Binary AND Operation
AND linking the bits of two integers
Bitwise AND Calculator
Bitwise AND Operation
This function performs a binary AND operation on the bits of two integers. The result bit is 1 only when both input bits are 1.
AND Operation Visualization
Example: CC AND 82 = 80
AND Truth Table
AND Properties
- Result is 1 only when both bits are 1
- Result is 0 when at least one bit is 0
- Commutative: A & B = B & A
- Perfect for bit masking and filtering
Mathematical Foundations of Bitwise AND Operation
The bitwise AND operation is applied bitwise to the corresponding positions:
Mathematical Definition
For each bit position i, logical conjunction is applied
Bitwise Application
Simultaneous AND operation on all corresponding bit pairs
AND Operation Formulas and Examples
General AND Formula
Bitwise AND operation of two binary numbers A and B
Step-by-Step Example: CC AND 82
1st Number A: CC₁₆ = 204₁₀ = 11001100₂
2nd Number B: 82₁₆ = 130₁₀ = 10000010₂
Position: 76543210 (from right)
AND Operation:
Position 7: 1 & 1 = 1
Position 6: 1 & 0 = 0
Position 5: 0 & 0 = 0
Position 4: 0 & 0 = 0
Position 3: 1 & 0 = 0
Position 2: 1 & 0 = 0
Position 1: 0 & 1 = 0
Position 0: 0 & 0 = 0
Result: 10000000₂ = 128₁₀ = 80₁₆
More AND Examples
A: 11111111₂ (255)
B: 11110000₂ (240)
Result: 11110000₂ (240)
A: 00111111₂ (63)
B: 00001111₂ (15)
Result: 00001111₂ (15)
Mathematical Properties of AND Operation
\[A \land B = B \land A\]
Order of operands is arbitrary
\[(A \land B) \land C = A \land (B \land C)\]
Parentheses are arbitrary
\[A \land 1 = A\]
AND with all 1s = original
\[A \land 0 = 0\]
AND with 0 always results in 0
AND Reference
Standard Example
AND Truth Table
Bitwise Operators
&: bitwise AND
|: bitwise OR
^: bitwise XOR
~: bitwise NOT
<<,>>: bit shift
Common Applications
Mask bits: x & mask
Check flags: status & flag
Extract bits: value & pattern
Filter register: reg & filter
Bitwise AND Operation - Detailed Description
AND Operation Fundamentals
The bitwise AND is applied to the bit sequences of two integers. The bits at the same position are linked with a logical AND (logical conjunction). For each pair, the result bit is 1 if both bits are 1.
• 0 & 0 = 0 (both bits are 0)
• 0 & 1 = 0 (at least one bit is 0)
• 1 & 0 = 0 (at least one bit is 0)
• 1 & 1 = 1 (both bits are 1)
Processing Logic
When one or both bits are 0, the result bit is set to 0. The AND operation is particularly useful for bit masking and filtering specific bits.
Processing Steps
1. Convert both numbers to binary representation
2. Process bit by bit from right to left
3. Apply AND operation to corresponding bit pairs
4. Convert result to desired format
Practical Applications
Bitwise AND operations are essential for digital logic and computer programming. They enable targeted bit extraction, data masking, and implementation of filters and security mechanisms.
• Bit masks for data extraction
• Hardware register filtering
• Permission checks
• Data validation and cleaning
Mathematical Properties
The AND operation follows important mathematical laws of Boolean algebra and has a zero element (0) that makes any AND operation result in 0.
Important Properties
- Commutative: A & B = B & A
- Associative: (A & B) & C = A & (B & C)
- Identity element: A & 1...1 = A
- Zero element: A & 0 = 0
Practical AND Operation Examples
Bit Masking
Scenario: Extract lower 4 bits
Data: 11010110₂ (214)
Mask: 00001111₂ (15)
Result: 00000110₂ (6)
Flag Checking
Scenario: Check read flag
Status: 11100110₂
Read Flag: 00000100₂
Result: 00000100₂ (flag set)
Hardware Filter
Scenario: Filter register
Register: 11011010₂
Filter: 11110000₂
Result: 11010000₂
Programming Tips
- Define masks: #define MASK_LOW4 0x0F
- Check flags: if (status & FLAG_READ)
- Extract bits: value = (data & mask)
- Register access: filtered = reg & access_mask
- Range checking: valid = (input & valid_mask)
- Performance: AND is very efficient
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