XOR Properties

XOR has properties that make it incredibly useful:

$1.$ $x \wedge x = 0$ (a number XORed with itself is 0)

$2.$ $x \wedge 0 = x$ (XORing with 0 does nothing)

$3.$ XOR is commutative and associative: $a \wedge b \wedge c = c \wedge a \wedge b$

These properties let you find unique elements, detect duplicates, and even swap variables without a temporary. If you XOR all numbers in a list where every number appears twice except one, the pairs cancel out ($x \wedge x = 0$) and only the unique number remains.