Give Boolean expression and Truth table for NOR gate.

Options

A. `C = A + B`
B. `C = bar(A+B)`
C. `C = A.B`
D. `C = bar(A.B)`

Answer

To solve the question regarding the Boolean expression and truth table for a NOR gate, we can follow these steps:

### Step 1: Understand the NOR Gate
The NOR gate is a digital logic gate that outputs true or 1 only when both of its inputs are false or 0. It is essentially a combination of an OR gate followed by a NOT gate.

### Step 2: Write the Boolean Expression
For a NOR gate with two inputs A and B, the output C can be expressed in Boolean algebra. The OR operation is represented by the plus sign (+), and the NOT operation is represented by a complement ('). Therefore, the Boolean expression for the NOR gate is:
\[ C = (A + B)' \]
This means that the output C is the complement of the logical sum of A and B.

### Step 3: Identify the Correct Option
From the given options:
1. \( C = A + B \)
2. \( C = (A + B)' \) (This is the correct option)
3. \( C = A \cdot B \)
4. \( C = (A \cdot B)' \)

The correct Boolean expression for the NOR gate is option 2: \( C = (A + B)' \).

### Step 4: Create the Truth Table
Now, we will create the truth table for the NOR gate. The truth table lists all possible combinations of inputs A and B and the corresponding output C.

| A | B | A + B | C = (A + B)' |
|---|---|-------|--------------|
| 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 |

### Explanation of the Truth Table:
- When both A and B are 0, \( A + B = 0 \) and \( C = (0)' = 1 \).
- When A is 0 and B is 1, \( A + B = 1 \) and \( C = (1)' = 0 \).
- When A is 1 and B is 0, \( A + B = 1 \) and \( C = (1)' = 0 \).
- When both A and B are 1, \( A + B = 1 \) and \( C = (1)' = 0 \).

### Final Answer
- **Boolean Expression**: \( C = (A + B)' \)
- **Truth Table**:
| A | B | C |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |