Question

Implement the following Boolean equation using fundamental gates: Y = ABC + A'B'C' + A'C'B.

Answer 1

To implement the Boolean equation \( Y = ABC + A'B'C' + A'C'B \) using fundamental logic gates, we need to use **AND, OR,** and **NOT** gates. Below is a step-by-step approach to build the circuit.

### 1. Break down the equation into smaller parts:

The equation consists of three main product terms (AND terms):
1. \( ABC \)
2. \( A'B'C' \)
3. \( A'C'B \)

We will need to implement each term individually and then combine them using an **OR** gate.

#### A. First Term: \( ABC \)
- This is the **AND** of \( A \), \( B \), and \( C \).
  - Use an **AND** gate with three inputs: \( A \), \( B \), and \( C \).

#### B. Second Term: \( A'B'C' \)
- This is the **AND** of \( A' \), \( B' \), and \( C' \) (the complements of \( A \), \( B \), and \( C \)).
  - First, generate the complement of \( A \), \( B \), and \( C \) using **NOT** gates:
    - \( A' \) from \( A \)
    - \( B' \) from \( B \)
    - \( C' \) from \( C \)
  - Then, use a three-input **AND** gate to combine \( A' \), \( B' \), and \( C' \).

#### C. Third Term: \( A'C'B \)
- This is the **AND** of \( A' \), \( C' \), and \( B \).
  - You already have \( A' \) and \( C' \) from the previous step.
  - Use a three-input **AND** gate to combine \( A' \), \( C' \), and \( B \).

### 2. Combine the results:
- After generating the three product terms \( ABC \), \( A'B'C' \), and \( A'C'B \), combine them using a **3-input OR** gate.

### 3. Circuit Diagram

Let’s outline the steps in a more visual way:

1. **NOT Gates**:
   - \( A' = \text{NOT}(A) \)
   - \( B' = \text{NOT}(B) \)
   - \( C' = \text{NOT}(C) \)

2. **AND Gates**:
   - \( ABC \) = AND of \( A \), \( B \), \( C \).
   - \( A'B'C' \) = AND of \( A' \), \( B' \), \( C' \).
   - \( A'C'B \) = AND of \( A' \), \( C' \), \( B \).

3. **OR Gate**:
   - Combine \( ABC \), \( A'B'C' \), and \( A'C'B \) using an OR gate.

### Final Gate-level Implementation:

1. Inputs: \( A \), \( B \), \( C \)
2. NOT gates:
   - \( A' \), \( B' \), \( C' \)
3. AND gates:
   - \( ABC \) (AND gate with inputs \( A \), \( B \), \( C \))
   - \( A'B'C' \) (AND gate with inputs \( A' \), \( B' \), \( C' \))
   - \( A'C'B \) (AND gate with inputs \( A' \), \( C' \), \( B \))
4. OR gate:
   - Combine outputs of the three AND gates to get the final output \( Y \).

### Conclusion:

By following these steps, you can implement the Boolean expression \( Y = ABC + A'B'C' + A'C'B \) using fundamental gates (NOT, AND, OR). The final circuit will consist of three NOT gates (to generate the complements), three AND gates (for each product term), and one OR gate to combine the results.

Answer 2

To implement the Boolean equation \( Y = ABC + A'B'C' + A'C'B \) using fundamental gates, we can follow these steps. The fundamental gates are AND, OR, and NOT gates. Here’s a detailed procedure:

### 1. **Simplify the Boolean Expression**

First, let’s break down and simplify the Boolean expression if possible.

Given:
\[ Y = ABC + A'B'C' + A'C'B \]

We can analyze and simplify this expression:

1. **Combine terms if possible:**

   We have:
   - \( ABC \)
   - \( A'B'C' \)
   - \( A'C'B \)

   This expression doesn’t simplify easily through Boolean algebraic identities, so we’ll implement it directly using fundamental gates.

### 2. **Draw the Logic Diagram**

To implement \( Y \) with fundamental gates:

#### **1. Implement Each Term:**

- **Term 1: \( ABC \)**

  Use three AND gates:
  - Input to the first AND gate: \( A \) and \( B \) (outputs \( AB \)).
  - Input to the second AND gate: \( AB \) and \( C \) (outputs \( ABC \)).

- **Term 2: \( A'B'C' \)**

  Use three NOT gates and one AND gate:
  - Input to the first NOT gate: \( A \) (outputs \( A' \)).
  - Input to the second NOT gate: \( B \) (outputs \( B' \)).
  - Input to the third NOT gate: \( C \) (outputs \( C' \)).
  - Input to the AND gate: \( A' \), \( B' \), and \( C' \) (outputs \( A'B'C' \)).

- **Term 3: \( A'C'B \)**

  Use two NOT gates and one AND gate:
  - Input to the first NOT gate: \( A \) (outputs \( A' \)).
  - Input to the second NOT gate: \( C \) (outputs \( C' \)).
  - Input to the AND gate: \( A' \), \( C' \), and \( B \) (outputs \( A'C'B \)).

#### **2. Combine the Terms:**

- Use one OR gate to combine the outputs of the three AND gates:
  - Inputs to the OR gate: outputs from AND gates of \( ABC \), \( A'B'C' \), and \( A'C'B \).

### **Logic Diagram**

Here’s a step-by-step description of the logic diagram:

1. **NOT Gates:**
   - Create \( A' \), \( B' \), and \( C' \) using three NOT gates.

2. **AND Gates:**
   - First AND gate for \( ABC \):
     - Input \( A \) and \( B \) into this gate. Output is \( AB \).
     - Input \( AB \) and \( C \) into another AND gate to get \( ABC \).
   - Second AND gate for \( A'B'C' \):
     - Input \( A' \), \( B' \), and \( C' \) into this gate. Output is \( A'B'C' \).
   - Third AND gate for \( A'C'B \):
     - Input \( A' \), \( C' \), and \( B \) into this gate. Output is \( A'C'B \).

3. **OR Gate:**
   - Input the outputs from the three AND gates into the OR gate. The output will be \( Y \), which is the result of \( ABC + A'B'C' + A'C'B \).

### **Summary**

You will need:

- **3 NOT gates** (to create \( A' \), \( B' \), and \( C' \))
- **3 AND gates** (one each for \( ABC \), \( A'B'C' \), and \( A'C'B \))
- **1 OR gate** (to combine the outputs of the AND gates)

The logic diagram for this Boolean equation will show how the inputs are processed through these gates to produce the output \( Y \).