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

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

### Step 2: Create the Truth Table
We can create a truth table for the XOR gate with inputs A and B:

| A | B | Q (Output) |
|---|---|------------|
| 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 Q is 1 when:
- A is 0 and B is 1 (A=0, B=1)
- A is 1 and B is 0 (A=1, B=0)

### Step 4: Write the Boolean Expression
To express the output Q in terms of A and B, we can write:
- For the case when A=0 and B=1: This can be represented as \( A'B \) (A NOT AND B).
- For the case when A=1 and B=0: This can be represented as \( AB' \) (A AND B NOT).

Combining these two cases, we get the Boolean expression for the XOR gate:
\[ Q = A'B + AB' \]

### Step 5: Conclusion
Thus, the Boolean expression for the XOR gate is:
\[ Q = A'B + AB' \]