Let's talk about the bitwise operators in python,

<<, >>, &, |, ~, ^.

They usually operate on numbers but instead of treating them like a single value, they treat them as a string of bits written in 2's complement binary. 2's complement is calculated by inverting the bits and adding 1.

• For a list of python operators read this.

2's Complement Binary form for Positive Integers

$ (2)_{10} = 010 \\ (5)_{10} = 0101 \\ (9)_{10} = 01001 \\ (20)_{10} = 010100 $

2's Complement Binary form for Negative Integers

A negative integers -x is written in binary as x-1 and all the bits complemented. For example, -4 is complement(4-1) = complement(3) = ...1111111100.

The binary form of negative numbers in written with leading 1s instead on 0s. Python uses infinite number of bits. For example, -5 is treated by the bitwise operators in python as ...111111111011. -4 is treated by the bitwise operators in python as ...11111111100.

Bitwise Operators

• x << y

  Returns the bits of x shifted to the left by y places (the new bits on the left are 0). The same as x * (2**y).

  $(11)_{10}$ << $(3)_{10}$ = 1011 << $(3)_{10}$ = 1011000

x = 11
y = 3
a = x << y
a

88

x*(2**y)

88

bin(x)

'0b1011'

bin(a)

'0b1011000'

bin(x*(2**y))

'0b1011000'

• x >> y

  Returns the bits of x shifted to the right by y places. The same as x // (2**y).

  $(11)_{10}$ >> $(3)_{10}$ = 1011 >> $(3)_{10}$ = 1

a = x >> y
a

1

x // (2**y)

1

bin(a)

'0b1'

bin(x//(2**y))

'0b1'

• x & y

  Performs a bitwise AND operation. Each bit in the output is 1 if the corresponding bits in x and y are 1, otherwise it's 0.

  $(11)_{10}$ & $(7)_{10}$ = 1011 & 111 = 11 = $(3)_{10}$

x = 11
y = 7
a = x & y
a

3

bin(a)

'0b11'

• x | y

  Performs a bitwise OR operation. Each bit in the output is 1 if the corresponding bits in x or y are 1, otherwise it's 0.

  $(11)_{10}$ & $(7)_{10}$ = 1011 & 111 = 1111 = $(15)_{10}$

a = x | y
a

15

bin(a)

'0b1111'

• ~x

  Returns the complement of x.

  ~$(11)_{10}$ = ~1011 = 10100 = $-(12)_{10}$

a = ~x
a

-12

• x ^ y

  Performs a bitwise exclusive OR. Each bit in the output is the same as the corresponding bit in x if that bit in y is 0. The output bit is the complement of the corresponding bit in x if that bit in y is 1.

  $(11)_{10}$ ^ $(7)_{10}$ = 1011 & 111 = 1100 = $(12)_{10}$

a = x ^ y
a

12

Set Operations

Assume we have two sets A and B. We have the following set operations in python,

• A | B returns the union.

• A & B returns the intersection.

• A - B returns the difference.

• A ^ B returns the symmetric difference.