8. Derived Logic Gates: XOR and XNOR
XOR and XNOR gates are essential for arithmetic circuits (like adders) and data transmission applications.
1. The XOR Gate (Exclusive OR)
- Function: The output is 1 if the inputs are different.
- Symbol: OR symbol with an extra curve on the input side.
- Equation: $Y = A \oplus B = \overline{A}B + A\overline{B}$
| A | B | Y = A $\oplus$ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Note: XOR is also known as an inequality detector.
2. The XNOR Gate (Exclusive NOR)
- Function: The output is 1 if the inputs are the same (inverse of XOR).
- Symbol: XOR symbol followed by a bubble.
- Equation: $Y = \overline{A \oplus B} = A B + \overline{A}\overline{B}$
| A | B | Y = A $\odot$ B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Note: XNOR is also known as an equality detector.
Practical Use
XOR gates are used extensively in binary addition (specifically calculating the sum bit in a half adder), parity checking, and data encryption.