Question 1191388: Simplify the following functions using a K-map
F(X, Y, Z) = m0 + m2 + m5 + m7
Could you please explain to me this homework question step-by-step? Thank you!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to simplify F(X, Y, Z) = m0 + m2 + m5 + m7 using a K-map:
1. **Set up the K-map:** Since there are three variables (X, Y, and Z), the K-map will be a 2x4 grid:
```
YZ
00 01 11 10
X 0 m0 m1 m3 m2
1 m4 m5 m7 m6
```
2. **Fill in the K-map based on the minterms:**
* m0: X=0, Y=0, Z=0 (top left cell)
* m2: X=0, Y=1, Z=0 (top right cell)
* m5: X=1, Y=0, Z=1 (bottom middle cell)
* m7: X=1, Y=1, Z=1 (bottom right cell)
```
YZ
00 01 11 10
X 0 1 0 0 1
1 0 1 1 0
```
3. **Group the 1s:** We want to make the largest possible groups of 1s, where the groups are powers of 2 (1, 2, 4, 8).
* **Group 1:** The two 1s in the top row can be grouped together. This represents X'Z' (X=0 and Z=0 for both cells).
* **Group 2:** The two 1s in the bottom right corner can be grouped together. This represents XZ (X=1 and Z=1 for both cells).
4. **Write the simplified expression:**
F(X, Y, Z) = X'Z' + XZ
Therefore, the simplified function is **F(X, Y, Z) = X'Z' + XZ**.
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