A   set   is a collection (container) of certain values,
without any particular order, and no repeated values.

It corresponds with a finite set in mathematics.

A set can be implemented as an associative array (partial mapping)
in which the value of each key-value pair is ignored.

Given a set S, the power set (or powerset) of S, written P(S), or 2S, is the set of all subsets of S.



Task

By using a library or built-in set type, or by defining a set type with necessary operations, write a function with a set S as input that yields the power set 2S of S.



For example, the power set of     {1,2,3,4}     is

 {{}, {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}}.
For a set which contains n elements, the corresponding power set has 2n elements, including the edge cases of empty set.


The power set of the empty set is the set which contains itself (20 = 1):


 





P




{\displaystyle {\mathcal {P}}}

(



∅


{\displaystyle \varnothing }

) = { 



∅


{\displaystyle \varnothing }

 }

And the power set of the set which contains only the empty set, has two subsets, the empty set and the set which contains the empty set (21 = 2):


 





P




{\displaystyle {\mathcal {P}}}

({



∅


{\displaystyle \varnothing }

}) = { 



∅


{\displaystyle \varnothing }

, { 



∅


{\displaystyle \varnothing }

 } }


Extra credit:  Demonstrate that your language supports these last two powersets.




#include <iostream>
#include <set>
#include <vector>
#include <iterator>
#include <algorithm>
typedef std::set<int> set_type;
typedef std::set<set_type> powerset_type;
 
powerset_type powerset(set_type const& set)
{
  typedef set_type::const_iterator set_iter;
  typedef std::vector<set_iter> vec;
  typedef vec::iterator vec_iter;
 
  struct local
  {
    static int dereference(set_iter v) { return *v; }
  };
 
  powerset_type result;
 
  vec elements;
  do
  {
    set_type tmp;
    std::transform(elements.begin(), elements.end(),
                   std::inserter(tmp, tmp.end()),
                   local::dereference);
    result.insert(tmp);
    if (!elements.empty() && ++elements.back() == set.end())
    {
      elements.pop_back();
    }
    else
    {
      set_iter iter;
      if (elements.empty())
      {
        iter = set.begin();
      }
      else
      {
        iter = elements.back();
        ++iter;
      }
      for (; iter != set.end(); ++iter)
      {
        elements.push_back(iter);
      }
    }
  } while (!elements.empty());
 
  return result;
}
 
int main()
{
  int values[4] = { 2, 3, 5, 7 };
  set_type test_set(values, values+4);
 
  powerset_type test_powerset = powerset(test_set);
 
  for (powerset_type::iterator iter = test_powerset.begin();
       iter != test_powerset.end();
       ++iter)
  {
    std::cout << "{ ";
    char const* prefix = "";
    for (set_type::iterator iter2 = iter->begin();
         iter2 != iter->end();
         ++iter2)
    {
      std::cout << prefix << *iter2;
      prefix = ", ";
    }
    std::cout << " }\n";
  }
}