Simplify the function y(A,B,C) = Σ (0,1,3,6,7) using K-map

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:

\[
\begin{array}{c|c|c|c|c}
A \backslash BC & 00 & 01 & 11 & 10 \\
\hline
0 & 1 (0) & 1 (1) & 0 (3) & 0 (2) \\
1 & 1 (4) & 0 (5) & 1 (7) & 1 (6) \\
\end{array}
\]

### 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 \( \overline{A} + B \).

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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.

Learn how to simplify the Boolean function y(A,B,C) = Σ(0, 1, 3, 6, 7) using a Karnaugh map. Follow step-by-step instructions to group minterms and derive the simplified Boolean expression. Suitable for advanced high school students studying Boolean algebra.

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