Simplify the following expression using 𝐾 −map;
i. 𝑓(𝑎, 𝑏, 𝑐, 𝑑) = ∑𝑚(0, 1, 2 ,3, 4, 6, 8, 9, 10, 11, 12, 14)

To simplify the given function \( f(a, b, c, d) = \sum m(0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 14) \) using a **Karnaugh Map (K-map)**, we need to follow these steps:

---

### Step 1: Construct the K-map
The K-map for a 4-variable function (\(a, b, c, d\)) is a 4x4 grid. Each cell corresponds to a minterm number. Here’s the typical layout of the cells:

| **CD\AB** | 00   | 01   | 11   | 10   |
|-----------|-------|-------|-------|-------|
| **00**    | 0     | 1     | 3     | 2     |
| **01**    | 4     | 5     | 7     | 6     |
| **11**    | 12    | 13    | 15    | 14    |
| **10**    | 8     | 9     | 11    | 10    |

---

### Step 2: Mark the minterms in the K-map
Place 1s in the cells corresponding to the minterms \(0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 14\). Remaining cells are filled with 0.

| **CD\AB** | 00   | 01   | 11   | 10   |
|-----------|-------|-------|-------|-------|
| **00**    | 1     | 1     | 1     | 1     |
| **01**    | 1     | 0     | 0     | 1     |
| **11**    | 1     | 0     | 0     | 1     |
| **10**    | 1     | 1     | 1     | 1     |

---

### Step 3: Group the 1s into rectangles
1. **Group 1 (8-cells):** Combine all cells in rows **10** and **00**. This group corresponds to \( a' + b \) (since \(c, d\) vary).
2. **Group 2 (4-cells):** Combine \(6, 14, 10, 2 \ ) .


Simplify the following expression using K-map: f(a, b, c, d) = ∑m(0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 14)

Learn how to simplify the Boolean expression f(a, b, c, d) = ∑m(0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 14) using a Karnaugh Map (K-map). This step-by-step guide covers marking minterms, grouping, and deriving the simplified expression. Ideal for Grades 11-12 and college-level students studying Boolean Algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation