The and operation x & y produces a number that has one bits in positions where both x and y have one bits. For example, 22 & 26 = 18, because

Using the and operation, we can check if a number x is even because x & 1 =0 if x is even, and x & 1 = 1 if x is odd. More generally, x is divisible by 2k exactly when x & (2k ‒ 1) = 0.

The or operation x | y produces a number that has one bits in positions where at least one of x and y have one bits. For example, 22 | 26 = 30, because

The xor operation x ^ y produces a number that has one bits in positions where exactly one of x and y have one bits. For example, 22 ^ 26 = 12, because

The not operation ~x produces a number where all the bits of x have been inverted. The formula ~x = ‒x ‒ 1 holds, for example, ~29 = ‒30.

The result of the not operation at the bit level depends on the length of the bit representation, because the operation inverts all bits. For example, if the numbers are 32-bit int numbers, the result is as follows:

The left bit shift x << k appends k zero bits to the number, and the right bit shift x >> k removes the k last bits from the number. For example, 14 << 2 = 56, because 14 and 56 correspond to 1110 and 111000. Similarly, 49 >> 3 = 6, because 49 and 6 correspond to 110001 and 110.

Note that x << k corresponds to multiplying x by 2k, and x >> k corresponds to dividing x by 2k rounded down to an integer.