Sets in Python – Real Python

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Perhaps you recall learning about sets and set theory at some point in your mathematical education. Maybe you even remember Venn diagrams:

If this doesn’t ring a bell, don’t worry! This tutorial should still be easily accessible for you.

In mathematics, a rigorous definition of a set can be abstract and difficult to grasp. Practically though, a set can be thought of simply as a well-defined collection of distinct objects, typically called elements or members.

Grouping objects into a set can be useful in programming as well, and Python provides a built-in set type to do so. Sets are distinguished from other object types by the unique operations that can be performed on them.

Here’s what you’ll learn in this tutorial: You’ll see how to define set objects in Python and discover the operations that they support. As with the earlier tutorials on lists and dictionaries, when you are finished with this tutorial, you should have a good feel for when a set is an appropriate choice. You will also learn about frozen sets, which are similar to sets except for one important detail.

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Defining a Set

Python’s built-in set type has the following characteristics:

Sets are unordered.
Set elements are unique. Duplicate elements are not allowed.
A set itself may be modified, but the elements contained in the set must be of an immutable type.

Let’s see what all that means, and how you can work with sets in Python.

A set can be created in two ways. First, you can define a set with the built-in set() function:

x = set(<iter>)

In this case, the argument <iter> is an iterable—again, for the moment, think list or tuple—that generates the list of objects to be included in the set. This is analogous to the <iter> argument given to the .extend() list method:

>>>

>>> x = set(['foo', 'bar', 'baz', 'foo', 'qux'])
>>> x
{'qux', 'foo', 'bar', 'baz'}

>>> x = set(('foo', 'bar', 'baz', 'foo', 'qux'))
>>> x
{'qux', 'foo', 'bar', 'baz'}

Strings are also iterable, so a string can be passed to set() as well. You have already seen that list(s) generates a list of the characters in the string s. Similarly, set(s) generates a set of the characters in s:

>>>

>>> s = 'quux'

>>> list(s)
['q', 'u', 'u', 'x']
>>> set(s)
{'x', 'u', 'q'}

You can see that the resulting sets are unordered: the original order, as specified in the definition, is not necessarily preserved. Additionally, duplicate values are only represented in the set once, as with the string 'foo' in the first two examples and the letter 'u' in the third.

Alternately, a set can be defined with curly braces ({}):

x = {<obj>, <obj>, ..., <obj>}

When a set is defined this way, each <obj> becomes a distinct element of the set, even if it is an iterable. This behavior is similar to that of the .append() list method.

Thus, the sets shown above can also be defined like this:

>>>

>>> x = {'foo', 'bar', 'baz', 'foo', 'qux'}
>>> x
{'qux', 'foo', 'bar', 'baz'}

>>> x = {'q', 'u', 'u', 'x'}
>>> x
{'x', 'q', 'u'}

To recap:

The argument to set() is an iterable. It generates a list of elements to be placed into the set.
The objects in curly braces are placed into the set intact, even if they are iterable.

Observe the difference between these two set definitions:

>>>

>>> {'foo'}
{'foo'}

>>> set('foo')
{'o', 'f'}

A set can be empty. However, recall that Python interprets empty curly braces ({}) as an empty dictionary, so the only way to define an empty set is with the set() function:

>>>

>>> x = set()
>>> type(x)
<class 'set'>
>>> x
set()

>>> x = {}
>>> type(x)
<class 'dict'>

An empty set is falsy in a Boolean context:

>>>

>>> x = set()
>>> bool(x)
False
>>> x or 1
1
>>> x and 1
set()

You might think the most intuitive sets would contain similar objects—for example, even numbers or surnames:

>>>

>>> s1 = {2, 4, 6, 8, 10}
>>> s2 = {'Smith', 'McArthur', 'Wilson', 'Johansson'}

Python does not require this, though. The elements in a set can be objects of different types:

>>>

>>> x = {42, 'foo', 3.14159, None}
>>> x
{None, 'foo', 42, 3.14159}

Don’t forget that set elements must be immutable. For example, a tuple may be included in a set:

>>>

>>> x = {42, 'foo', (1, 2, 3), 3.14159}
>>> x
{42, 'foo', 3.14159, (1, 2, 3)}

But lists and dictionaries are mutable, so they can’t be set elements:

>>>

>>> a = [1, 2, 3]
>>> {a}
Traceback (most recent call last):
  File "<pyshell#70>", line 1, in <module>
    {a}
TypeError: unhashable type: 'list'

>>> d = {'a': 1, 'b': 2}
>>> {d}
Traceback (most recent call last):
  File "<pyshell#72>", line 1, in <module>
    {d}
TypeError: unhashable type: 'dict'

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Set Size and Membership

The len() function returns the number of elements in a set, and the in and not in operators can be used to test for membership:

>>>

>>> x = {'foo', 'bar', 'baz'}

>>> len(x)
3

>>> 'bar' in x
True
>>> 'qux' in x
False

Operating on a Set

Many of the operations that can be used for Python’s other composite data types don’t make sense for sets. For example, sets can’t be indexed or sliced. However, Python provides a whole host of operations on set objects that generally mimic the operations that are defined for mathematical sets.

Operators vs. Methods

Most, though not quite all, set operations in Python can be performed in two different ways: by operator or by method. Let’s take a look at how these operators and methods work, using set union as an example.

Given two sets, x1 and x2, the union of x1 and x2 is a set consisting of all elements in either set.

Consider these two sets:

x1 = {'foo', 'bar', 'baz'}
x2 = {'baz', 'qux', 'quux'}

The union of x1 and x2 is {'foo', 'bar', 'baz', 'qux', 'quux'}.

Note: Notice that the element 'baz', which appears in both x1 and x2, appears only once in the union. Sets never contain duplicate values.

In Python, set union can be performed with the | operator:

>>>

>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'baz', 'qux', 'quux'}
>>> x1 | x2
{'baz', 'quux', 'qux', 'bar', 'foo'}

Set union can also be obtained with the .union() method. The method is invoked on one of the sets, and the other is passed as an argument:

>>>

>>> x1.union(x2)
{'baz', 'quux', 'qux', 'bar', 'foo'}

The way they are used in the examples above, the operator and method behave identically. But there is a subtle difference between them. When you use the | operator, both operands must be sets. The .union() method, on the other hand, will take any iterable as an argument, convert it to a set, and then perform the union.

Observe the difference between these two statements:

>>>

>>> x1 | ('baz', 'qux', 'quux')
Traceback (most recent call last):
  File "<pyshell#43>", line 1, in <module>
    x1 | ('baz', 'qux', 'quux')
TypeError: unsupported operand type(s) for |: 'set' and 'tuple'

>>> x1.union(('baz', 'qux', 'quux'))
{'baz', 'quux', 'qux', 'bar', 'foo'}

Both attempt to compute the union of x1 and the tuple ('baz', 'qux', 'quux'). This fails with the | operator but succeeds with the .union() method.

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Available Operators and Methods

Below is a list of the set operations available in Python. Some are performed by operator, some by method, and some by both. The principle outlined above generally applies: where a set is expected, methods will typically accept any iterable as an argument, but operators require actual sets as operands.

x1.union(x2[, x3 ...])

x1 | x2 [| x3 ...]

Compute the union of two or more sets.

x1.union(x2) and x1 | x2 both return the set of all elements in either x1 or x2:

>>>

>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'baz', 'qux', 'quux'}

>>> x1.union(x2)
{'foo', 'qux', 'quux', 'baz', 'bar'}

>>> x1 | x2
{'foo', 'qux', 'quux', 'baz', 'bar'}

More than two sets may be specified with either the operator or the method:

>>>

>>> a = {1, 2, 3, 4}
>>> b = {2, 3, 4, 5}
>>> c = {3, 4, 5, 6}
>>> d = {4, 5, 6, 7}

>>> a.union(b, c, d)
{1, 2, 3, 4, 5, 6, 7}

>>> a | b | c | d
{1, 2, 3, 4, 5, 6, 7}

The resulting set contains all elements that are present in any of the specified sets.

x1.intersection(x2[, x3 ...])

x1 & x2 [& x3 ...]

Compute the intersection of two or more sets.

x1.intersection(x2) and x1 & x2 return the set of elements common to both x1 and x2:

>>>

>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'baz', 'qux', 'quux'}

>>> x1.intersection(x2)
{'baz'}

>>> x1 & x2
{'baz'}

You can specify multiple sets with the intersection method and operator, just like you can with set union:

>>>

>>> a = {1, 2, 3, 4}
>>> b = {2, 3, 4, 5}
>>> c = {3, 4, 5, 6}
>>> d = {4, 5, 6, 7}

>>> a.intersection(b, c, d)
{4}

>>> a & b & c & d
{4}

The resulting set contains only elements that are present in all of the specified sets.

x1.difference(x2[, x3 ...])

x1 - x2 [- x3 ...]

Compute the difference between two or more sets.

x1.difference(x2) and x1 - x2 return the set of all elements that are in x1 but not in x2:

>>>

>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'baz', 'qux', 'quux'}

>>> x1.difference(x2)
{'foo', 'bar'}

>>> x1 - x2
{'foo', 'bar'}

Another way to think of this is that x1.difference(x2) and x1 - x2 return the set that results when any elements in x2 are removed or subtracted from x1.

Once again, you can specify more than two sets:

>>>

>>> a = {1, 2, 3, 30, 300}
>>> b = {10, 20, 30, 40}
>>> c = {100, 200, 300, 400}

>>> a.difference(b, c)
{1, 2, 3}

>>> a - b - c
{1, 2, 3}

When multiple sets are specified, the operation is performed from left to right. In the example above, a - b is computed first, resulting in {1, 2, 3, 300}. Then c is subtracted from that set, leaving {1, 2, 3}:

x1.symmetric_difference(x2)

x1 ^ x2 [^ x3 ...]

Compute the symmetric difference between sets.

x1.symmetric_difference(x2) and x1 ^ x2 return the set of all elements in either x1 or x2, but not both:

>>>

>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'baz', 'qux', 'quux'}

>>> x1.symmetric_difference(x2)
{'foo', 'qux', 'quux', 'bar'}

>>> x1 ^ x2
{'foo', 'qux', 'quux', 'bar'}

The ^ operator also allows more than two sets:

>>>