Simplify the following three- variable Boolean functions using karnaugh map
a) f_1 = ∑1, 2, 5, 6
b) f_2 = ∑0, 1, 2, 3, 7

To simplify the given Boolean functions using a Karnaugh Map (K-map), we follow these steps:

### Function (a) \( f_1 = \sum(1, 2, 5, 6) \)

We will map the minterms to a 3-variable K-map and simplify.

1. **Variables:** Let the three variables be \( A \), \( B \), and \( C \).
2. **K-map setup:** The K-map for three variables consists of 8 cells representing all possible combinations of \( A \), \( B \), and \( C \).
   - Rows represent combinations of \( A \) and \( B \), while columns represent \( C \).
   - The minterms are placed into their respective cells.

| AB \ C | 0 | 1 |
|--------|---|---|
| 00     | 0 | 1 |
| 01     | 2 | 3 |
| 11     | 6 | 7 |
| 10     | 4 | 5 |

- Minterms: \( f_1 = \sum(1, 2, 5, 6) \)

| AB \ C | 0 | 1 |
|--------|---|---|
| 00     | 0 | 1 |
| 01     | 1 | 0 |
| 11     | 1 | 0 |
| 10     | 0 | 1 |

### Grouping:

- Group 1: Minterms \( 1, 5 \) (form a vertical pair).
- Group 2: Minterms \( 2, 6 \) (form another vertical pair).

### Simplified Expression:
- Group \( 1, 5 \): \( \overline{A} C \)
- Group \( 2, 6 \): \( B \overline{C} \)

Thus, the simplified expression for \( f_1 \) is:
\[
f_1 = \overline{A}C + B\overline{C}
\]

---

### Function (b) \( f_2 = \sum(0, 1, 2, 3, 7) \)

1. **Variables:** Again, let the three variables be \( A \), \( B \), and \( C \).
2. **K-map setup:** Same as before.

| AB \ C | 0 | 1 |
|--------|---|---|
| 00     | 0 | 1 |
| 01     | 2 | 3 |
| 11     | 6 | 7 |
| 10     | 4 | 5 |

- Minterms: \( f_2 = \sum(0, 1, 2, 3, 7) \)

| AB \ C | 0 | 1 |
|--------|---|---|
| 00     | 1 | 1 |
| 01     | 1 | 1 |
| 11     | 0 | 1 |
| 10     | 0 | 0 |

### Grouping:

- Group 1: Minterms \( 0, 1, 2, 3 \) (form a 2x2 block).
- Group 2: Minterm \( 7 \) (stands alone).

### Simplified Expression:
- Group \( 0, 1, 2, 3 \): \( \overline{A} \)
- Minterm \( 7 \): \( AB C \)

Thus, the simplified expression for \( f_2 \) is:
\[
f_2 = \overline{A} + ABC
\]

---

### Final Simplified Expressions:
- \( f_1 = \overline{A}C + B\overline{C} \)
- \( f_2 = \overline{A} + ABC \)

---

Let me know if you'd like further details or clarifications!

Here are some related questions you might explore:
1. How do we derive Boolean expressions from a K-map with more than three variables?
2. How would the K-map look for a four-variable Boolean function?
3. What are the grouping rules when simplifying K-maps?
4. Can Karnaugh maps handle functions with "don't care" conditions?
5. What is the difference between minterms and maxterms in Boolean algebra?

**Tip:** Always check for the largest possible groupings in a K-map to achieve the most simplified expression.

Simplify the following three-variable Boolean functions using Karnaugh map
a) f_1 = ∑1, 2, 5, 6
b) f_2 = ∑0, 1, 2, 3, 7

Learn how to simplify the three-variable Boolean functions f_1 = ∑1, 2, 5, 6 and f_2 = ∑0, 1, 2, 3, 7 using Karnaugh Maps. This tutorial covers Karnaugh map grouping rules, Boolean algebra simplification, and provides step-by-step solutions for both functions. Suitable for undergraduate students in Computer Science or Electrical Engineering.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation