Use the K-map method to minimize a 4-variable (w, x, y, and z) function F
F =∑(0,2,5,7,10,13,14,15)

To minimize the 4-variable function \( F \) using a Karnaugh map (K-map), follow these steps:

### Step-by-Step Solution:

1. **List the Min Terms:**
   The min terms for \( F \) are \( m(0, 2, 5, 7, 10, 13, 14, 15) \).

2. **Draw the 4-variable K-map:**
   A 4-variable K-map is a 4x4 grid where each cell represents a unique combination of the variables \( w, x, y, \) and \( z \).

   |       | yz=00 | yz=01 | yz=11 | yz=10 |
   |-------|-------|-------|-------|-------|
   | wx=00 |   0   |   1   |   3   |   2   |
   | wx=01 |   4   |   5   |   7   |   6   |
   | wx=11 |  12   |  13   |  15   |  14   |
   | wx=10 |   8   |   9   |  11   |  10   |

3. **Fill in the K-map with 1s for the min terms:**

   |       | yz=00 | yz=01 | yz=11 | yz=10 |
   |-------|-------|-------|-------|-------|
   | wx=00 |   1   |   0   |   0   |   1   |
   | wx=01 |   0   |   1   |   1   |   0   |
   | wx=11 |   0   |   1   |   1   |   1   |
   | wx=10 |   0   |   0   |   0   |   1   |

4. **Group the 1s into the largest possible power-of-2 rectangles:**

   - A group of eight 1s covering cells (5, 7, 13, 15) in columns yz=01 and yz=11, wx=01 and wx=11.
   - A group of four 1s covering cells (13, 14, 15, 10) in row wx=11 and wx=10, columns yz=11 and yz=10.
   - A single group of two 1s covering cells (0, 2) in row wx=00, columns yz=00 and yz=10.

5. **Write the simplified Boolean expression for each group:**

   - Group of eight 1s (covering 5, 7, 13, 15): \( \overline{x}z \)
   - Group of four 1s (covering 13, 14, 15, 10): \( w\overline{y} \)
   - Group of two 1s (covering 0, 2): \( \overline{w}\overline{z} \)

6. **Combine the simplified expressions:**

   The simplified Boolean expression for \( F \) is:
   \[
   F = \overline{x}z + w\overline{y} + \overline{w}\overline{z}
   \]

### Final Simplified Expression:
\[ F = \overline{x}z + w\overline{y} + \overline{w}\overline{z} \]

Would you like any further details or have any questions?

#### Follow-up Questions:
1. Can you provide a detailed explanation of how to identify and group 1s in a K-map?
2. How would the K-map change if additional min terms were added or removed?
3. What are the benefits of using the K-map method over other simplification techniques?
4. How can we verify the correctness of the simplified Boolean expression?
5. How would the process differ for a 3-variable K-map?

#### Tip:
When grouping 1s in a K-map, always aim to form the largest possible groups that are powers of 2 (1, 2, 4, 8, etc.) to achieve the simplest expression.

Use the K-map method to minimize a 4-variable (w, x, y, and z) function F F =∑(0,2,5,7,10,13,14,15)

Learn how to minimize a 4-variable Boolean function F = ∑(0,2,5,7,10,13,14,15) using the Karnaugh Map (K-map) method. This solution simplifies the Boolean expression by grouping minterms and identifying the most efficient form. Suitable for high school students in Grades 10-12.

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