Q.No.

(a) (i) Determine a minimal SOP representation for F(W,X,Y,Z) =

6.

Em(0,1,3,4,8,9) + d(2,5,13) using K Map method. Draw the circuit of the minimal expression using logic gates.

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 \(W, X, Y, Z\).

   \[
   \begin{array}{|c|c|c|c|c|}
   \hline
   WX \backslash YZ & 00 & 01 & 11 & 10 \\
   \hline
   00 & 1 & 1 & 0 & 1 \\
   \hline
   01 & d & 1 & 0 & 0 \\
   \hline
   11 & 0 & 0 & d & 0 \\
   \hline
   10 & 1 & 1 & 0 & d \\
   \hline
   \end{array}
   \]

   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:

   ![Logic Circuit](https://latex.codecogs.com/svg.latex?%5Cbegin%7Barray%7D%7Bc%7D%5Ctext%7B%20AND%20gates%3A%20%7D%5Cbegin%7Barray%7D%7Bc%7D%28W%27%20%26%20Z%27%29%20%5C%5C%28X%27%20%26%20Y%27%29%20%5C%5C%28W%20%26%20X%27%20%26%20Y%29%20%5Cend%7Barray%7D%5C%5C%5Ctext%7B%20OR%20gate%3A%20%7D%5C%5C%28W%27%20%26%20Z%27%29%20%5C%2B%20%28X%27%20%26%20Y%27%29%20%5C%2B%20%28W%20%26%20X%27%20%26%20Y%29%5Cend%7Barray%7D)

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.

Learn how to determine the minimal Sum of Products (SOP) representation for the Boolean function F(W,X,Y,Z) = ∑m(0,1,3,4,8,9) + d(2,5,13) using Karnaugh Maps and logic gates. This problem involves Boolean algebra concepts and their application to circuit design.

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