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.

## 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 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'
```

## 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.

### 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:

```
>>> a = {1, 2, 3, 4, 5}
>>> b = {10, 2, 3, 4, 50}
>>> c = {1, 50, 100}
>>> a ^ b ^ c
{100, 5, 10}
```

As with the difference operator, when multiple sets are specified, the operation is performed from left to right.

Curiously, although the `^`

operator allows multiple sets, the `.symmetric_difference()`

method doesn’t:

```
>>> a = {1, 2, 3, 4, 5}
>>> b = {10, 2, 3, 4, 50}
>>> c = {1, 50, 100}
>>> a.symmetric_difference(b, c)
Traceback (most recent call last):
File "<pyshell#11>", line 1, in <module>
a.symmetric_difference(b, c)
TypeError: symmetric_difference() takes exactly one argument (2 given)
```

`x1.isdisjoint(x2)`

Determines whether or not two sets have any elements in common.

`x1.isdisjoint(x2)`

returns `True`

if `x1`

and `x2`

have no elements in common:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'baz', 'qux', 'quux'}
>>> x1.isdisjoint(x2)
False
>>> x2 - {'baz'}
{'quux', 'qux'}
>>> x1.isdisjoint(x2 - {'baz'})
True
```

If `x1.isdisjoint(x2)`

is `True`

, then `x1 & x2`

is the empty set:

```
>>> x1 = {1, 3, 5}
>>> x2 = {2, 4, 6}
>>> x1.isdisjoint(x2)
True
>>> x1 & x2
set()
```

**Note:** There is no operator that corresponds to the `.isdisjoint()`

method.

`x1.issubset(x2)`

`x1 <= x2`

Determine whether one set is a subset of the other.

In set theory, a set `x1`

is considered a subset of another set `x2`

if every element of `x1`

is in `x2`

.

`x1.issubset(x2)`

and `x1 <= x2`

return `True`

if `x1`

is a subset of `x2`

:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x1.issubset({'foo', 'bar', 'baz', 'qux', 'quux'})
True
>>> x2 = {'baz', 'qux', 'quux'}
>>> x1 <= x2
False
```

A set is considered to be a subset of itself:

```
>>> x = {1, 2, 3, 4, 5}
>>> x.issubset(x)
True
>>> x <= x
True
```

It seems strange, perhaps. But it fits the definition—every element of `x`

is in `x`

.

`x1 < x2`

Determines whether one set is a proper subset of the other.

A proper subset is the same as a subset, except that the sets can’t be identical. A set `x1`

is considered a proper subset of another set `x2`

if every element of `x1`

is in `x2`

, and `x1`

and `x2`

are not equal.

`x1 < x2`

returns `True`

if `x1`

is a proper subset of `x2`

:

```
>>> x1 = {'foo', 'bar'}
>>> x2 = {'foo', 'bar', 'baz'}
>>> x1 < x2
True
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'bar', 'baz'}
>>> x1 < x2
False
```

While a set is considered a subset of itself, it is not a proper subset of itself:

```
>>> x = {1, 2, 3, 4, 5}
>>> x <= x
True
>>> x < x
False
```

**Note:** The `<`

operator is the only way to test whether a set is a proper subset. There is no corresponding method.

`x1.issuperset(x2)`

`x1 >= x2`

Determine whether one set is a superset of the other.

A superset is the reverse of a subset. A set `x1`

is considered a superset of another set `x2`

if `x1`

contains every element of `x2`

.

`x1.issuperset(x2)`

and `x1 >= x2`

return `True`

if `x1`

is a superset of `x2`

:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x1.issuperset({'foo', 'bar'})
True
>>> x2 = {'baz', 'qux', 'quux'}
>>> x1 >= x2
False
```

You have already seen that a set is considered a subset of itself. A set is also considered a superset of itself:

```
>>> x = {1, 2, 3, 4, 5}
>>> x.issuperset(x)
True
>>> x >= x
True
```

`x1 > x2`

Determines whether one set is a proper superset of the other.

A proper superset is the same as a superset, except that the sets can’t be identical. A set `x1`

is considered a proper superset of another set `x2`

if `x1`

contains every element of `x2`

, and `x1`

and `x2`

are not equal.

`x1 > x2`

returns `True`

if `x1`

is a proper superset of `x2`

:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'bar'}
>>> x1 > x2
True
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'bar', 'baz'}
>>> x1 > x2
False
```

A set is not a proper superset of itself:

```
>>> x = {1, 2, 3, 4, 5}
>>> x > x
False
```

**Note:** The `>`

operator is the only way to test whether a set is a proper superset. There is no corresponding method.

## Modifying a Set

Although the elements contained in a set must be of immutable type, sets themselves can be modified. Like the operations above, there are a mix of operators and methods that can be used to change the contents of a set.

### Augmented Assignment Operators and Methods

Each of the union, intersection, difference, and symmetric difference operators listed above has an augmented assignment form that can be used to modify a set. For each, there is a corresponding method as well.

`x1.update(x2[, x3 ...])`

`x1 |= x2 [| x3 ...]`

Modify a set by union.

`x1.update(x2)`

and `x1 |= x2`

add to `x1`

any elements in `x2`

that `x1`

does not already have:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'baz', 'qux'}
>>> x1 |= x2
>>> x1
{'qux', 'foo', 'bar', 'baz'}
>>> x1.update(['corge', 'garply'])
>>> x1
{'qux', 'corge', 'garply', 'foo', 'bar', 'baz'}
```

`x1.intersection_update(x2[, x3 ...])`

`x1 &= x2 [& x3 ...]`

Modify a set by intersection.

`x1.intersection_update(x2)`

and `x1 &= x2`

update `x1`

, retaining only elements found in both `x1`

and `x2`

:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'baz', 'qux'}
>>> x1 &= x2
>>> x1
{'foo', 'baz'}
>>> x1.intersection_update(['baz', 'qux'])
>>> x1
{'baz'}
```

`x1.difference_update(x2[, x3 ...])`

`x1 -= x2 [| x3 ...]`

Modify a set by difference.

`x1.difference_update(x2)`

and `x1 -= x2`

update `x1`

, removing elements found in `x2`

:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'baz', 'qux'}
>>> x1 -= x2
>>> x1
{'bar'}
>>> x1.difference_update(['foo', 'bar', 'qux'])
>>> x1
set()
```

`x1.symmetric_difference_update(x2)`

`x1 ^= x2`

Modify a set by symmetric difference.

`x1.symmetric_difference_update(x2)`

and `x1 ^= x2`

update `x1`

, retaining elements found in either `x1`

or `x2`

, but not both:

```
>>> x1 = {'foo', 'bar', 'baz'}
>>> x2 = {'foo', 'baz', 'qux'}
>>>
>>> x1 ^= x2
>>> x1
{'bar', 'qux'}
>>>
>>> x1.symmetric_difference_update(['qux', 'corge'])
>>> x1
{'bar', 'corge'}
```

### Other Methods For Modifying Sets

Aside from the augmented operators above, Python supports several additional methods that modify sets.

`x.add(<elem>)`

Adds an element to a set.

`x.add(<elem>)`

adds `<elem>`

, which must be a single immutable object, to `x`

:

```
>>> x = {'foo', 'bar', 'baz'}
>>> x.add('qux')
>>> x
{'bar', 'baz', 'foo', 'qux'}
```

`x.remove(<elem>)`

Removes an element from a set.

`x.remove(<elem>)`

removes `<elem>`

from `x`

. Python raises an exception if `<elem>`

is not in `x`

:

```
>>> x = {'foo', 'bar', 'baz'}
>>> x.remove('baz')
>>> x
{'bar', 'foo'}
>>> x.remove('qux')
Traceback (most recent call last):
File "<pyshell#58>", line 1, in <module>
x.remove('qux')
KeyError: 'qux'
```

`x.discard(<elem>)`

Removes an element from a set.

`x.discard(<elem>)`

also removes `<elem>`

from `x`

. However, if `<elem>`

is not in `x`

, this method quietly does nothing instead of raising an exception:

```
>>> x = {'foo', 'bar', 'baz'}
>>> x.discard('baz')
>>> x
{'bar', 'foo'}
>>> x.discard('qux')
>>> x
{'bar', 'foo'}
```

`x.pop()`

Removes a random element from a set.

`x.pop()`

removes and returns an arbitrarily chosen element from `x`

. If `x`

is empty, `x.pop()`

raises an exception:

```
>>> x = {'foo', 'bar', 'baz'}
>>> x.pop()
'bar'
>>> x
{'baz', 'foo'}
>>> x.pop()
'baz'
>>> x
{'foo'}
>>> x.pop()
'foo'
>>> x
set()
>>> x.pop()
Traceback (most recent call last):
File "<pyshell#82>", line 1, in <module>
x.pop()
KeyError: 'pop from an empty set'
```

`x.clear()`

Clears a set.

`x.clear()`

removes all elements from `x`

:

```
>>> x = {'foo', 'bar', 'baz'}
>>> x
{'foo', 'bar', 'baz'}
>>>
>>> x.clear()
>>> x
set()
```

## Frozen Sets

Python provides another built-in type called a **frozenset**, which is in all respects exactly like a set, except that a frozenset is immutable. You can perform non-modifying operations on a frozenset:

```
>>> x = frozenset(['foo', 'bar', 'baz'])
>>> x
frozenset({'foo', 'baz', 'bar'})
>>> len(x)
3
>>> x & {'baz', 'qux', 'quux'}
frozenset({'baz'})
```

But methods that attempt to modify a frozenset fail:

```
>>> x = frozenset(['foo', 'bar', 'baz'])
>>> x.add('qux')
Traceback (most recent call last):
File "<pyshell#127>", line 1, in <module>
x.add('qux')
AttributeError: 'frozenset' object has no attribute 'add'
>>> x.pop()
Traceback (most recent call last):
File "<pyshell#129>", line 1, in <module>
x.pop()
AttributeError: 'frozenset' object has no attribute 'pop'
>>> x.clear()
Traceback (most recent call last):
File "<pyshell#131>", line 1, in <module>
x.clear()
AttributeError: 'frozenset' object has no attribute 'clear'
>>> x
frozenset({'foo', 'bar', 'baz'})
```

Deep Dive: Frozensets and Augmented Assignment

Since a frozenset is immutable, you might think it can’t be the target of an augmented assignment operator. But observe:

>>> f = frozenset(['foo', 'bar', 'baz']) >>> s = {'baz', 'qux', 'quux'} >>> f &= s >>> f frozenset({'baz'})What gives?

Python does not perform augmented assignments on frozensets in place. The statement

`x &= s`

is effectively equivalent to`x = x & s`

. It isn’t modifying the original`x`

. It is reassigning`x`

to a new object, and the object`x`

originally referenced is gone.You can verify this with the

`id()`

function:>>> f = frozenset(['foo', 'bar', 'baz']) >>> id(f) 56992872 >>> s = {'baz', 'qux', 'quux'} >>> f &= s >>> f frozenset({'baz'}) >>> id(f) 56992152

`f`

has a different integer identifier following the augmented assignment. It has been reassigned, not modified in place.Some objects in Python are modified in place when they are the target of an augmented assignment operator. But frozensets aren’t.

Frozensets are useful in situations where you want to use a set, but you need an immutable object. For example, you can’t define a set whose elements are also sets, because set elements must be immutable:

```
>>> x1 = set(['foo'])
>>> x2 = set(['bar'])
>>> x3 = set(['baz'])
>>> x = {x1, x2, x3}
Traceback (most recent call last):
File "<pyshell#38>", line 1, in <module>
x = {x1, x2, x3}
TypeError: unhashable type: 'set'
```

If you really feel compelled to define a set of sets (hey, it could happen), you can do it if the elements are frozensets, because they are immutable:

```
>>> x1 = frozenset(['foo'])
>>> x2 = frozenset(['bar'])
>>> x3 = frozenset(['baz'])
>>> x = {x1, x2, x3}
>>> x
{frozenset({'bar'}), frozenset({'baz'}), frozenset({'foo'})}
```

Likewise, recall from the previous tutorial on dictionaries that a dictionary key must be immutable. You can’t use the built-in set type as a dictionary key:

```
>>> x = {1, 2, 3}
>>> y = {'a', 'b', 'c'}
>>>
>>> d = {x: 'foo', y: 'bar'}
Traceback (most recent call last):
File "<pyshell#3>", line 1, in <module>
d = {x: 'foo', y: 'bar'}
TypeError: unhashable type: 'set'
```

If you find yourself needing to use sets as dictionary keys, you can use frozensets:

```
>>> x = frozenset({1, 2, 3})
>>> y = frozenset({'a', 'b', 'c'})
>>>
>>> d = {x: 'foo', y: 'bar'}
>>> d
{frozenset({1, 2, 3}): 'foo', frozenset({'c', 'a', 'b'}): 'bar'}
```

## Conclusion

In this tutorial, you learned how to define **set** objects in Python, and you became familiar with the functions, operators, and methods that can be used to work with sets.

You should now be comfortable with the basic built-in data types that Python provides.

Next, you will begin to explore how the code that operates on those objects is organized and structured in a Python program.