To draw the logic network for \( A + B + C \) using only NAND gates, we need to understand how to convert the OR operation into a network of NAND gates. The logic for \( A + B + C \) can be built step-by-step.

### Steps to Convert OR to NAND Logic:

1. **Convert the OR to AND using De Morgan's theorem:**
   \[
   A + B + C = \neg(\neg(A + B + C))
   \]

2. **Apply De Morgan's theorem to break down the OR operation:**
   \[
   \neg(A + B + C) = \neg(A) \cdot \neg(B) \cdot \neg(C)
   \]

3. **Use NAND gates to perform the NOT and AND operations:**

### Implementing the NAND logic:

1. **NOT using NAND:**
   - \(\neg A\) can be implemented as \(A \text{ NAND } A\).
   - \(\neg B\) can be implemented as \(B \text{ NAND } B\).
   - \(\neg C\) can be implemented as \(C \text{ NAND } C\).

2. **AND using NAND:**
   - \(A \text{ AND } B = \neg(\neg A \text{ OR } \neg B) = (A \text{ NAND } A) \text{ NAND } (B \text{ NAND } B)\).
   - Similarly, for three inputs, we can extend it.

3. **Combining the NOT and AND operations to get the final OR:**

Let's draw this step-by-step in NAND logic:

### Step-by-Step Diagram:

1. **Generate NOT gates using NAND gates:**
   - \( \neg A = A \text{ NAND } A \)
   - \( \neg B = B \text{ NAND } B \)
   - \( \neg C = C \text{ NAND } C \)

2. **Create AND gates using NAND gates:**
   - \( \text{AND1} = (\neg A) \text{ NAND } (\neg B) \)
   - \( \text{AND2} = \text{AND1} \text{ NAND } (\neg C) \)

3. **Combine the AND results to produce OR using NAND gates:**
   - \( \neg(\neg(A) \cdot \neg(B) \cdot \neg(C)) = \text{AND2} \text{ NAND } \text{AND2} \)

### Final NAND Logic Network:

```plaintext
   A ------|       |       |--------|       |
          NAND     |       |       NAND      |
         /    \    |      NAND     /    \    |
        A     A    |     /    \    \     \   |
                   |    /      \    \     \  |
                   |   /        \    \     \ |
                   |  |          |    \     ||
                   |  |          |     \    ||
    B ------|       |          AND1   \AND2 ||
          NAND     |         /    \    \    ||
         /    \    |        \    /      \   ||
        B     B    |         \  /        \  ||
                   |          NAND        NAND
                   |          /  \       /    \
                   |         /    \     /      \
    C ------|      |        /      \   /        \
          NAND    |       AND2     AND2
         /    \   |      /    \   /    \
        C     C   |     /      \ /      \
                  |    NAND    NAND
                  |   /    \  /    \
                  |  /      \/      \
                  | OR using NAND    \
                  |     (final)       \
```

### Explanation:
- Each input \( A, B, \) and \( C \) is first negated using NAND gates.
- The intermediate NAND gates are then used to implement the AND operation as described.
- Finally, a NAND gate is used to invert the result of the AND operation, effectively implementing the OR operation.

Would you like more details or have any questions?

Here are 8 related questions to consider:

1. How can De Morgan's theorem be applied to other logic operations?
2. What are the benefits of using NAND gates exclusively in digital circuits?
3. How can this NAND logic network be simplified further, if possible?
4. How does the NAND gate universal property work?
5. What are some practical applications of using NAND-only circuits?
6. How can you implement a full adder using only NAND gates?
7. How do you convert other basic logic gates (AND, OR, NOT) using only NAND gates?
8. How do logic gates contribute to the functioning of microprocessors?

**Tip:** Always verify your logic circuit step-by-step using truth tables to ensure its accuracy.

Learn how to draw the logic network for A + B + C using only NAND gates and applying De Morgan's theorem. This detailed guide covers the conversion of OR operations to NAND gates, suitable for undergraduate students and digital logic enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation