Binary Exclusive OR Operation
Exclusive OR linking the bits of two integers
Bitwise XOR Calculator
Bitwise XOR Operation
This function performs a binary exclusive OR operation on the bits of two integers. The result bit is 1 when the input bits are different, and 0 when they are the same.
XOR Operation Visualization
Example: F0 XOR 42 = B2
XOR Truth Table
XOR Properties
- Result is 1 when bits are different
- Result is 0 when bits are the same
- Commutative: A ⊕ B = B ⊕ A
- Self-inverse: A ⊕ A = 0
Mathematical Foundations of Bitwise XOR Operation
The bitwise XOR operation is applied bitwise to the corresponding positions:
Mathematical Definition
For each bit position i, exclusive OR is applied
Bitwise Application
Simultaneous XOR operation on all corresponding bit pairs
XOR Operation Formulas and Examples
General XOR Formula
Bitwise exclusive OR operation of two binary numbers A and B
Step-by-Step Example: F0 XOR 42
1st Number A: F0₁₆ = 240₁₀ = 11110000₂
2nd Number B: 42₁₆ = 66₁₀ = 01000010₂
Position: 76543210 (from right)
XOR Operation:
Position 7: 1 ⊕ 0 = 1
Position 6: 1 ⊕ 1 = 0
Position 5: 1 ⊕ 0 = 1
Position 4: 1 ⊕ 0 = 1
Position 3: 0 ⊕ 0 = 0
Position 2: 0 ⊕ 0 = 0
Position 1: 0 ⊕ 1 = 1
Position 0: 0 ⊕ 0 = 0
Result: 10110010₂ = 178₁₀ = B2₁₆
More XOR Examples
A: 11111111₂ (255)
B: 00000000₂ (0)
Result: 11111111₂ (255)
A: 10101010₂ (170)
B: 01010101₂ (85)
Result: 11111111₂ (255)
Mathematical Properties of XOR Operation
\[A \oplus B = B \oplus A\]
Order of operands is arbitrary
\[(A \oplus B) \oplus C = A \oplus (B \oplus C)\]
Parentheses are arbitrary
\[A \oplus 0 = A\]
XOR with 0 changes nothing
\[A \oplus A = 0\]
XOR with itself = 0
XOR Reference
Standard Example
XOR Truth Table
Bitwise Operators
^: bitwise XOR
|: bitwise OR
&: bitwise AND
~: bitwise NOT
<<,>>: bit shift
Common Applications
Toggle bits: x ^ mask
Encryption: data ^ key
Checksum: byte1 ^ byte2
Swap values: a^=b; b^=a; a^=b;
Bitwise XOR Operation - Detailed Description
XOR Operation Fundamentals
The bitwise exclusive OR (XOR) is applied to the bit sequences of two integers. The bits at the same position are linked with a logical XOR operation. The result bit is 1 if the two bits are different, and 0 if they are the same.
• 0 ⊕ 0 = 0 (both bits are 0)
• 0 ⊕ 1 = 1 (bits are different)
• 1 ⊕ 0 = 1 (bits are different)
• 1 ⊕ 1 = 0 (both bits are 1)
Processing Logic
When bits are different, the result bit is set to 1. When bits are the same, the result is 0. XOR is particularly useful for toggling bits, encryption, and detecting differences between data.
Processing Steps
1. Convert both numbers to binary representation
2. Process bit by bit from right to left
3. Apply XOR operation to corresponding bit pairs
4. Convert result to desired format
Practical Applications
Bitwise XOR operations are essential in cryptography, data integrity checking, and algorithmic programming. They provide unique properties that make them invaluable for many computational tasks.
• Encryption and decryption
• Error detection and correction
• Bit toggling and masking
• Hash functions and checksums
Mathematical Properties
The XOR operation has unique mathematical properties including being its own inverse, making it particularly useful for reversible operations like encryption where A ⊕ B ⊕ B = A.
Important Properties
- Commutative: A ⊕ B = B ⊕ A
- Associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)
- Identity element: A ⊕ 0 = A
- Self-inverse: A ⊕ A = 0
Practical XOR Operation Examples
Bit Toggling
Scenario: Toggle specific bits
Data: 11001100₂
Mask: 00110011₂
Result: 11111111₂ (toggled)
Simple Encryption
Scenario: XOR cipher
Message: 01001000₂ (H)
Key: 10101010₂
Encrypted: 11100010₂
Checksum
Scenario: Data integrity
Byte 1: 11110000₂
Byte 2: 10101010₂
Checksum: 01011010₂
Programming Tips
- Toggle bits: value ^= mask
- Swap variables: a^=b; b^=a; a^=b;
- Simple cipher: encrypted = data ^ key
- Find unique: result = a ^ b ^ c
- Parity check: odd = (data ^ (data >> 4)) & 1
- Performance: XOR is very fast
|
|