Overview
The XOR (Exclusive OR) gate outputs HIGH only when its inputs are different from each other. When both inputs are the same — both 0 or both 1 — the output is LOW. This "exclusive" behaviour distinguishes XOR from the ordinary OR gate, which outputs 1 when both inputs are 1. XOR is the "either, but not both" gate.
The boolean expression for XOR is Y = A ⊕ B, where the ⊕ symbol denotes exclusive OR. This can also be written in standard boolean form as Y = A'B + AB' — the output is 1 when A is 0 and B is 1, or when A is 1 and B is 0. XOR is critically important in digital arithmetic because it computes the sum bit of two binary digits without carrying.
XOR gates appear in adders, parity generators, error detection/correction circuits, hash functions, and cryptographic operations. The 74HC86 is a classic quad 2-input XOR gate IC. XOR is also the basis of many secret-sharing and one-time-pad encryption schemes because XORing a message with a random key produces perfectly random ciphertext.
Truth Table
| A | B | Output (Y) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
How It Works
A 2-input XOR gate requires more transistors than AND or OR gates — typically 8 transistors in CMOS. It is constructed from a combination of NAND and inverter stages, or from a transmission gate topology.
The key insight is that XOR can be decomposed: Y = A ⊕ B = A'B + AB'. This means: use two AND gates (one with A inverted, one with B inverted), and then OR the results. This 4-gate implementation (two ANDs + two NOTs + one OR) is common in educational contexts, though real ICs use optimised transistor-level designs.
XOR has the remarkable property of self-inverse: A ⊕ B ⊕ B = A. XORing any value with itself gives 0; XORing with 0 preserves the value; XORing with 1 inverts the value. This makes XOR perfect for parity and cryptographic operations.
Real-World Applications
The sum output of a half adder is computed using XOR: Sum = A ⊕ B. A full adder extends this with carry-in.
XORing all bits of a word generates a parity bit. Re-XORing at the receiver detects single-bit errors.
CRC (Cyclic Redundancy Check) and Hamming codes rely on XOR operations to detect and correct transmission errors.
One-time pad encryption XORs plaintext with a key. XOR ciphers are reversible: the same operation decrypts the message.
In software, XOR with a bitmask toggles specific bits without affecting others — used in toggle switches and bitfield manipulation.
Try It in the Interactive Simulator
Build XOR Gate circuits in real time — drag gates, connect wires, toggle inputs, and see outputs update instantly.
Frequently Asked Questions
- What is the difference between XOR and OR?
OR outputs 1 when any input is 1, including when both inputs are 1. XOR outputs 1 only when inputs are different — when both are 1, XOR outputs 0.
- What is the boolean expression for XOR?
Y = A ⊕ B, which expands to Y = A'B + AB'. The output is 1 when exactly one input is 1.
- Why is XOR used in binary adders?
The sum of two single bits (without carry) is exactly the XOR: 0+0=0, 0+1=1, 1+0=1, 1+1=0 (sum bit only, carry is separate). XOR computes the least significant bit of addition.
- What is the IC number for XOR gate?
The 74HC86 is a quad 2-input XOR gate in CMOS. The 74LS86 is its TTL equivalent.
- What is the relationship between XOR and XNOR?
XNOR is the complement of XOR. XNOR outputs 1 when inputs are the same; XOR outputs 1 when inputs are different. Y_XNOR = (A ⊕ B)'.