The Boolean expression for Exculsive OR gate ( X-OR gate) is

Options

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

Answer

To derive the Boolean expression for an Exclusive OR (XOR) gate, we can follow these steps:

### Step 1: Understand the definition of XOR
The Exclusive OR gate outputs true (1) only when the inputs are different. If both inputs are the same, the output is false (0).

### Step 2: Write the truth table for XOR
We can create a truth table to visualize the behavior of the XOR gate:

| A | B | A XOR B |
|---|---|---------|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |

### Step 3: Analyze the truth table
From the truth table, we can see that the output is 1 when:
- A is 0 and B is 1 (first row)
- A is 1 and B is 0 (third row)

### Step 4: Write the Boolean expression
Based on the analysis, we can express the output of the XOR gate using the following Boolean expression:
\[ A \oplus B = (A \cdot \overline{B}) + (\overline{A} \cdot B) \]

Where:
- \( A \cdot \overline{B} \) represents the case when A is true and B is false.
- \( \overline{A} \cdot B \) represents the case when A is false and B is true.
- The plus sign (+) denotes the OR operation.

### Step 5: Final expression
Thus, the final Boolean expression for the Exclusive OR gate (XOR) is:
\[ A \oplus B = (A \cdot \overline{B}) + (\overline{A} \cdot B) \]