XOR Gate

The XOR gate outputs 1 when its inputs differ, implementing exclusive-OR and modulo-2 addition of bits.

What it does

XOR, exclusive-OR, outputs 1 when its inputs are different and 0 when they are the same. For two inputs it answers the question, does exactly one input equal 1.

Truth table

Kronos motion — when

A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

Algebraic form

XOR is written with a plus in a circle. It equals A AND NOT B, OR NOT A AND B. It is commutative and associative, and it satisfies A XOR 0 = A and A XOR A = 0. XOR with all-ones inverts a value, so XOR is a controllable inverter.

Modulo-2 addition

XOR is addition without carry in base two. This makes it the sum bit of a half adder and the workhorse of parity, checksums, and linear-feedback shift registers.

In code

python

out = a ^ b        # bitwise XOR
parity = 0
for bit in bits:  # parity is XOR of all bits
    parity ^= bit

Where it appears

Parity generation and checking for error detection.
Comparing two values: any difference makes the XOR output 1.
Adders, where XOR forms the sum and AND forms the carry.
Cryptographic mixing and toggle logic.

XOR needs several simpler gates to build, but its distinctive difference behavior earns it a dedicated symbol.