# 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?

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
``````

```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 : 3 ^ 4
Out: 7

In : 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 : x = Foo(3)

In : x
Out: 3.0

In : x ^ 4
Out: 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)
``````
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