2 Simplify the following Boolean function, using four variables K-map. F(A, B, C, D) = Σ(2, 3, 6, 7, 12, 13, 14)
2 Simplify the following Boolean function, using four variables K-map. F(A, B, C, D) = Σ(2, 3, 6, 7, 12, 13, 14)
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter6: Modularity Using Functions
Section6.4: A Case Study: Rectangular To Polar Coordinate Conversion
Problem 6E
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Please use four variables K-map and show your work on a paper
![### Simplification of a Boolean Function Using a Four-Variable K-map
#### Problem Statement:
Simplify the following Boolean function using a four-variable Karnaugh map (K-map).
\[ F(A, B, C, D) = \Sigma(2, 3, 6, 7, 12, 13, 14) \]
### Explanation:
In this problem, we are given a Boolean function \( F \) defined by the sum of minterms. The minterm indices provided are 2, 3, 6, 7, 12, 13, and 14. These indices represent the binary combinations of the variables \( A \), \( B \), \( C \), and \( D \).
### Steps for Simplification Using K-map:
1. **Construct the K-map**:
A four-variable K-map has 16 cells, corresponding to each combination of \( A, B, C, \) and \( D \).
2. **Populate the K-map**:
- Mark ‘1’ in the cells corresponding to the minterms given in the function.
- The K-map for this problem will look like this:
| AB\CD | 00 | 01 | 11 | 10 |
|------|----|----|----|----|
| 00 | 0 | 0 | 0 | 0 |
| 01 | 0 | 1 | 1 | 0 |
| 11 | 0 | 1 | 1 | 0 |
| 10 | 0 | 1 | 1 | 1 |
3. **Simplify**:
- Group adjacent 1s in the K-map into rectangles containing 1, 2, 4, or 8 cells.
- The goal is to cover all 1s using the fewest number of groups.
4. **Write the simplified Boolean expression**:
- Each group you form in the K-map translates to a term in the simplified Boolean expression.
### Final Simplified Expression:
After mapping and grouping the minterms from the K-map, you derive the simplified Boolean expression.
\[ F(A, B, C, D) = AC' + AD' \]
This simplification process minimizes the original Boolean function to its simplest form for](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6b89b23-87fd-4ddb-835f-d9a4143d61af%2Fc3326361-f4a3-4484-9eba-f329ec1c602f%2F3g1yzzj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Simplification of a Boolean Function Using a Four-Variable K-map
#### Problem Statement:
Simplify the following Boolean function using a four-variable Karnaugh map (K-map).
\[ F(A, B, C, D) = \Sigma(2, 3, 6, 7, 12, 13, 14) \]
### Explanation:
In this problem, we are given a Boolean function \( F \) defined by the sum of minterms. The minterm indices provided are 2, 3, 6, 7, 12, 13, and 14. These indices represent the binary combinations of the variables \( A \), \( B \), \( C \), and \( D \).
### Steps for Simplification Using K-map:
1. **Construct the K-map**:
A four-variable K-map has 16 cells, corresponding to each combination of \( A, B, C, \) and \( D \).
2. **Populate the K-map**:
- Mark ‘1’ in the cells corresponding to the minterms given in the function.
- The K-map for this problem will look like this:
| AB\CD | 00 | 01 | 11 | 10 |
|------|----|----|----|----|
| 00 | 0 | 0 | 0 | 0 |
| 01 | 0 | 1 | 1 | 0 |
| 11 | 0 | 1 | 1 | 0 |
| 10 | 0 | 1 | 1 | 1 |
3. **Simplify**:
- Group adjacent 1s in the K-map into rectangles containing 1, 2, 4, or 8 cells.
- The goal is to cover all 1s using the fewest number of groups.
4. **Write the simplified Boolean expression**:
- Each group you form in the K-map translates to a term in the simplified Boolean expression.
### Final Simplified Expression:
After mapping and grouping the minterms from the K-map, you derive the simplified Boolean expression.
\[ F(A, B, C, D) = AC' + AD' \]
This simplification process minimizes the original Boolean function to its simplest form for
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