The XOR (exclusive OR) bitwise operator, denoted by the caret symbol (^), is a binary operator that performs a logical exclusive OR operation on each pair of corresponding bits of two operands. The result of the XOR operation is 1 if the corresponding bits of the operands are different, and 0 if they are the same.
How XOR Works
Basic Operation:
The XOR operator compares each bit of the first operand to the corresponding bit of the second operand.
If the bits are different, the result bit is set to 1.
If the bits are the same, the result bit is set to 0.
The truth table for XOR is as follows:
x
y
x ^ y
0
0
0
0
1
1
1
0
1
1
1
0
Example:
Consider the numbers 5 (binary 0101) and 3 (binary 0011).
Performing XOR on these numbers:
0101 (5)
^ 0011 (3)
------
0110 (6)
The result is 6 (binary 0110) because the bits differ in the second and third positions.
Properties of XOR:
Identity: $$ x \oplus 0 = x $$
Self-inverse: $$ x \oplus x = 0 $$
Commutative: $$ x \oplus y = y \oplus x $$
Associative: $$ (x \oplus y) \oplus z = x \oplus (y \oplus z) $$
Applications:
Swapping Values: XOR can be used to swap two variables without using a temporary variable.
function swapWithXOR(a, b) {
a = a ^ b;
b = a ^ b;
a = a ^ b;
console.log("a: " + a + ", b: " + b);
}
swapWithXOR(4, 1); // a: 1, b: 4
Finding Odd Occurrences: XOR is useful in problems where you need to find the element that appears an odd number of times in an array.
Cryptography: XOR is used in simple encryption algorithms like the XOR cipher.
Special Cases
XOR with Negative Numbers:
When dealing with negative numbers, the XOR operation considers the two's complement representation of the numbers.
