Which of the following is the boolean expression for XOR gate?

Options

A. `y = A underlineB + underlineAB`
B. `y = underline(AB). underline(A B)`
C. `y = (A + underline B).(underline A + B)`
D. `y = (AB)(underline AB)`

Answer

To determine the Boolean expression for an XOR gate, we can follow these steps:

### Step 1: Understand the XOR Gate
An XOR (exclusive OR) gate outputs true (1) only when the inputs are different. In other words, it outputs true if one input is true and the other is false.

### Step 2: Create the Truth Table
The truth table for an XOR gate with two inputs A and B is as follows:

| A | B | Output (A XOR B) |
|---|---|------------------|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |

From the truth table, we can see that the output is 1 when A and B are different.

### Step 3: Derive the Boolean Expression
Based on the truth table, we can derive the Boolean expression for the XOR operation. The output is true (1) when:
- A is true (1) and B is false (0), which can be represented as A AND NOT B (A * B')
- A is false (0) and B is true (1), which can be represented as NOT A AND B (A' * B)

Thus, the Boolean expression for the XOR gate can be written as:
\[ Z = A \cdot \overline{B} + \overline{A} \cdot B \]
Where:
- \( \cdot \) represents the AND operation
- \( + \) represents the OR operation
- \( \overline{A} \) represents NOT A

### Step 4: Verify the Expression
To ensure the correctness of the expression, we can check it against the truth table:
- For (0, 0): \( 0 \cdot 1 + 1 \cdot 0 = 0 + 0 = 0 \)
- For (0, 1): \( 0 \cdot 0 + 1 \cdot 1 = 0 + 1 = 1 \)
- For (1, 0): \( 1 \cdot 1 + 0 \cdot 0 = 1 + 0 = 1 \)
- For (1, 1): \( 1 \cdot 0 + 0 \cdot 1 = 0 + 0 = 0 \)

The outputs match the truth table for an XOR gate.

### Conclusion
The Boolean expression for the XOR gate is:
\[ Z = A \cdot \overline{B} + \overline{A} \cdot B \]