw the K-Map:**

| \AB\C | 0 | 1 |
|-------|---|---|
| 00    |   |   |
| 01    |   |   |
| 11    |   |   |
| 10    |   |   |

**2. Fill in the K-Map:**

- Minterm 0: A'B'C'
- Minterm 1: A'B'C
- Minterm 4: AB'C'
- Minterm 5: AB'C
- Minterm 6: ABC'
- Minterm 7: ABC

Fill the map with 1s at the positions corresponding to these minterms:

| \AB\C | 0 | 1 |
|-------|---|---|
| 00    | 1 | 1 |
| 01    | 1 | 1 |
| 11    | 1 | 1 |
| 10    | 0 | 0 |

**3. Group the 1s into the largest possible groups:**

- **Group 1:** The four 1s in the top row covering minterms 0, 1, 4, 5. This simplifies to \(B'\) (since \(A\) and \(C\) are changing).

- **Group 2:** The four 1s in the bottom row covering minterms 6 and 7. This simplifies to \(A\) (since \(B\) and \(C\) are changing).

**4. Write the simplified expression:**

Combining these groups, we get:

\[ F(A, B, C) = B' + A \]

So, the simplified expressions for the given SOP expressions using K-Maps are:

1. \( F(A, B, C, D) = A'D' + A'BC \)
2. \( F(A, B, C) = B' + A \)