Question 1192134: Use a k-map to simplify the following function:
F(W, X, Y, Z) = X'Y' + XYZ' + WXY + W'X'Y' + WZ
Can I get an explanation of this homework question step-by-step please? Thank you very much!!!
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to simplify the function F(W, X, Y, Z) = X'Y' + XYZ' + WXY + W'X'Y' + WZ using a K-map:
1. **Set up the K-map:**
```
WZ
00 01 11 10
WX 00 1 1 1 1 (X'Y')
01 0 0 1 0 (XYZ')
11 0 1 1 0 (WXY)
10 1 1 1 1 (W'X'Y' and WZ)
```
2. **Fill in the K-map:**
* X'Y' (W=0, X=0, Y=0): This covers the first row (00) entirely, regardless of Z.
* XYZ' (X=1, Y=1, Z=0): This is the cell at WX=01, YZ=10.
* WXY (W=1, X=1, Y=1): This is the cell at WX=11, YZ=11.
* W'X'Y' (W=0, X=0, Y=0): This is already covered by X'Y'.
* WZ: This term covers all cells where W=1 *or* Z=1. The cells where W=1 are already covered by WXY and a portion of X'Y'. The cells where Z=1 are on the 01 and 11 columns.
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, 16).
* **Group 1:** The entire first row (X'Y') is a group of 4. This represents X'Y'.
* **Group 2:** The 1s in the bottom row (W=1) are best grouped as a group of 4 (covering WZ). This represents W.
* **Group 3:** The remaining '1' at WX=01 and YZ=10 is covered by XYZ'.
4. **Write the simplified expression:**
By combining the groups, we get the simplified Boolean expression:
F(W, X, Y, Z) = X'Y' + W + XYZ'
Therefore, the simplified function is **F(W, X, Y, Z) = X'Y' + W + XYZ'**.
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