# What does the caret (^) operator do?

## Question:

I ran across the caret operator in python today and trying it out, I got the following output:

```
>>> 8^3
11
>>> 8^4
12
>>> 8^1
9
>>> 8^0
8
>>> 7^1
6
>>> 7^2
5
>>> 7^7
0
>>> 7^8
15
>>> 9^1
8
>>> 16^1
17
>>> 15^1
14
>>>
```

It seems to be based on 8, so I’m guessing some sort of byte operation? I can’t seem to find much about this searching sites other than it behaves oddly for floats, does anybody have a link to what this operator does or can you explain it here?

## Answers:

It’s a bit-by-bit exclusive-or. Binary bitwise operators are documented in chapter 5 of the Python Language Reference.

It’s a bitwise XOR (exclusive OR).

It evaluates to `True`

if and only if its arguments differ (one is `True`

, the other is `False`

).

To demonstrate:

```
>>> 0^0
0
>>> 1^1
0
>>> 1^0
1
>>> 0^1
1
```

To explain one of your own examples:

```
>>> 8^3
11
```

Think about it this way:

1000 # 8 (binary) 0011 # 3 (binary) ---- # APPLY XOR ('vertically') 1011 # result = 11 (binary)

It invokes the `__xor__()`

or `__rxor__()`

method of the object as needed, which for integer types does a bitwise exclusive-or.

Generally speaking, the symbol `^`

is an infix version of the `__xor__`

or `__rxor__`

methods. Whatever data types are placed to the right and left of the symbol must implement this function in a compatible way. For integers, it is the common `XOR`

operation, but for example there is not a built-in definition of the function for type `float`

with type `int`

:

```
In [12]: 3 ^ 4
Out[12]: 7
In [13]: 3.3 ^ 4
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-13-858cc886783d> in <module>()
----> 1 3.3 ^ 4
TypeError: unsupported operand type(s) for ^: 'float' and 'int'
```

One neat thing about Python is that you can override this behavior in a class of your own. For example, in some languages the `^`

symbol means exponentiation. You could do that this way, just as one example:

```
class Foo(float):
def __xor__(self, other):
return self ** other
```

Then something like this will work, and now, *for instances of Foo only*, the

`^`

symbol will mean exponentiation.```
In [16]: x = Foo(3)
In [17]: x
Out[17]: 3.0
In [18]: x ^ 4
Out[18]: 81.0
```

When you use the `^`

operator, behind the curtains the method ** __xor__** is called.

`a^b`

is equivalent to `a.__xor__(b)`

.

Also, `a ^= b`

is equivalent to `a = a.__ixor__(b)`

(where `__xor__`

is used as a fallback when `__ixor__`

is implicitly called via using `^=`

but does not exist).

In principle, what `__xor__`

does is completely up to its implementation. Common use cases in Python are:

**Symmetric Difference**of sets (all elements present in exactly one of two sets)

Demo:

```
>>> a = {1, 2, 3}
>>> b = {1, 4, 5}
>>> a^b
{2, 3, 4, 5}
>>> a.symmetric_difference(b)
{2, 3, 4, 5}
```

**Bitwise Non-Equal**for the bits of two integers

Demo:

```
>>> a = 5
>>> b = 6
>>> a^b
3
```

Explanation:

```
101 (5 decimal)
XOR 110 (6 decimal)
-------------------
011 (3 decimal)
```