To determine the minimal Sum of Products (SOP) representation for the function $F(W,X,Y,Z) = \sum m(0,1,3,4,8,9) + d(2,5,13)$ using the Karnaugh Map (K-map) method, follow these steps:

1. Construct the K-map: The K-map for a 4-variable function has 16 cells corresponding to all combinations of the variables $W, X, Y, Z$.

   \hline WX \backslash YZ & 00 & 01 & 11 & 10 \\ \hline 00 & 1 & 1 & 0 & 1 \\ \hline 01 & d & 1 & 0 & 0 \\ \hline 11 & 0 & 0 & d & 0 \\ \hline 10 & 1 & 1 & 0 & d \\ \hline \end{array}$$ Where: - '1' indicates the minterms (0, 1, 3, 4, 8, 9). - 'd' indicates the don't care terms (2, 5, 13). - '0' indicates the other minterms not present in the function.

2. Group the ones and don't care terms:

   • Group all the adjacent ones and don't care terms to form the largest possible groups of 1, 2, 4, or 8 cells.
   • Use the don't care terms to help form larger groups.

   Groupings:

   • Group 1: $m(0, 1, 4, 5)$ -> $W'Z'$
   • Group 2: $m(0, 1, 2, 3)$ -> $X'Y'$
   • Group 3: $m(8, 9)$ -> $WX'Y$

   These groupings cover all the minterms and use the don't care terms to minimize the expression.

3. Write the minimal SOP expression: From the groupings: $F(W, X, Y, Z) = W'Z' + X'Y' + WX'Y$

4. Draw the circuit:

   • AND Gates: Each product term (AND) in the expression will have its own AND gate.
   • OR Gate: The outputs of the AND gates will be fed into a single OR gate to form the final output.

   Here’s the circuit diagram:

   

Would you like more details or have any questions about this process?

Here are 5 numbered questions you might find useful:

1. How do you simplify boolean expressions using the K-map method?
2. What is the significance of 'don't care' conditions in K-map simplification?
3. How do you draw a circuit from a given boolean expression?
4. What are the rules for grouping minterms in K-map?
5. Can you explain the difference between SOP and POS forms?

Tip: When using K-maps for simplification, always look for the largest possible groups of 1s to minimize the expression efficiently.