XOR Properties

Key properties: $1.$ $a \oplus a = 0$ (self-inverse)

$2.$ $a \oplus 0 = a$ (identity)

$3.$ Commutative: $a \oplus b = b \oplus a$

$4.$ Associative: $(a \oplus b) \oplus c = a \oplus (b \oplus c)$

Application: Finding the unique element.

If all elements appear twice except one, XOR them all. Pairs cancel out, leaving the unique element.

result = 0
for num in nums:
    result = result ^ num
return result

Why it works: $a \oplus a = 0$, so pairs vanish. Only the unique element remains.

Time: $O(n)$.

Space: $O(1)$ (excluding input array).