#####   ######  #####    ###    #   #  ###  #   #  ######
##  ##  ##      ##  ##  ## ##   #   #   #   #   #  ##
#####   ####    #####   #   #   #   #   #   #   #  ####
##  #   ##      ##      ## ##    # #    #    # #   ##
##   #  ######  ##       ###      #    ###    #    ######

$ curl repovive.com/roadmaps/fundamental-algorithms/bit-manipulation/xor-properties

░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░███████████████████████████████████████████████████████████████████████████████████

#####   ######  #####    ###    #   #  ###  #   #  ######
##  ##  ##      ##  ##  ## ##   #   #   #   #   #  ##
#####   ####    #####   #   #   #   #   #   #   #  ####
##  #   ##      ##      ## ##    # #    #    # #   ##
##   #  ######  ##       ###      #    ###    #    ######

$ curl repovive.com/roadmaps/fundamental-algorithms/bit-manipulation/xor-properties

Repovive

XOR Properties - Bit Manipulation | Fundamental Algorithms | Repovive

Repovive.

Fundamental Algorithms8 sections · 294 units2h total

Open in Course

XOR Properties

XOR is self-inverse and associative.

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).