To simplify the Boolean function $y(A,B,C) = \Sigma(0, 1, 3, 6, 7)$ using a Karnaugh map (K-map), follow these steps:

1. Create the Karnaugh Map (K-map):

The function $y(A,B,C)$ is a three-variable function, so we'll create a 2x4 K-map. The variables $A$, $B$, and $C$ will be represented as follows:

• Rows: Represent $A$ (0 for the first row, 1 for the second row).
• Columns: Represent $BC$ in the following order: 00, 01, 11, 10.

2. Populate the K-map with the given minterms:

The minterms are given by the indices $0, 1, 3, 6, 7$. We’ll place a 1 in the corresponding cells of the K-map.

Here’s how the minterms map to the K-map:

A \backslash BC & 00 & 01 & 11 & 10 \\ \hline 0 & 1 (0) & 1 (1) & 0 (3) & 0 (2) \\ 1 & 1 (4) & 0 (5) & 1 (7) & 1 (6) \\ \end{array}$$ ### 3. **Simplified K-map with minterms**: $$\begin{array}{c|c|c|c|c} A \backslash BC & 00 & 01 & 11 & 10 \\ \hline 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 \\ \end{array}$$ ### 4. **Group the 1s into the largest possible groups**: We now group the 1s together. The goal is to cover all the 1s using the fewest and largest possible groups (rectangles that are powers of 2: 1, 2, or 4 cells). #### Step-by-Step Grouping: 1. **Group 1**: The cells in positions (0,0), (0,1), and (0,2) form a horizontal group of 3. 2. **Group 2**: The cells in positions (1,0), (1,2), and (1,3) form another group. ### 5. **Determine the simplified Boolean expression**: - **Group 1** (covers minterms 0, 1, 3): This group covers the rows where $$A = 0$$ and columns where $$B = 0$$. Therefore, the term for this group is $$\overline{A}$$. - **Group 2** (covers minterms 4, 7, 6): This group covers the columns where $$C = 0$$ and the row where $$B = 1$$. Therefore, the term for this group is $$B$$. Thus, the simplified Boolean expression is: $$y(A,B,C) = \overline{A} + B$$ ### Conclusion: The simplified expression for $$y(A,B,C) = \Sigma(0, 1, 3, 6, 7)$$ using the Karnaugh map is $$\overline{A} + B$$. --- Would you like any further details on this process or have any other questions? ### 5 Related Questions: 1. How can Karnaugh maps be used for four-variable functions? 2. What are the rules for grouping cells in a Karnaugh map? 3. How does Karnaugh mapping compare to Boolean algebra simplification? 4. Can Karnaugh maps be used for functions with more than four variables? 5. What are don’t-care conditions and how are they used in K-maps? **Tip:** When grouping in K-maps, always aim for the largest possible groups (power of 2) to simplify the Boolean expression effectively.