XOR and XNOR: difference, equality and parity

Difference and equality

Two-input XOR is 1 when its inputs differ; XNOR is 1 when they agree:

OR and XOR differ at 11. In these symbols, the extra input-side curve marks XOR and an output circle makes it XNOR.

More inputs mean odd parity

For three inputs, $a\oplus b\oplus c$ is 1 when an odd number of inputs are 1. It is not an “exactly one” test: 111 also gives 1. The full-adder sum is this parity^{iparity Whether the number of one bits is odd or even. Parity detects an odd number of bit flips but can miss an even number.} function, while its carry^{icarry The value carried from one addition column to the next. The final carry and signed overflow are different conditions. Learn more} is a majority function.

At 111, sum=1 and cout=1 represent the integer sum $1+2\times1=3$.

What parity can detect

For data 1011, the even-parity bit $p=d_3\oplus d_2\oplus d_1\oplus d_0$ is 1. XOR of the data and parity bit is therefore zero. Any odd number of flips changes that check to one. Even numbers of flips can escape detection, and one parity bit cannot locate the error.

Vector equality needs an OR of bit mismatches, followed by inversion^{iinversion The NOT operation exchanges 0 and 1. An overline applies NOT to the entire expression covered by the line. Learn more}:

XOR-reducing the mismatches instead would allow two differences to cancel.