Python also provides two set types, set and frozenset. The set type is mutable, while frozenset is immutable. They are unordered collections of immutable objects. Hashability is a characteristic that allows an object to be used as a set member as well as a key for a dictionary, as we'll see very soon.

Objects that compare equally must have the same hash value. Sets are very commonly used to test for membership, so let's introduce the in operator in the following example:

>>> small_primes = set()  # empty set
>>> small_primes.add(2)  # adding one element at a time
>>> small_primes.add(3)
>>> small_primes.add(5)
>>> small_primes
{2, 3, 5}
>>> small_primes.add(1)  # Look what I've done, 1 is not a prime!
>>> small_primes
{1, 2, 3, 5}
>>> small_primes.remove(1)  # so let's remove it
>>> 3 in small_primes # membership test
True
>>> 4 in small_primes
False
>>> 4 not in small_primes  # negated membership test
True
>>> small_primes.add(3)  # trying to add 3 again
>>> small_primes
{2, 3, 5}  # no change, duplication is not allowed
>>> bigger_primes = set([5, 7, 11, 13])  # faster creation
>>> small_primes | bigger_primes # union operator `|`
{2, 3, 5, 7, 11, 13}
>>> small_primes & bigger_primes  # intersection operator `&`
{5}
>>> small_primes - bigger_primes  # difference operator `-`
{2, 3}

In the preceding code, you can see two different ways to create a set. One creates an empty set and then adds elements one at a time. The other creates the set using a list of numbers as an argument to the constructor, which does all the work for us. Of course, you can create a set from a list or tuple (or any iterable) and then you can add and remove members from the set as you please.

Another way of creating a set is by simply using the curly braces notation, like this:

>>> small_primes = {2, 3, 5, 5, 3}
>>> small_primes
{2, 3, 5}

Notice I added some duplication to emphasize that the resulting set won't have any. Let's see an example about the immutable counterpart of the set type, frozenset:

>>> small_primes = frozenset([2, 3, 5, 7])
>>> bigger_primes = frozenset([5, 7, 11])
>>> small_primes.add(11)  # we cannot add to a frozenset
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'frozenset' object has no attribute 'add'
>>> small_primes.remove(2)  # neither we can remove
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'frozenset' object has no attribute 'remove'
>>> small_primes & bigger_primes  # intersect, union, etc. allowed
frozenset({5, 7})

As you can see, frozenset objects are quite limited in respect of their mutable counterpart. They still prove very effective for membership test, union, intersection, and difference operations, and for performance reasons.