What does the XOR operator do

Question

What mathematical operation does XOR perform

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is the Python bitwise XOR operator  It is how you spell XOR in python    gt  gt  gt  0   0 0  gt  gt  gt  0   1 1  gt  gt  gt  1   0 1  gt  gt  gt  1   1 0   XOR stands for exclusive OR  It is used in cryptography because it let s you  flip  the bits using a mask in a reversable operation    gt  gt  gt  10   5 15  gt  gt  gt  15   5 10   where 5 is the mask   input XOR mask  XOR mask gives you the input again

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XOR is a binary operation  it stands for  exclusive or   that is to say the resulting bit evaluates to one if only exactly one of the bits is set   This is its function table   a   b   a   b -- --- ------ 0   0   0 0   1   1 1   0   1 1   1   0   This operation is performed between every two corresponding bits of a number   Example  7   10 In binary  0111   1010    0111   1010          1101   13   Properties  The operation is commutative  associative and self-inverse   It is also the same as addition modulo 2

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A little more information on XOR operation    XOR a number with itself odd number of times the result is number itself  XOR a number even number of times with itself  the result is 0  Also XOR with 0 is always the number itself

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One thing that other answers don t mention here is XOR with negative numbers -   a     b    a   b ---- ----- ------  0     0     0  0     1     1  1     0     1  1     1     0   While you could easily understand how XOR will work using the above functional table  it doesn t tell how it will work on negative numbers     How XOR works with Negative Numbers    Since this question is also tagged as python  I will be answering it with that in mind  The XOR       is an logical operator that will return 1 when the bits are different and 0 elsewhere   A negative number is stored in binary as two s complement  In 2 s complement  The leftmost bit position is reserved for the sign of the value  positive or negative  and doesn t contribute towards the value of number      In  Python  negative numbers are written with a leading one instead of a leading   zero  So if you are using only 8 bits for your two s-complement   numbers  then you treat patterns from 00000000 to 01111111 as the   whole numbers from 0 to 127  and reserve 1xxxxxxx for writing negative   numbers    With that in mind  lets understand how XOR works on negative number with an example  Lets consider the expression -   -5   -3        The binary representation of -5 can be considered as 1000   101 and  Binary representation of -3 can be considered as 1000   011     Here      denotes all 0s  the number of which depends on bits used for representation  32-bit  64-bit  etc   The 1 at the MSB   Most Significant Bit   denotes that the number represented by the binary representation is negative  The XOR operation will be done on all bits as usual   XOR Operation          -5      10000101                                                     -3      10000011                                                    Result    00000110     6                                                 -5   -3   6   Since  the MSB becomes 0 after the XOR operation  so the resultant number we get is a positive number  Similarly  for all negative numbers  we consider their representation in binary format using 2 s complement  one of most commonly used  and do simple XOR on their binary representation    The MSB bit of result will denote the sign and the rest of the bits will denote the value  of the final result   The following table could be useful in determining the sign of result     a       b     a   b ------ ------- ------                               -       -   -               -   -       -           The basic rules of XOR remains same for negative XOR operations as well  but how the operation really works in negative numbers could be useful for someone someday

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The     XOR operator generates 1 when it is applied on two different bits  0 and 1   It generates 0 when it is applied on two same bits  0 and 0 or 1 and 1

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Another application for XOR is in circuits  It is used to sum bits    When you look at a truth table    x   y   x y --- --- -----  0   0    0        0 plus 0   0   0   1    1        0 plus 1   1  1   0    1        1 plus 0   1  1   1    0        1 plus 1   0   binary math with 1 bit   You can notice that the result of XOR is x added with y  without keeping track of the carry bit  the carry bit is obtained from the AND between x and y   x y    is actually  xy    yx        Which is the  negated x ANDed with y  OR   negated y ANDed with x

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