Given a set

{0, 1, 2, 3}

How can I produce the subsets:

[set(), {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}, {0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}, {0, 1, 2, 3}]

The Python itertools page has exactly a powerset recipe for this:

from itertools import chain, combinationsdef powerset(iterable):    "powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"    s = list(iterable)    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

Output:

>>> list(powerset("abcd"))[(), ('a',), ('b',), ('c',), ('d',), ('a', 'b'), ('a', 'c'), ('a', 'd'), ('b', 'c'), ('b', 'd'), ('c', 'd'), ('a', 'b', 'c'), ('a', 'b', 'd'), ('a', 'c', 'd'), ('b', 'c', 'd'), ('a', 'b', 'c', 'd')]

If you don’t like that empty tuple at the beginning, you can just change the range statement to range(1, len(s)+1) to avoid a 0-length combination.

Here is more code for a powerset. This is written from scratch:

>>> def powerset(s):...     x = len(s)...     for i in range(1 << x):...         print [s[j] for j in range(x) if (i & (1 << j))]...>>> powerset([4,5,6])[][4][5][4, 5][6][4, 6][5, 6][4, 5, 6]

Mark Rushakoff’s comment is applicable here: “If you don’t like that empty tuple at the beginning, on.”you can just change the range statement to range(1, len(s)+1) to avoid a 0-length combination”, except in my case you change for i in range(1 << x) to for i in range(1, 1 << x).

Returning to this years later, I’d now write it like this:

def powerset(s):    x = len(s)    masks = [1 << i for i in range(x)]    for i in range(1 << x):        yield [ss for mask, ss in zip(masks, s) if i & mask]

And then the test code would look like this, say:

print(list(powerset([4, 5, 6])))

Using yield means that you do not need to calculate all results in a single piece of memory. Precalculating the masks outside the main loop is assumed to be a worthwhile optimization.

If you’re looking for a quick answer, I just searched “python power set” on google and came up with this: Python Power Set Generator

Here’s a copy-paste from the code in that page:

def powerset(seq):    """    Returns all the subsets of this set. This is a generator.    """    if len(seq) <= 1:        yield seq        yield []    else:        for item in powerset(seq[1:]):            yield [seq[0]]+item            yield item

This can be used like this:

 l = [1, 2, 3, 4] r = [x for x in powerset(l)]

Now r is a list of all the elements you wanted, and can be sorted and printed:

r.sort()print r[[], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]

def powerset(lst):    return reduce(lambda result, x: result + [subset + [x] for subset in result],                  lst, [[]])

There is a refinement of powerset:

def powerset(seq):    """    Returns all the subsets of this set. This is a generator.    """    if len(seq) <= 0:        yield []    else:        for item in powerset(seq[1:]):            yield [seq[0]]+item            yield item

TL;DR (go directly to Simplification)

I know I have previously added an answer, but I really like my new implementation. I am taking a set as input, but it actually could be any iterable, and I am returning a set of sets which is the power set of the input. I like this approach because it is more aligned with the mathematical definition of power set (set of all subsets).

def power_set(A):    """A is an iterable (list, tuple, set, str, etc)    returns a set which is the power set of A."""    length = len(A)    l = [a for a in A]    ps = set()    for i in range(2 ** length):        selector = f'{i:0{length}b}'        subset = {l[j] for j, bit in enumerate(selector) if bit == '1'}        ps.add(frozenset(subset))    return ps

If you want exactly the output you posted in your answer use this:

>>> [set(s) for s in power_set({1, 2, 3, 4})][{3, 4}, {2}, {1, 4}, {2, 3, 4}, {2, 3}, {1, 2, 4}, {1, 2}, {1, 2, 3}, {3}, {2, 4}, {1}, {1, 2, 3, 4}, set(), {1, 3}, {1, 3, 4}, {4}]

Explanation

It is known that the number of elements of the power set is 2 ** len(A), so that could clearly be seen in the for loop.

I need to convert the input (ideally a set) into a list because by a set is a data structure of unique unordered elements, and the order will be crucial to generate the subsets.

selector is key in this algorithm. Note that selector has the same length as the input set, and to make this possible it is using an f-string with padding. Basically, this allows me to select the elements that will be added to each subset during each iteration. Let’s say the input set has 3 elements {0, 1, 2}, so selector will take values between 0 and 7 (inclusive), which in binary are:

000 # 0001 # 1010 # 2011 # 3100 # 4101 # 5110 # 6111 # 7

So, each bit could serve as an indicator if an element of the original set should be added or not. Look at the binary numbers, and just think of each number as an element of the super set in which 1 means that an element at index j should be added, and 0 means that this element should not be added.

I am using a set comprehension to generate a subset at each iteration, and I convert this subset into a frozenset so I can add it to ps (power set). Otherwise, I won’t be able to add it because a set in Python consists only of immutable objects.

Simplification

You can simplify the code using some python comprehensions, so you can get rid of those for loops. You can also use zip to avoid using j index and the code will end up as the following:

def power_set(A):    length = len(A)    return {        frozenset({e for e, b in zip(A, f'{i:{length}b}') if b == '1'})        for i in range(2 ** length)    }

That’s it. What I like of this algorithm is that is clearer and more intuitive than others because it looks quite magical to rely on itertools even though it works as expected.

def get_power_set(s):  power_set=[[]]  for elem in s:    # iterate over the sub sets so far    for sub_set in power_set:      # add a new subset consisting of the subset at hand added elem      power_set=power_set+[list(sub_set)+[elem]]  return power_set

For example:

get_power_set([1,2,3])

yield

[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]

I have found the following algorithm very clear and simple:

def get_powerset(some_list):    """Returns all subsets of size 0 - len(some_list) for some_list"""    if len(some_list) == 0:        return [[]]    subsets = []    first_element = some_list[0]    remaining_list = some_list[1:]    # Strategy: get all the subsets of remaining_list. For each    # of those subsets, a full subset list will contain both    # the original subset as well as a version of the subset    # that contains first_element    for partial_subset in get_powerset(remaining_list):        subsets.append(partial_subset)        subsets.append(partial_subset[:] + [first_element])    return subsets

Another way one can generate the powerset is by generating all binary numbers that have n bits. As a power set the amount of number with n digits is 2 ^ n. The principle of this algorithm is that an element could be present or not in a subset as a binary digit could be one or zero but not both.

def power_set(items):    N = len(items)    # enumerate the 2 ** N possible combinations    for i in range(2 ** N):        combo = []        for j in range(N):            # test bit jth of integer i            if (i >> j) % 2 == 1:                combo.append(items[j])        yield combo

I found both algorithms when I was taking MITx: 6.00.2x Introduction to Computational Thinking and Data Science, and I consider it is one of the easiest algorithms to understand I have seen.

I just wanted to provide the most comprehensible solution, the anti code-golf version.

from itertools import combinationsl = ["x", "y", "z", ]def powerset(items):    combo = []    for r in range(len(items) + 1):        #use a list to coerce a actual list from the combinations generator        combo.append(list(combinations(items,r)))    return combol_powerset = powerset(l)for i, item in enumerate(l_powerset):    print "All sets of length ", i    print item

The results

All sets of length 0

[()]

All sets of length 1

[('x',), ('y',), ('z',)]

All sets of length 2

[('x', 'y'), ('x', 'z'), ('y', 'z')]

All sets of length 3

[('x', 'y', 'z')]

For more see the itertools docs, also the wikipedia entry on power sets

Just a quick power set refresher !

Power set of a set X, is simply the set of all subsets of X including the empty set

Example set X = (a,b,c)

Power Set = { { a , b , c } , { a , b } , { a , c } , { b , c } , { a } , { b } , { c } , { } }

Here is another way of finding power set:

def power_set(input):
    # returns a list of all subsets of the list a
    if (len(input) == 0):
        return [[]]
    else:
        main_subset = [ ]
        for small_subset in power_set(input[1:]):
            main_subset += [small_subset]
            main_subset += [[input[0]] + small_subset]
        return main_subset

print(power_set([0,1,2,3]))

full credit to source

This can be done very naturally with itertools.product:

import itertools

def powerset(l):
    for sl in itertools.product(*[[[], [i]] for i in l]):
        yield {j for i in sl for j in i}

A simple way would be to harness the internal representation of integers under 2’s complement arithmetic.

Binary representation of integers is as {000, 001, 010, 011, 100, 101, 110, 111} for numbers ranging from 0 to 7. For an integer counter value, considering 1 as inclusion of corresponding element in collection and ‘0’ as exclusion we can generate subsets based on the counting sequence. Numbers have to be generated from 0 to pow(2,n) -1 where n is the length of array i.e. number of bits in binary representation.

A simple Subset Generator Function based on it can be written as below. It basically relies

def subsets(array):
    if not array:
        return
    else:
        length = len(array)
        for max_int in range(0x1 << length):
            subset = []
            for i in range(length):
                if max_int & (0x1 << i):
                    subset.append(array[i])
            yield subset

and then it can be used as

def get_subsets(array):
    powerset = []
    for i in subsets(array):
        powerser.append(i)
    return powerset

Testing

Adding following in local file

if __name__ == '__main__':
    sample = ['b',  'd',  'f']

    for i in range(len(sample)):
        print "Subsets for " , sample[i:], " are ", get_subsets(sample[i:])

gives following output

Subsets for  ['b', 'd', 'f']  are  [[], ['b'], ['d'], ['b', 'd'], ['f'], ['b', 'f'], ['d', 'f'], ['b', 'd', 'f']]
Subsets for  ['d', 'f']  are  [[], ['d'], ['f'], ['d', 'f']]
Subsets for  ['f']  are  [[], ['f']]

With empty set, which is part of all the subsets, you could use:

def subsets(iterable):
    for n in range(len(iterable) + 1):
        yield from combinations(iterable, n)

Almost all of these answers use list rather than set, which felt like a bit of a cheat to me. So, out of curiosity I tried to do a simple version truly on set and summarize for other “new to Python” folks.

I found there’s a couple oddities in dealing with Python’s set implementation. The main surprise to me was handling empty sets. This is in contrast to Ruby’s Set implementation, where I can simply do Set[Set[]] and get a Set containing one empty Set, so I found it initially a little confusing.

To review, in doing powerset with sets, I encountered two problems:

1. set() takes an iterable, so set(set()) will return set() because the empty set iterable is empty (duh I guess :))
2. to get a set of sets, set({set()}) and set.add(set) won’t work because set() isn’t hashable

To solve both issues, I made use of frozenset(), which means I don’t quite get what I want (type is literally set), but makes use of the overall set interace.

def powerset(original_set):
  # below gives us a set with one empty set in it
  ps = set({frozenset()}) 
  for member in original_set:
    subset = set()
    for m in ps:
      # to be added into subset, needs to be
      # frozenset.union(set) so it's hashable
      subset.add(m.union(set([member]))
    ps = ps.union(subset)
  return ps

Below we get 2² (16) frozensets correctly as output:

In [1]: powerset(set([1,2,3,4]))
Out[2]:
{frozenset(),
 frozenset({3, 4}),
 frozenset({2}),
 frozenset({1, 4}),
 frozenset({3}),
 frozenset({2, 3}),
 frozenset({2, 3, 4}),
 frozenset({1, 2}),
 frozenset({2, 4}),
 frozenset({1}),
 frozenset({1, 2, 4}),
 frozenset({1, 3}),
 frozenset({1, 2, 3}),
 frozenset({4}),
 frozenset({1, 3, 4}),
 frozenset({1, 2, 3, 4})}

As there’s no way to have a set of sets in Python, if you want to turn these frozensets into sets, you’ll have to map them back into a list (list(map(set, powerset(set([1,2,3,4])))) ) or modify the above.

Perhaps the question is getting old, but I hope my code will help someone.

def powSet(set):
    if len(set) == 0:
       return [[]]
    return addtoAll(set[0],powSet(set[1:])) + powSet(set[1:])

def addtoAll(e, set):
   for c in set:
       c.append(e)
   return set

Use function powerset() from package more_itertools.

Yields all possible subsets of the iterable

>>> list(powerset([1, 2, 3]))
[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]

If you want sets, use:

list(map(set, powerset(iterable)))

Getting all the subsets with recursion. Crazy-ass one-liner

from typing import List

def subsets(xs: list) -> List[list]:
    return subsets(xs[1:]) + [x + [xs[0]] for x in subsets(xs[1:])] if xs else [[]]

Based on a Haskell solution

subsets :: [a] -> [[a]]
subsets [] = [[]]
subsets (x:xs) = map (x:) (subsets xs) ++ subsets xs

def findsubsets(s, n): 
    return list(itertools.combinations(s, n)) 

def allsubsets(s) :
    a = []
    for x in range(1,len(s)+1):
        a.append(map(set,findsubsets(s,x)))      
    return a

You can do it like this:

def powerset(x):
    m=[]
    if not x:
        m.append(x)
    else:
        A = x[0]
        B = x[1:]
        for z in powerset(B):
            m.append(z)
            r = [A] + z
            m.append(r)
    return m

print(powerset([1, 2, 3, 4]))

Output:

[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]

I know this is too late

There are many other solutions already but still…

def power_set(lst):
    pw_set = [[]]

    for i in range(0,len(lst)):
        for j in range(0,len(pw_set)):
            ele = pw_set[j].copy()
            ele = ele + [lst[i]]
            pw_set = pw_set + [ele]

    return pw_set

This is wild because none of these answers actually provide the return of an actual Python set. Here is a messy implementation that will give a powerset that actually is a Python set.

test_set = set(['yo', 'whatup', 'money'])
def powerset( base_set ):
    """ modified from pydoc's itertools recipe shown above"""
    from itertools import chain, combinations
    base_list = list( base_set )
    combo_list = [ combinations(base_list, r) for r in range(len(base_set)+1) ]

    powerset = set([])
    for ll in combo_list:
        list_of_frozensets = list( map( frozenset, map( list, ll ) ) ) 
        set_of_frozensets = set( list_of_frozensets )
        powerset = powerset.union( set_of_frozensets )

    return powerset

print powerset( test_set )
# >>> set([ frozenset(['money','whatup']), frozenset(['money','whatup','yo']), 
#        frozenset(['whatup']), frozenset(['whatup','yo']), frozenset(['yo']),
#        frozenset(['money','yo']), frozenset(['money']), frozenset([]) ])

I’d love to see a better implementation, though.

Here is my quick implementation utilizing combinations but using only built-ins.

def powerSet(array):
    length = str(len(array))
    formatter = '{:0' + length + 'b}'
    combinations = []
    for i in xrange(2**int(length)):
        combinations.append(formatter.format(i))
    sets = set()
    currentSet = []
    for combo in combinations:
        for i,val in enumerate(combo):
            if val=='1':
                currentSet.append(array[i])
        sets.add(tuple(sorted(currentSet)))
        currentSet = []
    return sets

All subsets in range n as set:

n = int(input())
l = [i for i in range (1, n + 1)]

for number in range(2 ** n) :
    binary = bin(number)[: 1 : -1]
    subset = [l[i] for i in range(len(binary)) if binary[i] == "1"]
    print(set(sorted(subset)) if number > 0 else "{}")

import math    
def printPowerSet(set,set_size): 
    pow_set_size =int(math.pow(2, set_size))
    for counter in range(pow_set_size):
    for j in range(set_size):  
        if((counter & (1 << j)) > 0):
            print(set[j], end = "")
    print("")
set = ['a', 'b', 'c']
printPowerSet(set,3)

A variation of the question, is an exercise I see on the book “Discovering Computer Science: Interdisciplinary Problems, Principles, and Python Programming. 2015 edition”. In that exercise 10.2.11, the input is just an integer number, and the output should be the power sets. Here is my recursive solution (not using anything else but basic python3 )

def powerSetR(n):
    assert n >= 0
    if n == 0:
        return [[]]
    else:
        input_set = list(range(1, n+1)) # [1,2,...n]
        main_subset = [ ]
        for small_subset in powerSetR(n-1):
            main_subset += [small_subset]
            main_subset += [ [input_set[-1]] + small_subset]
        return main_subset

superset = powerSetR(4)
print(superset)       
print("Number of sublists:", len(superset))

And the output is

[[], [4], [3], [4, 3], [2], [4, 2], [3, 2], [4, 3, 2], [1], [4, 1], [3, 1], [4, 3, 1], [2, 1], [4, 2, 1], [3, 2, 1], [4, 3, 2, 1]] Number of sublists: 16

I hadn’t come across the more_itertools.powerset function and would recommend using that. I also recommend not using the default ordering of the output from itertools.combinations, often instead you want to minimise the distance between the positions and sort the subsets of items with shorter distance between them above/before the items with larger distance between them.

The itertools recipes page shows it uses chain.from_iterable

• Note that the r here matches the standard notation for the lower part of a binomial coefficient, the s is usually referred to as n in mathematics texts and on calculators (“n Choose r”)

def powerset(iterable):
    "powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

The other examples here give the powerset of [1,2,3,4] in such a way that the 2-tuples are listed in “lexicographic” order (when we print the numbers as integers). If I write the distance between the numbers alongside it (i.e. the difference), it shows my point:

12 ⇒ 1
13 ⇒ 2
14 ⇒ 3
23 ⇒ 1
24 ⇒ 2
34 ⇒ 1

The correct order for subsets should be the order which ‘exhausts’ the minimal distance first, like so:

12 ⇒ 1
23 ⇒ 1
34 ⇒ 1
13 ⇒ 2
24 ⇒ 2
14 ⇒ 3

Using numbers here makes this ordering look ‘wrong’, but consider for example the letters ["a","b","c","d"] it is clearer why this might be useful to obtain the powerset in this order:

ab ⇒ 1
bc ⇒ 1
cd ⇒ 1
ac ⇒ 2
bd ⇒ 2
ad ⇒ 3

This effect is more pronounced with more items, and for my purposes it makes the difference between being able to describe the ranges of the indexes of the powerset meaningfully.

(There is a lot written on Gray codes etc. for the output order of algorithms in combinatorics, I don’t see it as a side issue).

I actually just wrote a fairly involved program which used this fast integer partition code to output the values in the proper order, but then I discovered more_itertools.powerset and for most uses it’s probably fine to just use that function like so:

from more_itertools import powerset
from numpy import ediff1d

def ps_sorter(tup):
    l = len(tup)
    d = ediff1d(tup).tolist()
    return l, d

ps = powerset([1,2,3,4])

ps = sorted(ps, key=ps_sorter)

for x in ps:
    print(x)

⇣

()
(1,)
(2,)
(3,)
(4,)
(1, 2)
(2, 3)
(3, 4)
(1, 3)
(2, 4)
(1, 4)
(1, 2, 3)
(2, 3, 4)
(1, 2, 4)
(1, 3, 4)
(1, 2, 3, 4)

I wrote some more involved code which will print the powerset nicely (see the repo for pretty printing functions I’ve not included here: print_partitions, print_partitions_by_length, and pprint_tuple).

• Repo: ordered-powerset, specifically pset_partitions.py

This is all pretty simple, but still might be useful if you want some code that’ll let you get straight to accessing the different levels of the powerset:

from itertools import permutations as permute
from numpy import cumsum

# http://jeromekelleher.net/generating-integer-partitions.html
# via
# https://stackoverflow.com/questions/10035752/elegant-python-code-for-integer-partitioning#comment25080713_10036764

def asc_int_partitions(n):
    a = [0 for i in range(n + 1)]
    k = 1
    y = n - 1
    while k != 0:
        x = a[k - 1] + 1
        k -= 1
        while 2 * x <= y:
            a[k] = x
            y -= x
            k += 1
        l = k + 1
        while x <= y:
            a[k] = x
            a[l] = y
            yield tuple(a[:k + 2])
            x += 1
            y -= 1
        a[k] = x + y
        y = x + y - 1
        yield tuple(a[:k + 1])

# https://stackoverflow.com/a/6285330/2668831
def uniquely_permute(iterable, enforce_sort=False, r=None):
    previous = tuple()
    if enforce_sort: # potential waste of effort (default: False)
        iterable = sorted(iterable)
    for p in permute(iterable, r):
        if p > previous:
            previous = p
            yield p

def sum_min(p):
    return sum(p), min(p)

def partitions_by_length(max_n, sorting=True, permuting=False):
    partition_dict = {0: ()}
    for n in range(1,max_n+1):
        partition_dict.setdefault(n, [])
        partitions = list(asc_int_partitions(n))
        for p in partitions:
            if permuting:
                perms = uniquely_permute(p)
                for perm in perms:
                    partition_dict.get(len(p)).append(perm)
            else:
                partition_dict.get(len(p)).append(p)
    if not sorting:
        return partition_dict
    for k in partition_dict:
        partition_dict.update({k: sorted(partition_dict.get(k), key=sum_min)})
    return partition_dict

def print_partitions_by_length(max_n, sorting=True, permuting=True):
    partition_dict = partitions_by_length(max_n, sorting=sorting, permuting=permuting)
    for k in partition_dict:
        if k == 0:
            print(tuple(partition_dict.get(k)), end="")
        for p in partition_dict.get(k):
            print(pprint_tuple(p), end=" ")
        print()
    return

def generate_powerset(items, subset_handler=tuple, verbose=False):
    """
    Generate the powerset of an iterable `items`.

    Handling of the elements of the iterable is by whichever function is passed as
    `subset_handler`, which must be able to handle the `None` value for the
    empty set. The function `string_handler` will join the elements of the subset
    with the empty string (useful when `items` is an iterable of `str` variables).
    """
    ps = {0: [subset_handler()]}
    n = len(items)
    p_dict = partitions_by_length(n-1, sorting=True, permuting=True)
    for p_len, parts in p_dict.items():
        ps.setdefault(p_len, [])
        if p_len == 0:
            # singletons
            for offset in range(n):
                subset = subset_handler([items[offset]])
                if verbose:
                    if offset > 0:
                        print(end=" ")
                    if offset == n - 1:
                        print(subset, end="\n")
                    else:
                        print(subset, end=",")
                ps.get(p_len).append(subset)
        for pcount, partition in enumerate(parts):
            distance = sum(partition)
            indices = (cumsum(partition)).tolist()
            for offset in range(n - distance):
                subset = subset_handler([items[offset]] + [items[offset:][i] for i in indices])
                if verbose:
                    if offset > 0:
                        print(end=" ")
                    if offset == n - distance - 1:
                        print(subset, end="\n")
                    else:
                        print(subset, end=",")
                ps.get(p_len).append(subset)
        if verbose and p_len < n-1:
            print()
    return ps

As an example, I wrote a CLI demo program which takes a string as a command line argument:

python string_powerset.py abcdef

⇣

a, b, c, d, e, f

ab, bc, cd, de, ef
ac, bd, ce, df
ad, be, cf
ae, bf
af

abc, bcd, cde, def
abd, bce, cdf
acd, bde, cef
abe, bcf
ade, bef
ace, bdf
abf
aef
acf
adf

abcd, bcde, cdef
abce, bcdf
abde, bcef
acde, bdef
abcf
abef
adef
abdf
acdf
acef

abcde, bcdef
abcdf
abcef
abdef
acdef

abcdef

If you want any specific length of subsets you can do it like this:

from itertools import combinations
someSet = {0, 1, 2, 3}
([x for i in range(len(someSet)+1) for x in combinations(someSet,i)])

More generally for arbitary length subsets you can modify the range arugment. The output is

[(), (0,), (1,), (2,), (3,), (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (0, 1, 2), (0, 1, 3), (0, 2, 3), (1, 2, 3), (0, 1, 2, 3)]

def powerset(some_set):
    res = [(a,b) for a in some_set for b in some_set]
    return res