Question

Minimize the following Boolean equation using 'K' map: Y = A'B'C' + A'B'C + A'BC + ABC, and implement using basic gates.

Answer 1

To minimize the Boolean expression \( Y = A'B'C' + A'B'C + A'BC + ABC \) using a Karnaugh map (K-map), follow these steps:

### Step 1: Create the K-map

Since there are three variables (A, B, C), we use a 3-variable K-map.

The K-map looks like this:

\[
\begin{array}{c|cc|cc}
\text{AB} \backslash \text{C} & 0 & 0 & 1 & 1 \\
\hline
00 & A'B'C' & A'B'C & A'BC & A'BC \\
01 & A'B'C' & A'B'C & A'BC & A'BC \\
10 & AB'C' & AB'C & ABC & ABC \\
11 & AB'C' & AB'C & ABC & ABC \\
\end{array}
\]

Where the rows are for values of AB and columns are for values of C.

### Step 2: Populate the K-map

Fill in the K-map with the values from the Boolean expression.

- \( A'B'C' \) corresponds to cell \( (A=0, B=0, C=0) \)
- \( A'B'C \) corresponds to cell \( (A=0, B=0, C=1) \)
- \( A'BC \) corresponds to cell \( (A=0, B=1, C=1) \)
- \( ABC \) corresponds to cell \( (A=1, B=1, C=1) \)

The K-map will look like this:

\[
\begin{array}{c|cc|cc}
\text{AB} \backslash \text{C} & 0 & 1 \\
\hline
00 & 1 & 1 \\
01 & 0 & 1 \\
10 & 0 & 0 \\
11 & 0 & 1 \\
\end{array}
\]

### Step 3: Group the 1s

Group the 1s in the K-map to form the largest possible groups (1, 2, 4, 8... cells):

1. **Group 1**: The group of 1s in the top row (A'B'C' and A'B'C)
   - This group simplifies to \( A'B' \).

2. **Group 2**: The group of 1s in the right column (A'BC and ABC)
   - This group simplifies to \( BC \).

### Step 4: Write the simplified Boolean expression

Combine the simplified groups:

\[ Y = A'B' + BC \]

### Step 5: Implement using basic gates

To implement \( Y = A'B' + BC \) using basic gates:

1. **For \( A'B' \):**
   - Use a NOT gate to get \( A' \) and another NOT gate to get \( B' \).
   - Use an AND gate to combine \( A' \) and \( B' \).

2. **For \( BC \):**
   - Use an AND gate to combine \( B \) and \( C \).

3. **Combine \( A'B' \) and \( BC \):**
   - Use an OR gate to combine the outputs of the two AND gates.

The circuit will look like this:

1. **NOT Gates**:
   - Input A to a NOT gate to get \( A' \).
   - Input B to a NOT gate to get \( B' \).

2. **AND Gates**:
   - First AND gate: \( A' \) and \( B' \).
   - Second AND gate: \( B \) and \( C \).

3. **OR Gate**:
   - Combine the outputs of the two AND gates.

This circuit will give you the minimized output \( Y \).

Answer 2

To minimize the Boolean equation \( Y = A'B'C' + A'B'C + A'BC + ABC \) using a Karnaugh Map (K-map) and implement it using basic gates, follow these steps:

### 1. Construct the Karnaugh Map

A Karnaugh Map for three variables (A, B, C) is a 2x4 grid. Here’s how to set it up:

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

- The rows represent different combinations of \( A \) and \( B \).
- The columns represent the values of \( C \).

#### Populate the K-map

From the Boolean equation:

1. \( A'B'C' \) corresponds to A=0, B=0, C=0.
2. \( A'B'C \) corresponds to A=0, B=0, C=1.
3. \( A'BC \) corresponds to A=0, B=1, C=1.
4. \( ABC \) corresponds to A=1, B=1, C=1.

Place 1s in the K-map according to these terms:

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

### 2. Group Ones

Next, identify and group the 1s in the K-map into the largest possible groups of 1, 2, or 4.

- Group the four 1s in column C=1 (covers all rows, i.e., \( C \)).
- There are no additional groups of 2 or 4, so the K-map simplifies to just one group.

### 3. Derive the Simplified Boolean Expression

From the grouping:
- The simplified Boolean expression is just \( C \).

### 4. Implement Using Basic Gates

The simplified Boolean function \( Y = C \) can be implemented directly with a single wire connected to the input \( C \). Here’s how you would implement it:

- **Connect the input \( C \) directly to the output \( Y \)**.

Since \( Y = C \) means the output is simply the same as the input \( C \), you don’t need additional gates in this case.

### Summary

1. **K-map Grouping**: Group the 1s in the K-map to get the simplified expression.
2. **Simplified Expression**: \( Y = C \).
3. **Implementation**: Directly connect the input \( C \) to the output \( Y \).

This approach ensures that your Boolean equation is minimized effectively and implemented with the least complexity.