To minimize a four-variable Boolean function $f(w, x, y, z)$ using the Karnaugh map (K-map) method, follow these steps. The given function is:

$f(w, x, y, z) = \sum(0, 2, 5, 7, 10, 13, 14, 15)$

Step 1: Create a 4-variable K-map

The K-map for a 4-variable function will have a 4x4 grid, with two variables on the horizontal axis and two on the vertical axis. Let's assign the variables as follows:

• Rows: $wy$ combination (00, 01, 11, 10)
• Columns: $xz$ combination (00, 01, 11, 10)

The K-map looks like this:

	00	01	11	10
00
01
11
10

Step 2: Plot the minterms

Next, we fill in the K-map with 1's corresponding to the minterms $\sum(0, 2, 5, 7, 10, 13, 14, 15)$. The binary equivalents of these minterms are:

• Minterm 0: $(0000)$
• Minterm 2: $(0010)$
• Minterm 5: $(0101)$
• Minterm 7: $(0111)$
• Minterm 10: $(1010)$
• Minterm 13: $(1101)$
• Minterm 14: $(1110)$
• Minterm 15: $(1111)$

Now, plot the 1's in the K-map at these locations:

	00	01	11	10
00	1			1
01		1	1
11			1	1
10		1	1

Step 3: Group the 1's into largest possible groups

We group adjacent 1's in powers of 2 (1, 2, 4, or 8 cells), where adjacency includes wrapping around the edges.

1. A group of four covering minterms $(0, 2, 10, 14)$.
2. A group of four covering minterms $(5, 7, 13, 15)$.

Step 4: Write the simplified Boolean expression

For each group, we find the common variables:

1. Group covering (0, 2, 10, 14):

   • Common in rows (00, 10): $w'$
   • Common in columns (00, 10): $z'$
   • So, the term for this group is $w'z'$.

2. Group covering (5, 7, 13, 15):

   • Common in rows (01, 11): $y$
   • Common in columns (01, 11): $x$
   • So, the term for this group is $xy$.

Step 5: Final simplified expression

The minimized Boolean expression is:

$f(w, x, y, z) = w'z' + xy$

This is the simplified expression for the given function using the K-map method.

Would you like further details or have any questions?

Here are 5 related questions:

1. How does wrapping work in a K-map?
2. What happens if a minterm is not included in any group?
3. Can K-maps be used for more than 4 variables?
4. How would the solution change if we included "don't care" terms?
5. Can we use K-maps for functions of fewer variables?

Tip: Always try to form the largest possible groups in the K-map to minimize the number of terms in the expression.