Bitwise Operations Reference — Truth Tables, Bit Hacks & Shift Guide

Truth Tables — Core Operations

AND (&)

A	B	A&B
1	1	1
1	0	0
0	1	0
0	0	0

1 only if both are 1
Use: bit masking / flag check

OR (|)

A	B	A\|B
1	1	1
1	0	1
0	1	1
0	0	0

1 if either is 1
Use: set bits / combine flags

XOR (^)

A	B	A^B
1	1	0
1	0	1
0	1	1
0	0	0

1 only if bits differ
Use: toggle / swap / parity

NOT (~)

A	~A
1	0
0	1

Flips every bit
Use: bitwise complement

NAND

A	B	~(A&B)
1	1	0
1	0	1
0	1	1
0	0	1

Universal gate in hardware
Use: CPU logic circuits

NOR

A	B	~(\|)
1	1	0
1	0	0
0	1	0
0	0	1

Also a universal gate
Use: CPU logic circuits

XNOR

A	B	~(^)
1	1	1
1	0	0
0	1	0
0	0	1

1 only if bits are equal
Use: equality check

Bit Shift — Visual Guide (8-bit example: 0b00101101 = 45)

Original

00101101

= 45

45 << 1

01011010

= 90 (×2)

45 << 2

10110100

= 180 (×4)

45 >> 1

00010110

= 22 (÷2, trunc)

45 >> 2

00001011

= 11 (÷4, trunc)

Arithmetic right shift (>>) preserves sign bit (MSB). Logical right shift (>>>) always fills with 0.
Left shift (<<) always fills with 0 regardless of signedness.

Operator Precedence (C/Java/JS — highest to lowest)

Precedence	Operator	Description	Associativity
1	~	Bitwise NOT	Right-to-left
2	<< >> >>>	Bit shifts	Left-to-right
3	&	Bitwise AND	Left-to-right
4	^	Bitwise XOR	Left-to-right
5	\|	Bitwise OR	Left-to-right

Common pitfall: x & 1 == 0 parses as x & (1 == 0), not (x & 1) == 0. Always parenthesize.

20 Essential Bit Manipulation Tricks

Set nth bitx | (1 << n)Set bit n to 1

Clear nth bitx & ~(1 << n)Set bit n to 0

Toggle nth bitx ^ (1 << n)Flip bit n

Check nth bit(x >> n) & 1Returns 0 or 1

Is power of 2?x && !(x & (x-1))x>0 and only 1 bit set

Multiply by 2ⁿx << nFast x·2ⁿ

Divide by 2ⁿx >> nFast floor(x/2ⁿ)

Modulo 2ⁿx & ((1<<n) - 1)x % 2ⁿ (non-negative)

Is even?(x & 1) == 0LSB is 0

Is odd?x & 1LSB is 1

Swap without tempa^=b; b^=a; a^=bXOR swap (integers only)

Clear lowest set bitx & (x - 1)Brian Kernighan

Isolate lowest set bitx & -xTwo's complement trick

Count set bits__builtin_popcount(x)POPCNT instruction

Set all bits to 1~0All-ones mask

Extract lowest bytex & 0xFFLow 8 bits

Merge bits from a,b(a & mask) | (b & ~mask)mask=1→from a

Sign extend (byte→int)(x ^ 0x80) - 0x808 to 32-bit sign ext

Next power of 21 << (32 - __builtin_clz(x-1))Round up to next 2ⁿ

Upper/lowercase flipc ^ 32'A'↔'a' due to ASCII offset

Signed Integer Ranges

Type	Bits	Min	Max	Unsigned Max
int8_t	8	-128	127	255 (uint8_t)
int16_t	16	-32,768	32,767	65,535 (uint16_t)
int32_t	32	-2,147,483,648	2,147,483,647	4,294,967,295
int64_t	64	−9.22×10¹⁸	9.22×10¹⁸	1.84×10¹⁹

Bitwise Operations: Complete Reference