Explain how XOR (^) bit operator works

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

1. 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

2. Example:

   • Consider the numbers 5 (binary 0101) and 3 (binary 0011).

   • Performing XOR on these numbers:

     Copy

     0101 (5)
     ^ 0011 (3)
     ------
     0110 (6)

   • The result is 6 (binary 0110) because the bits differ in the second and third positions.

3. Properties of XOR:

   • Identity: $$ x \oplus 0 = x $$
   • Self-inverse: $$ x \oplus x = 0 $$
   • Commutative: $$ x \oplus y = y \oplus x $$
   • Associative: $$ (x \oplus y) \oplus z = x \oplus (y \oplus z) $$

4. Applications:

   • Swapping Values: XOR can be used to swap two variables without using a temporary variable.
     javascript Copy

     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

1. XOR with Negative Numbers:
   • When dealing with negative numbers, the XOR operation considers the two's complement representation of the numbers.
   • Example: $$ -4 \oplus -2 $$
     • Convert to binary and find the two's complement:
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       -4: 1000 0100 (binary), two's complement: 1111 1100
       -2: 1000 0010 (binary), two's co...