Questions tagged [powerset]

For a given set S: the powerset P(S) the set of all subsets of that set S.

• P(S) = {T: T ⊆ S) where T is a set.

Given a set with three elements {1, 2, 3}, the powerset P({1, 2, 3}) would contain the eight subsets of the set, including itself and the empty set:

• {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

For a set S of finite size n, the size of the powerset P(S) is 2ⁿ. The cardinality of S is always strictly less than the cardinality of P(S): the set of natural numbers N is countably infinite, but the set of the powerset of N is uncountable.

Algorithm to create a Powerset

Python offers itertools which can be used to create a powerset of a given set or list. Here is the documentation: https://docs.python.org/3/library/itertools.html#recipes

def powerset(s: set):
    s = list(s)
    ps = chain.from_iterable(combinations(s, r) for r in range(len(s) + 1))
    return list(ps) # set(ps) also works

Here is a test run:

s = {1,2,3}
ps = powerset(s)
# ps = [(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]

There exists an iterative method to create a powerset. The size of the powerset grows exponentially and reaches over two billion for sets of size larger than 32. For a set S of size n:

• Create an initially empty list L which will eventually represent the powerset
• For each element, give it an index, ranging from 0 to n-1.
  • Suppose we have {A, B, C}. Let the index of A be 0, B be 1, C be 2.
• Loop through all the numbers from 0 to 2ⁿ-1 and express each in binary form.
  • The loop counter i goes from 0 (000 in binary) to 7 (111 in binary).
• Create a subset T_i based on the binary number, which contains the elements such that the binary digit position of the elements' index for i is 1.
  • The ones digit is given position 0, the fours digit (third digit) is given position 2.
  • When we have 0, the subset T₀ is the empty set.
  • When we have 5, the binary digits are 101, the digits with position 2 and 0 are ones, and so the subset T₅ is {A, C}.
• Add that subset to the original list L.

Read more

• Wikipedia
• set, a mathematical object and a datastructure which represents a collection of unique objects.