Questions: Problem 3 For the SOP equation: z=A̅ B C̅ + A̅ C D̅ + A̅ B C̅ D̅ + A B̅ C Use a K map to simplify this equation.

To simplify the given Sum of Products (SOP) equation using a Karnaugh map (K-map), we need to follow these steps:

1. Identify the Variables: The given equation involves four variables: \( A, B, C, \) and \( D \).

2. Write the Minterms: The given SOP expression is: \[ z = \bar{A} B \bar{C} + \bar{A} C \bar{D} + \bar{A} B \bar{C} \bar{D} + A \bar{B} C \]

   We can identify the minterms from this expression:

   • \(\bar{A} B \bar{C}\) corresponds to the minterm where \( A = 0, B = 1, C = 0 \).
   • \(\bar{A} C \bar{D}\) corresponds to the minterm where \( A = 0, C = 1, D = 0 \).
   • \(\bar{A} B \bar{C} \bar{D}\) corresponds to the minterm where \( A = 0, B = 1, C = 0, D = 0 \).
   • \(A \bar{B} C\) corresponds to the minterm where \( A = 1, B = 0, C = 1 \).

3. Plot the Minterms on the K-map: A 4-variable K-map has 16 cells, each representing a minterm. The K-map is organized as follows:

   \[ \begin{array}{c|c|c|c|c} & 00 & 01 & 11 & 10 \\ \hline 00 & m_0 & m_1 & m_3 & m_2 \\ \hline 01 & m_4 & m_5 & m_7 & m_6 \\ \hline 11 & m_{12} & m_{13} & m_{15} & m_{14} \\ \hline 10 & m_8 & m_9 & m_{11} & m_{10} \\ \end{array} \]

   Here, the rows represent the values of \( A \) and \( B \) (00, 01, 11, 10), and the columns represent the values of \( C \) and \( D \) (00, 01, 11, 10).

   • \(\bar{A} B \bar{C}\) corresponds to \( m_4 \).
   • \(\bar{A} C \bar{D}\) corresponds to \( m_1 \).
   • \(\bar{A} B \bar{C} \bar{D}\) corresponds to \( m_0 \).
   • \(A \bar{B} C\) corresponds to \( m_{12} \).

4. Fill the K-map: Place 1s in the cells corresponding to the minterms:

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

5. Group the 1s: Look for groups of 1s that can be combined. The goal is to form the largest possible groups of 1, 2, 4, or 8 cells. Each group must be a power of two and can wrap around the edges of the K-map.

   • Group 1: \( m_0, m_1 \) (covers \(\bar{A} \bar{C}\))
   • Group 2: \( m_0, m_4 \) (covers \(\bar{A} B\))
   • Group 3: \( m_{12} \) (covers \(A \bar{B} C\))

6. Write the Simplified Expression: From the groups, derive the simplified expression:

   • Group 1: \(\bar{A} \bar{C}\)
   • Group 2: \(\bar{A} B\)
   • Group 3: \(A \bar{B} C\)

   The simplified expression is: \[ z = \bar{A} \bar{C} + \bar{A} B + A \bar{B} C \]

In summary, the simplified SOP expression for the given equation is \( z = \bar{A} \bar{C} + \bar{A} B + A \bar{B} C \).