To determine the Sum of Products (SOP) form of the given Boolean function from the Karnaugh map, we proceed with the following steps:

Step 1: Identify cells with value 1

The cells in the Karnaugh map with 1 represent the minterms of the function. We'll analyze these based on the variable names w, x, y, z and their combinations.

Variable Representation:

• wx (rows):

  • 00 → $w' x'$
  • 01 → $w' x$
  • 11 → $w x$
  • 10 → $w x'$

• yz (columns):

  • 00 → $y' z'$
  • 01 → $y' z$
  • 11 → $y z$
  • 10 → $y z'$

Values of 1 in the table:

• From the map, we observe the following cells with 1:
  1. $wx = 01, yz = 00$ → $w'x y' z'$
  2. $wx = 01, yz = 01$ → $w'x y' z$
  3. $wx = 01, yz = 11$ → $w'x y z$
  4. $wx = 01, yz = 10$ → $w'x y z'$
  5. $wx = 11, yz = 10$ → $w x y z'$

Ignore "Don't Care" (x) Cells:

The "don't care" (x) cells can be used for further minimization, but they do not directly contribute to the SOP unless grouped for simplification.

Step 2: Write the SOP Expression

The expression derived directly from the ones without minimization is:

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

Step 3: Minimize (if needed)

Using grouping in the Karnaugh map:

• The 4 adjacent 1s in the row $wx = 01$ (covering all columns $yz = 00, 01, 11, 10$) simplify to $w'x y$.
• The single 1 in $wx = 11, yz = 10$ remains as $w x y z'$.

Thus, the simplified SOP form is:

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

Final Answer:

The SOP form of the Boolean function is:

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

Would you like a detailed breakdown of how I simplified this? Here are some related questions to deepen your understanding:

1. How do "don't care" conditions help in simplification?
2. What is the difference between SOP and POS forms in Boolean algebra?
3. How are minterms and maxterms represented in Karnaugh maps?
4. Can you explain the process of grouping 1s in K-maps?
5. How would the answer change if one "don't care" value was forced to 1?

Tip: Always group as many 1s as possible (including "don't care" cells) to simplify your Boolean function.