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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 2^S, 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 2^S 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):

{\mathcal {P}}

(

\varnothing

) = {

\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):

{\mathcal {P}}

({

\varnothing

}) = {

\varnothing

, {

\varnothing

} }

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

11l

Translation of: Python

F list_powerset(lst)
   V result = [[Int]()]
   L(x) lst
      result.extend(result.map(subset -> subset [+] [@x]))
   R result

print(list_powerset([1, 2, 3]))

Output:

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

ABAP

This works for ABAP Version 7.40 and above

report z_powerset.

interface set.
  methods:
    add_element
      importing
        element_to_be_added type any
      returning
        value(new_set)      type ref to set,

    remove_element
      importing
        element_to_be_removed type any
      returning
        value(new_set)        type ref to set,

    contains_element
      importing
        element_to_be_found type any
      returning
        value(contains)     type abap_bool,

    get_size
      returning
        value(size) type int4,

    is_equal
      importing
        set_to_be_compared_with type ref to set
      returning
        value(equal)            type abap_bool,

    get_elements
      exporting
        elements type any table,

    stringify
      returning
        value(stringified_set) type string.
endinterface.


class string_set definition.
  public section.
    interfaces:
      set.


    methods:
      constructor
        importing
          elements type stringtab optional,

      build_powerset
        returning
          value(powerset) type ref to string_set.


  private section.
    data elements type stringtab.
endclass.


class string_set implementation.
  method constructor.
    loop at elements into data(element).
      me->set~add_element( element ).
    endloop.
  endmethod.


  method set~add_element.
    if not line_exists( me->elements[ table_line = element_to_be_added ] ).
      append element_to_be_added to me->elements.
    endif.

    new_set = me.
  endmethod.


  method set~remove_element.
    if line_exists( me->elements[ table_line = element_to_be_removed ] ).
      delete me->elements where table_line = element_to_be_removed.
    endif.

    new_set = me.
  endmethod.


  method set~contains_element.
    contains = cond abap_bool(
      when line_exists( me->elements[ table_line = element_to_be_found ] )
      then abap_true
      else abap_false ).
  endmethod.


  method set~get_size.
    size = lines( me->elements ).
  endmethod.


  method set~is_equal.
    if set_to_be_compared_with->get_size( ) ne me->set~get_size( ).
      equal = abap_false.

      return.
    endif.

    loop at me->elements into data(element).
      if not set_to_be_compared_with->contains_element( element ).
        equal = abap_false.

        return.
      endif.
    endloop.

    equal = abap_true.
  endmethod.


  method set~get_elements.
    elements = me->elements.
  endmethod.


  method set~stringify.
    stringified_set = cond string(
      when me->elements is initial
      then `∅`
      when me->elements eq value stringtab( ( `∅` ) )
      then `{ ∅ }`
      else reduce string(
        init result = `{ `
        for element in me->elements
        next result = cond string(
          when element eq ``
          then |{ result }∅, |
          when strlen( element ) eq 1 and element ne `∅`
          then |{ result }{ element }, |
          else |{ result }\{{ element }\}, | ) ) ).

    stringified_set = replace(
      val = stringified_set
      regex = `, $`
      with = ` }`).
  endmethod.


  method build_powerset.
    data(powerset_elements) = value stringtab( ( `` ) ).

    loop at me->elements into data(element).
      do lines( powerset_elements ) times.
        if powerset_elements[ sy-index ] ne `∅`.
          append |{ powerset_elements[ sy-index ] }{ element }| to powerset_elements.
        else.
          append element to powerset_elements.
        endif.
      enddo.
    endloop.

    powerset = new string_set( powerset_elements ).
  endmethod.
endclass.


start-of-selection.
  data(set1) = new string_set( ).
  data(set2) = new string_set( ).
  data(set3) = new string_set( ).

  write: |𝑷( { set1->set~stringify( ) } ) -> { set1->build_powerset( )->set~stringify( ) }|, /.

  set2->set~add_element( `∅` ).
  write: |𝑷( { set2->set~stringify( ) } ) -> { set2->build_powerset( )->set~stringify( ) }|, /.

  set3->set~add_element( `1` )->add_element( `2` )->add_element( `3` )->add_element( `3` )->add_element( `4`
    )->add_element( `4` )->add_element( `4` ).
  write: |𝑷( { set3->set~stringify( ) } ) -> { set3->build_powerset( )->set~stringify( ) }|, /.

Output:

𝑷( ∅ ) -> { ∅ }

𝑷( { ∅ } ) -> { ∅, {∅} }

𝑷( { 1, 2, 3, 4 } ) -> { ∅, 1, 2, {12}, 3, {13}, {23}, {123}, 4, {14}, {24}, {124}, {34}, {134}, {234}, {1234} }

A solution (without recursion) that prints the power set of the n arguments passed by the command line. The idea is that the i'th bit of a natural between 0 and $2^{n}-1$ indicates whether or not we should put the i'th element of the command line inside the set.

with Ada.Text_IO, Ada.Command_Line;
use Ada.Text_IO, Ada.Command_Line;
 
procedure powerset is
begin
	for set in 0..2**Argument_Count-1 loop
		Put ("{");			
		declare
			k : natural := set;
			first : boolean := true;
		begin
			for i in 1..Argument_Count loop
				if k mod 2 = 1 then
					Put ((if first then "" else ",") & Argument (i));
					first := false;
			  	end if;
				k := k / 2; -- we go to the next bit of "set"
			end loop;
		end;
		Put_Line("}");
	end loop;
end powerset;

Output:

>./powerset a b c d
{}
{a}
{b}
{a,b}
{c}
{a,c}
{b,c}
{a,b,c}
{d}
{a,d}
{b,d}
{a,b,d}
{c,d}
{a,c,d}
{b,c,d}
{a,b,c,d}

ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

Requires: ALGOL 68g mk14.1+

MODE MEMBER = INT;

PROC power set = ([]MEMBER s)[][]MEMBER:(
  [2**UPB s]FLEX[1:0]MEMBER r;
  INT upb r := 0;
  r[upb r +:= 1] := []MEMBER(());
  FOR i TO UPB s DO
    MEMBER e = s[i];
    FOR j TO upb r DO
      [UPB r[j] + 1]MEMBER x;
      x[:UPB x-1] := r[j];
      x[UPB x] := e; # append to the end of x #
      r[upb r +:= 1] := x # append to end of r #
    OD
  OD;
  r[upb r] := s;
  r    
);
# Example: #
test:(
  [][]MEMBER set = power set((1, 2, 4));
  FOR member TO UPB set DO
    INT upb = UPB set[member];
    FORMAT repr set = $"("f( upb=0 | $$ | $n(upb-1)(d", ")d$ )");"$;
    printf(($"set["d"] = "$,member, repr set, set[member],$l$))
  OD
)

Output:

set[1] = ();
set[2] = (1);
set[3] = (2);
set[4] = (1, 2);
set[5] = (4);
set[6] = (1, 4);
set[7] = (2, 4);
set[8] = (1, 2, 4);

APL

Works with: Dyalog APL

ps←(↓∘⍉(2/⍨≢)⊤(⍳2*≢))(/¨)⊂

Output:

      ps 1 2 3 4
┌─┬─┬───┬─┬───┬───┬─────┬─┬───┬───┬─────┬───┬─────┬─────┬───────┬┐
│4│3│3 4│2│2 4│2 3│2 3 4│1│1 4│1 3│1 3 4│1 2│1 2 4│1 2 3│1 2 3 4││
└─┴─┴───┴─┴───┴───┴─────┴─┴───┴───┴─────┴───┴─────┴─────┴───────┴┘

      ps ⍬
┌┐
││
└┘

      ps ,⊂⍬
┌──┬┐
│┌┐││
│││││
│└┘││
└──┴┘

AppleScript

Translation of: JavaScript

(functional composition examples)

Translation of: Haskell

-- POWER SET -----------------------------------------------------------------

-- powerset :: [a] -> [[a]]
on powerset(xs)
    script subSet
        on |λ|(acc, x)
            script cons
                on |λ|(y)
                    {x} & y
                end |λ|
            end script
            
            acc & map(cons, acc)
        end |λ|
    end script
    
    foldr(subSet, {{}}, xs)
end powerset


-- TEST ----------------------------------------------------------------------
on run
    script test
        on |λ|(x)
            set {setName, setMembers} to x
            {setName, powerset(setMembers)}
        end |λ|
    end script
    
    map(test, [¬
        ["Set [1,2,3]", {1, 2, 3}], ¬
        ["Empty set", {}], ¬
        ["Set containing only empty set", {{}}]])
    
    --> {{"Set [1,2,3]", {{}, {3}, {2}, {2, 3}, {1}, {1, 3}, {1, 2}, {1, 2, 3}}}, 
    -->  {"Empty set", {{}}}, 
    -->  {"Set containing only empty set", {{}, {{}}}}}
end run

-- GENERIC FUNCTIONS ---------------------------------------------------------

-- foldr :: (a -> b -> a) -> a -> [b] -> a
on foldr(f, startValue, xs)
    tell mReturn(f)
        set v to startValue
        set lng to length of xs
        repeat with i from lng to 1 by -1
            set v to |λ|(v, item i of xs, i, xs)
        end repeat
        return v
    end tell
end foldr

-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
    tell mReturn(f)
        set lng to length of xs
        set lst to {}
        repeat with i from 1 to lng
            set end of lst to |λ|(item i of xs, i, xs)
        end repeat
        return lst
    end tell
end map

-- Lift 2nd class handler function into 1st class script wrapper 
-- mReturn :: Handler -> Script
on mReturn(f)
    if class of f is script then
        f
    else
        script
            property |λ| : f
        end script
    end if
end mReturn

Output:

{{"Set [1,2,3]", {{}, {3}, {2}, {2, 3}, {1}, {1, 3}, {1, 2}, {1, 2, 3}}}, 
 {"Empty set", {{}}}, 
 {"Set containing only empty set", {{}, {{}}}}}

Arturo

print powerset [1 2 3 4]

Output:

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

ATS

(* ****** ****** *)
//
#include
"share/atspre_define.hats" // defines some names
#include
"share/atspre_staload.hats" // for targeting C
#include
"share/HATS/atspre_staload_libats_ML.hats" // for ...
//
(* ****** ****** *)
//
extern
fun
Power_set(xs: list0(int)): void
//
(* ****** ****** *)

// Helper: fast power function.
fun power(n: int, p: int): int =
	if p = 1 then n
	else if p = 0 then 1
	else if p % 2 = 0 then power(n*n, p/2)
	else n * power(n, p-1)

fun print_list(list: list0(int)): void =
  case+ list of
  | nil0() => println!(" ")
  | cons0(car, crd) =>
    let
      val () = begin print car; print ','; end
      val () = print_list(crd)
    in
    end

fun get_list_length(list: list0(int), length: int): int =
  case+ list of
  | nil0() => length
  | cons0(car, crd) => get_list_length(crd, length+1)


fun get_list_from_bit_mask(mask: int, list: list0(int), result: list0(int)): list0(int) =
  if mask = 0 then result
  else
    case+ list of
    | nil0() => result
    | cons0(car, crd) =>
      let
        val current: int = mask % 2
      in
        if current = 0 then
          get_list_from_bit_mask(mask >> 1, crd, result)
        else
          get_list_from_bit_mask(mask >> 1, crd, list0_cons(car, result))
      end


implement
Power_set(xs) = let
  val len: int = get_list_length(xs, 0)
  val pow: int = power(2, len)
  fun loop(mask: int, list: list0(int)): void =
    if mask > 0 && mask >= pow then ()
    else
      let
        val () = print_list(get_list_from_bit_mask(mask, list, list0_nil()))
      in
        loop(mask+1, list)
      end
  in
    loop(0, xs)
  end

(* ****** ****** *)

implement
main0() =
let
  val xs: list0(int) = cons0(1, list0_pair(2, 3))
in
  Power_set(xs)
end (* end of [main0] *)

(* ****** ****** *)

AutoHotkey

ahk discussion

a = 1,a,--             ; elements separated by commas
StringSplit a, a, `,   ; a0 = #elements, a1,a2,... = elements of the set

t = {
Loop % (1<<a0) {       ; generate all 0-1 sequences
   x := A_Index-1
   Loop % a0
      t .= (x>>A_Index-1) & 1 ? a%A_Index% "," : ""
   t .= "}`n{"         ; new subsets in new lines
}
MsgBox % RegExReplace(SubStr(t,1,StrLen(t)-1),",}","}")

AWK

cat power_set.awk
#!/usr/local/bin/gawk -f

# User defined function
function tochar(l,n,	r) {
 while (l) { n--; if (l%2 != 0) r = r sprintf(" %c ",49+n); l = int(l/2) }; return r
}

# For each input
{ for (i=0;i<=2^NF-1;i++) if (i == 0) printf("empty\n"); else printf("(%s)\n",tochar(i,NF)) }

Output:

$ gawk -f power_set.awk 
1 2 3 4
empty
( 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 )

BASIC

BBC BASIC

The elements of a set are represented as the bits in an integer (hence the maximum size of set is 32).

      DIM list$(3) : list$() = "1", "2", "3", "4"
      PRINT FNpowerset(list$())
      END
      
      DEF FNpowerset(list$())
      IF DIM(list$(),1) > 31 ERROR 100, "Set too large to represent as integer"
      LOCAL i%, j%, s$
      s$ = "{"
      FOR i% = 0 TO (2 << DIM(list$(),1)) - 1
        s$ += "{"
        FOR j% = 0 TO DIM(list$(),1)
          IF i% AND (1 << j%) s$ += list$(j%) + ","
        NEXT
        IF RIGHT$(s$) = "," s$ = LEFT$(s$)
        s$ += "},"
      NEXT i%
      = LEFT$(s$) + "}"

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}}

BQN

P ← (⥊·↕2⥊˜≠)/¨<

Output:

   P 1‿2‿3‿4‿5
⟨ ⟨⟩ ⟨ 5 ⟩ ⟨ 4 ⟩ ⟨ 4 5 ⟩ ⟨ 3 ⟩ ⟨ 3 5 ⟩ ⟨ 3 4 ⟩ ⟨ 3 4 5 ⟩ ⟨ 2 ⟩ ⟨ 2 5 ⟩ ⟨ 2 4 ⟩ ⟨ 2 4 5 ⟩ ⟨ 2 3 ⟩ ⟨ 2 3 5 ⟩ ⟨ 2 3 4 ⟩ ⟨ 2 3 4 5 ⟩ ⟨ 1 ⟩ ⟨ 1 5 ⟩ ⟨ 1 4 ⟩ ⟨ 1 4 5 ⟩ ⟨ 1 3 ⟩ ⟨ 1 3 5 ⟩ ⟨ 1 3 4 ⟩ ⟨ 1 3 4 5 ⟩ ⟨ 1 2 ⟩ ⟨ 1 2 5 ⟩ ⟨ 1 2 4 ⟩ ⟨ 1 2 4 5 ⟩ ⟨ 1 2 3 ⟩ ⟨ 1 2 3 5 ⟩ ⟨ 1 2 3 4 ⟩ ⟨ 1 2 3 4 5 ⟩ ⟩

Bracmat

( ( powerset
  =   done todo first
    .   !arg:(?done.?todo)
      & (   !todo:%?first ?todo
          & (powerset$(!done !first.!todo),powerset$(!done.!todo))
        | !done
        )
  )
& out$(powerset$(.1 2 3 4))
);

Output:

Burlesque

blsq ) {1 2 3 4}R@
{{} {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}}

C

#include <stdio.h>

struct node {
	char *s;
	struct node* prev;
};

void powerset(char **v, int n, struct node *up)
{
	struct node me;

	if (!n) {
		putchar('[');
		while (up) {
			printf(" %s", up->s);
			up = up->prev;
		}
		puts(" ]");
	} else {
		me.s = *v;
		me.prev = up;
		powerset(v + 1, n - 1, up);
		powerset(v + 1, n - 1, &me);
	}
}

int main(int argc, char **argv)
{
	powerset(argv + 1, argc - 1, 0);
	return 0;
}

Output:

% ./a.out 1 2 3
[ ]
[ 3 ]
[ 2 ]
[ 3 2 ]
[ 1 ]
[ 3 1 ]
[ 2 1 ]
[ 3 2 1 ]

C#

public IEnumerable<IEnumerable<T>> GetPowerSet<T>(List<T> list)
{
    return from m in Enumerable.Range(0, 1 << list.Count)
                  select
                      from i in Enumerable.Range(0, list.Count)
                      where (m & (1 << i)) != 0
                      select list[i];
}

public void PowerSetofColors()
{
    var colors = new List<KnownColor> { KnownColor.Red, KnownColor.Green, 
        KnownColor.Blue, KnownColor.Yellow };
    
    var result = GetPowerSet(colors);
    
    Console.Write( string.Join( Environment.NewLine, 
        result.Select(subset => 
            string.Join(",", subset.Select(clr => clr.ToString()).ToArray())).ToArray()));
}

Output:

  Red
  Green
  Red,Green
  Blue
  Red,Blue
  Green,Blue
  Red,Green,Blue
  Yellow
  Red,Yellow
  Green,Yellow
  Red,Green,Yellow
  Blue,Yellow
  Red,Blue,Yellow
  Green,Blue,Yellow
  Red,Green,Blue,Yellow

An alternative implementation for an arbitrary number of elements:

  public IEnumerable<IEnumerable<T>> GetPowerSet<T>(IEnumerable<T> input) {
    var seed = new List<IEnumerable<T>>() { Enumerable.Empty<T>() }
      as IEnumerable<IEnumerable<T>>;

    return input.Aggregate(seed, (a, b) =>
      a.Concat(a.Select(x => x.Concat(new List<T>() { b }))));
  }

Non-recursive version

  using System;
  class Powerset
  {
    static int count = 0, n = 4;
    static int [] buf = new int [n];
  
    static void Main()
    {
  	int ind = 0;
  	int n_1 = n - 1;
  	for (;;)
  	{
  	  for (int i = 0; i <= ind; ++i) Console.Write("{0, 2}", buf [i]);
  	  Console.WriteLine();
  	  count++;
  
  	  if (buf [ind] < n_1) { ind++; buf [ind] = buf [ind - 1] + 1; }
  	  else if (ind > 0) { ind--; buf [ind]++; }
  	  else break;
  	}
  	Console.WriteLine("n=" + n + "   count=" + count);
    }
  }

Recursive version

using System;
class Powerset
{
  static int n = 4;
  static int [] buf = new int [n];

  static void Main()
  {
    rec(0, 0);
  }

  static void rec(int ind, int begin)
  {
    for (int i = begin; i < n; i++)
    {
      buf [ind] = i;
      for (int j = 0; j <= ind; j++) Console.Write("{0, 2}", buf [j]);
      Console.WriteLine();
      rec(ind + 1, buf [ind] + 1);
    }
  }
}

C++

Non-recursive version

#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";
  }
}

Output:

{  }
{ 2 }
{ 2, 3 }
{ 2, 3, 5 }
{ 2, 3, 5, 7 }
{ 2, 3, 7 }
{ 2, 5 }
{ 2, 5, 7 }
{ 2, 7 }
{ 3 }
{ 3, 5 }
{ 3, 5, 7 }
{ 3, 7 }
{ 5 }
{ 5, 7 }
{ 7 }

C++14 version

This simplified version has identical output to the previous code.

#include <set>
#include <iostream>

template <class S>
auto powerset(const S& s)
{
    std::set<S> ret;
    ret.emplace();
    for (auto&& e: s) {
        std::set<S> rs;
        for (auto x: ret) {
            x.insert(e);
            rs.insert(x);
        }
        ret.insert(begin(rs), end(rs));
    }
    return ret;
}

int main()
{
    std::set<int> s = {2, 3, 5, 7};
    auto pset = powerset(s);

    for (auto&& subset: pset) {
        std::cout << "{ ";
        char const* prefix = "";
        for (auto&& e: subset) {
            std::cout << prefix << e;
            prefix = ", ";
        }
        std::cout << " }\n";
    }
}

Recursive version

#include <iostream>
#include <set>

template<typename Set> std::set<Set> powerset(const Set& s, size_t n)
{
    typedef typename Set::const_iterator SetCIt;
    typedef typename std::set<Set>::const_iterator PowerSetCIt;
    std::set<Set> res;
    if(n > 0) {
        std::set<Set> ps = powerset(s, n-1);
        for(PowerSetCIt ss = ps.begin(); ss != ps.end(); ss++)
            for(SetCIt el = s.begin(); el != s.end(); el++) {
                Set subset(*ss);
                subset.insert(*el);
                res.insert(subset);
            }
        res.insert(ps.begin(), ps.end());
    } else
        res.insert(Set());
    return res;
}
template<typename Set> std::set<Set> powerset(const Set& s)
{
    return powerset(s, s.size());
}

Clojure

(use '[clojure.math.combinatorics :only [subsets] ])

(def S #{1 2 3 4})

user> (subsets S)
(() (1) (2) (3) (4) (1 2) (1 3) (1 4) (2 3) (2 4) (3 4) (1 2 3) (1 2 4) (1 3 4) (2 3 4) (1 2 3 4))

Alternate solution, with no dependency on third-party library:

(defn powerset [coll] 
  (reduce (fn [a x]
            (into a (map #(conj % x)) a))
          #{#{}} coll))

(powerset #{1 2 3})

#{#{} #{1} #{2} #{1 2} #{3} #{1 3} #{2 3} #{1 2 3}}

Using bit-test: see: https://clojuredocs.org/clojure.core/bit-test#example-5d401face4b0ca44402ef78b

(defn powerset [coll]
  (let [cnt (count coll)
        bits (Math/pow 2 cnt)]
    (for [i (range bits)]
      (for [j (range i)
            :while (< j cnt)
            :when (bit-test i j)]
         (nth coll j)))))

(powerset [1 2 3])

(() (1) (2) (1 2) (3) (1 3) (2 3) (1 2 3))

CoffeeScript

print_power_set = (arr) ->
  console.log "POWER SET of #{arr}"
  for subset in power_set(arr)
    console.log subset
    
power_set = (arr) ->  
  result = []
  binary = (false for elem in arr)
  n = arr.length
  while binary.length <= n
    result.push bin_to_arr binary, arr
    i = 0
    while true
      if binary[i]
        binary[i] = false
        i += 1
      else
        binary[i] = true
        break
    binary[i] = true
  result

bin_to_arr = (binary, arr) ->
  (arr[i] for i of binary when binary[arr.length - i  - 1])

print_power_set []
print_power_set [4, 2, 1] 
print_power_set ['dog', 'c', 'b', 'a']

Output:

> coffee power_set.coffee 
POWER SET of 
[]
POWER SET of 4,2,1
[]
[ 1 ]
[ 2 ]
[ 2, 1 ]
[ 4 ]
[ 4, 1 ]
[ 4, 2 ]
[ 4, 2, 1 ]
POWER SET of dog,c,b,a
[]
[ 'a' ]
[ 'b' ]
[ 'b', 'a' ]
[ 'c' ]
[ 'c', 'a' ]
[ 'c', 'b' ]
[ 'c', 'b', 'a' ]
[ 'dog' ]
[ 'dog', 'a' ]
[ 'dog', 'b' ]
[ 'dog', 'b', 'a' ]
[ 'dog', 'c' ]
[ 'dog', 'c', 'a' ]
[ 'dog', 'c', 'b' ]
[ 'dog', 'c', 'b', 'a' ]

ColdFusion

Port from the JavaScript version, compatible with ColdFusion 8+ or Railo 3+

public array function powerset(required array data)
{
  var ps = [""];
  var d = arguments.data;
  var lenData = arrayLen(d);
  var lenPS = 0;
  for (var i=1; i LTE lenData; i++)
  {
    lenPS = arrayLen(ps);
    for (var j = 1; j LTE lenPS; j++)
    {
      arrayAppend(ps, listAppend(ps[j], d[i]));
    }
  }
  return ps;
}

var res = 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"]

Common Lisp

(defun powerset (s) 
  (if s (mapcan (lambda (x) (list (cons (car s) x) x)) 
                (powerset (cdr s))) 
      '(())))

Output:

> (powerset '(l i s p))
((L I S P) (I S P) (L S P) (S P) (L I P) (I P) (L P) (P) (L I S) (I S) (L S) (S) (L I) (I) (L) NIL)

(defun power-set (s)
  (reduce #'(lambda (item ps)
              (append (mapcar #'(lambda (e) (cons item e))
                              ps)
                      ps))
          s
          :from-end t
          :initial-value '(())))

Output:

>(power-set '(1 2 3))
((1 2 3) (1 2) (1 3) (1) (2 3) (2) (3) NIL)

Alternate, more recursive (same output):

(defun powerset (l)
  (if (null l)
      (list nil)
      (let ((prev (powerset (cdr l))))
	(append (mapcar #'(lambda (elt) (cons (car l) elt)) prev)
		prev))))

Imperative-style using LOOP:

(defun powerset (xs)
  (loop for i below (expt 2 (length xs)) collect
       (loop for j below i for x in xs if (logbitp j i) collect x)))

Output:

>(powerset '(1 2 3)
(NIL (1) (2) (1 2) (3) (1 3) (2 3) (1 2 3))

Yet another imperative solution, this time with dolist.

(defun power-set (list)
    (let ((pow-set (list nil)))
      (dolist (element (reverse list) pow-set)
        (dolist (set pow-set)
          (push (cons element set) pow-set)))))

Output:

>(power-set '(1 2 3))
((1) (1 3) (1 2 3) (1 2) (2) (2 3) (3) NIL)

D

This implementation defines a range which *lazily* enumerates the power set.

import std.algorithm;
import std.range;

auto powerSet(R)(R r)
{
	return
		(1L<<r.length)
		.iota
		.map!(i =>
			r.enumerate
			.filter!(t => (1<<t[0]) & i)
			.map!(t => t[1])
		);
}

unittest
{
	int[] emptyArr;
	assert(emptyArr.powerSet.equal!equal([emptyArr]));
	assert(emptyArr.powerSet.powerSet.equal!(equal!equal)([[], [emptyArr]]));
}

void main(string[] args)
{
	import std.stdio;
	args[1..$].powerSet.each!writeln;
}

An alternative version, which implements the range construct from scratch:

import std.range;

struct PowerSet(R)
	if (isRandomAccessRange!R)
{
	R r;
	size_t position;

	struct PowerSetItem
	{
		R r;
		size_t position;

		private void advance()
		{
			while (!(position & 1))
			{
				r.popFront();
				position >>= 1;
			}
		}

		@property bool empty() { return position == 0; }
		@property auto front()
		{
			advance();
			return r.front;
		}
		void popFront()
		{
			advance();
			r.popFront();
			position >>= 1;
		}
	}

	@property bool empty() { return position == (1 << r.length); }
	@property PowerSetItem front() { return PowerSetItem(r.save, position); }
	void popFront() { position++; }
}

auto powerSet(R)(R r) { return PowerSet!R(r); }

Output:

$ rdmd powerset a b c
[]
["a"]
["b"]
["a", "b"]
["c"]
["a", "c"]
["b", "c"]
["a", "b", "c"]

Alternative: using folds

An almost verbatim translation of the Haskell code in D.

Since D doesn't foldr, I've also copied Haskell's foldr implementation here.

Main difference from the Haskell:

It isn't lazy (but it could be made so by implementing this as a generator)

Main differences from the version above:

It isn't lazy
It doesn't rely on integer bit fiddling, so it should work on arrays larger than size_t.

// Haskell definition:
// foldr f z []     = z
// foldr f z (x:xs) = x `f` foldr f z xs
S foldr(T, S)(S function(T, S) f, S z, T[] rest) {
    return (rest.length == 0) ? z : f(rest[0], foldr(f, z, rest[1..$]));
}

// Haskell definition:
//powerSet = foldr (\x acc -> acc ++ map (x:) acc) [[]]
T[][] powerset(T)(T[] set) {
    import std.algorithm;
    import std.array;
    // Note: The types before x and acc aren't needed, so this could be made even more concise, but I think it helps 
    // to make the algorithm slightly clearer.
    return foldr( (T x, T[][] acc) => acc ~ acc.map!(accx => x ~ accx).array , [[]], set );
}

Déjà Vu

In Déjà Vu, sets are dictionaries with all values true and the default set to false.

powerset s:
	local :out [ set{ } ]
	for value in keys s:
		for subset in copy out:
			local :subset+1 copy subset
			set-to subset+1 value true
			push-to out subset+1
	out

!. powerset set{ 1 2 3 4 }

Output:

[ set{ } set{ 4 } set{ 3 4 } set{ 3 } set{ 2 3 } set{ 2 3 4 } set{ 2 4 } set{ 2 } set{ 1 2 } set{ 1 2 4 } set{ 1 2 3 4 } set{ 1 2 3 } set{ 1 3 } set{ 1 3 4 } set{ 1 4 } set{ 1 } ]

Delphi

Library: System.SysUtils

Translation of: C#

program Power_set;

{$APPTYPE CONSOLE}

uses
  System.SysUtils;

const
  n = 4;

var
  buf: TArray<Integer>;

procedure rec(ind, bg: Integer);
begin
  for var i := bg to n - 1 do
  begin
    buf[ind] := i;
    for var j := 0 to ind do
      write(buf[j]: 2);
    writeln;
    rec(ind + 1, buf[ind] + 1);
  end;
end;

begin
  SetLength(buf, n);
  rec(0,0);
  {$IFNDEF UNIX}readln;{$ENDIF}
end.

Dyalect

Translation of: C#

let n = 4
let buf = Array.Empty(n)
 
func rec(idx, begin) {
    for i in begin..<n {
        buf[idx] = i
        for j in 0..idx {
            print("{0, 2}".Format(buf[j]), terminator: "")
        }
        print("")
        rec(idx + 1, buf[idx] + 1)
    }
}

rec(0, 0)

E

pragma.enable("accumulator")

def powerset(s) {
  return accum [].asSet() for k in 0..!2**s.size() {
    _.with(accum [].asSet() for i ? ((2**i & k) > 0) => elem in s {
      _.with(elem)
    })
  }
}

It would also be possible to define an object which is the powerset of a provided set without actually instantiating all of its members immediately.

EchoLisp

(define (set-cons a A) 
    (make-set (cons a A)))

(define (power-set e)
    (cond ((null? e)
       (make-set (list ∅)))
    (else (let [(ps (power-set (cdr e)))]
       (make-set
       (append ps (map set-cons (circular-list (car e)) ps)))))))

(define B (make-set ' ( 🍎 🍇 🎂 🎄 )))
(power-set B)
    → { ∅ { 🍇 } { 🍇 🍎 } { 🍇 🍎 🎂 } { 🍇 🍎 🎂 🎄 } { 🍇 🍎 🎄 } { 🍇 🎂 } { 🍇 🎂 🎄 }
      { 🍇 🎄 } { 🍎 } { 🍎 🎂 } { 🍎 🎂 🎄 } { 🍎 🎄 } { 🎂 } { 🎂 🎄 } { 🎄 } }

;; The Von Neumann universe

(define V0 (power-set null)) ;; null and ∅ are the same
       → { ∅ }
(define V1 (power-set V0))
       → { ∅ { ∅ } }
(define V2 (power-set V1))
       → { ∅ { ∅ } { ∅ { ∅ } } { { ∅ } } }
(define V3 (power-set V2))
       → { ∅ { ∅ } { ∅ { ∅ } } …🔃 )
(length V3) → 16
(define V4 (power-set V3))
(length V4)  → 65536
;; length V5 = 2^65536 : out of bounds

Elixir

Translation of: Erlang

defmodule RC do
  use Bitwise
  def powerset1(list) do
    n = length(list)
    max = round(:math.pow(2,n))
    for i <- 0..max-1, do: (for pos <- 0..n-1, band(i, bsl(1, pos)) != 0, do: Enum.at(list, pos) )
  end
  
  def powerset2([]), do: [[]]
  def powerset2([h|t]) do
    pt = powerset2(t)
    (for x <- pt, do: [h|x]) ++ pt
  end
  
  def powerset3([]), do: [[]]
  def powerset3([h|t]) do
    pt = powerset3(t)
    powerset3(h, pt, pt)
  end
  
  defp powerset3(_, [], acc), do: acc
  defp powerset3(x, [h|t], acc), do: powerset3(x, t, [[x|h] | acc])
end

IO.inspect RC.powerset1([1,2,3])
IO.inspect RC.powerset2([1,2,3])
IO.inspect RC.powerset3([1,2,3])
IO.inspect RC.powerset1([])
IO.inspect RC.powerset1(["one"])

Output:

[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]
[[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2], [3], []]
[[1], [1, 3], [1, 2, 3], [1, 2], [2], [2, 3], [3], []]
[[]]
[[], ["one"]]

Erlang

Generates all subsets of a list with the help of binary:

For [1 2 3]:
    [     ] | 0 0 0 | 0
    [    3] | 0 0 1 | 1
    [  2  ] | 0 1 0 | 2
    [  2 3] | 0 1 1 | 3
    [1    ] | 1 0 0 | 4
    [1   3] | 1 0 1 | 5
    [1 2  ] | 1 1 0 | 6
    [1 2 3] | 1 1 1 | 7
    ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

powerset(Lst) ->
    N = length(Lst),
    Max = trunc(math:pow(2,N)),
    [[lists:nth(Pos+1,Lst) || Pos <- lists:seq(0,N-1), I band (1 bsl Pos) =/= 0]
      || I <- lists:seq(0,Max-1)].

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]]

Alternate shorter and more efficient version:

powerset([]) -> [[]];
powerset([H|T]) -> PT = powerset(T),
  [ [H|X] || X <- PT ] ++ PT.

or even more efficient version:

powerset([]) -> [[]];
powerset([H|T]) -> PT = powerset(T),
  powerset(H, PT, PT).

powerset(_, [], Acc) -> Acc;
powerset(X, [H|T], Acc) -> powerset(X, T, [[X|H]|Acc]).

F#

almost exact copy of OCaml version

let subsets xs = List.foldBack (fun x rest -> rest @ List.map (fun ys -> x::ys) rest) xs [[]]

alternatively with list comprehension

let rec pow = 
    function
    | [] -> [[]]
    | x::xs -> [for i in pow xs do yield! [i;x::i]]

Factor

We use hash sets, denoted by HS{ } brackets, for our sets. members converts from a set to a sequence, and <hash-set> converts back.

USING: kernel prettyprint sequences arrays sets hash-sets ;
IN: powerset

: add ( set elt -- newset ) 1array <hash-set> union ;
: powerset ( set -- newset ) members { HS{ } } [ dupd [ add ] curry map append ] reduce <hash-set> ;

Usage:

( scratchpad ) HS{ 1 2 3 4 } powerset .
HS{
    HS{ 1 2 3 4 }
    HS{ 1 2 }
    HS{ 1 3 }
    HS{ 2 3 }
    HS{ 1 2 3 }
    HS{ 1 4 }
    HS{ 2 4 }
    HS{ }
    HS{ 1 }
    HS{ 2 }
    HS{ 3 }
    HS{ 4 }
    HS{ 1 2 4 }
    HS{ 3 4 }
    HS{ 1 3 4 }
    HS{ 2 3 4 }
}

Forth

Works with: 4tH version 3.61.0

.

Translation of: C

: ?print dup 1 and if over args type space then ;
: .set begin dup while ?print >r 1+ r> 1 rshift repeat drop drop ;
: .powerset 0 do ." ( " 1 i .set ." )" cr loop ;
: check-none dup 2 < abort" Usage: powerset [val] .. [val]" ;
: check-size dup /cell 8 [*] >= abort" Set too large" ;
: powerset 1 argn check-none check-size 1- lshift .powerset ;

powerset

Output:

$ 4th cxq powerset.4th 1 2 3 4
( )
( 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 )

FreeBASIC

Los elementos de un conjunto se representan como bits en un número entero (por lo tanto, el tamaño máximo del conjunto es 32).

Function ConjuntoPotencia(set() As String) As String
    If Ubound(set,1) > 31 Then Print "Set demasiado grande para representarlo como un entero" : Exit Function
    If Ubound(set,1) < 0 Then Print "{}": Exit Function ' Set vacío
    Dim As Integer i, j
    Dim As String s = "{"
    For i = Lbound(set) To (2 Shl Ubound(set,1)) - 1
        s += "{"
        For j = Lbound(set) To Ubound(set,1)
            If i And (1 Shl j) Then s += set(j) + ","
        Next j
        If Right(s,1) = "," Then s = Left(s,Len(s)-1)
        s += "},"
    Next i    
    Return Left(s,Len(s)-1) + "}"
End Function

Print "El power set de [1, 2, 3, 4] comprende:"
Dim As String set(3) = {"1", "2", "3", "4"}
Print ConjuntoPotencia(set())
Print !"\nEl power set de [] comprende:"
Dim As String set0()
Print ConjuntoPotencia(set0())
Print "El power set de [[]] comprende:"
Dim As String set1(0) = {""}
Print ConjuntoPotencia(set1())
Sleep

Output:

El power set de [1, 2, 3, 4] comprende:
{{},{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}}

El power set de [] comprende:
{}

El power set de [[]] comprende:
{{},{}}

Frink

Frink's set and array classes have built-in subsets[] methods that return all subsets. If called with an array, the results are arrays. If called with a set, the results are sets.

a = new set[1,2,3,4]  
a.subsets[]

FunL

FunL uses Scala type scala.collection.immutable.Set as it's set type, which has a built-in method subsets returning an (Scala) iterator over subsets.

def powerset( s ) = s.subsets().toSet()

The powerset function could be implemented in FunL directly as:

def
  powerset( {} ) = {{}}
  powerset( s ) =
    acc = powerset( s.tail() )
    acc + map( x -> {s.head()} + x, acc )

or, alternatively as:

import lists.foldr

def powerset( s ) = foldr( \x, acc -> acc + map( a -> {x} + a, acc), {{}}, s )

println( powerset({1, 2, 3, 4}) )

Output:

{{}, {4}, {1, 2}, {1, 3}, {2, 3, 4}, {3}, {1, 2, 3, 4}, {1, 4}, {1, 2, 3}, {2}, {1, 2, 4}, {1}, {3, 4}, {2, 3}, {2, 4}, {1, 3, 4}}

Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website.

In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.

Solution

No program needed. Power set is intrinsically supported in Fōrmulæ.

Case 1. Power set of the set {1, 2, 3, 4}

Case 2. The power set of the empty set is the set which contains itself.

Case 3. 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

Case 4. Even when it is intrinsically supported, a program can be written:

GAP

# Built-in
Combinations([1, 2, 3]);                                           
# [ [  ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ], [ 2 ], [ 2, 3 ], [ 3 ] ]

# Note that it handles duplicates
Combinations([1, 2, 3, 1]);
# [ [  ], [ 1 ], [ 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 2, 3 ], [ 1, 1, 3 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ], 
#   [ 2 ], [ 2, 3 ], [ 3 ] ]

Go

No native set type in Go. While the associative array trick mentioned in the task description works well in Go in most situations, it does not work here because we need sets of sets, and converting a general set to a hashable value for a map key is non-trivial.

Instead, this solution uses a simple (non-associative) slice as a set representation. To ensure uniqueness, the element interface requires an equality method, which is used by the set add method. Adding elements with the add method ensures the uniqueness property.

While the "add" and "has" methods make a usable set type, the power set method implemented here computes a result directly without using the add method. The algorithm ensures that the result will be a valid set as long as the input is a valid set. This allows the more efficient append function to be used.

package main

import (
    "fmt"
    "strconv"
    "strings"
)

// types needed to implement general purpose sets are element and set

// element is an interface, allowing different kinds of elements to be
// implemented and stored in sets.
type elem interface {
    // an element must be distinguishable from other elements to satisfy
    // the mathematical definition of a set.  a.eq(b) must give the same
    // result as b.eq(a).
    Eq(elem) bool
    // String result is used only for printable output.  Given a, b where
    // a.eq(b), it is not required that a.String() == b.String().
    fmt.Stringer
}

// integer type satisfying element interface
type Int int

func (i Int) Eq(e elem) bool {
    j, ok := e.(Int)
    return ok && i == j
}

func (i Int) String() string {
    return strconv.Itoa(int(i))
}

// a set is a slice of elem's.  methods are added to implement
// the element interface, to allow nesting.
type set []elem

// uniqueness of elements can be ensured by using add method
func (s *set) add(e elem) {
    if !s.has(e) {
        *s = append(*s, e)
    }
}

func (s *set) has(e elem) bool {
    for _, ex := range *s {
        if e.Eq(ex) {
            return true
        }
    }
    return false
}

func (s set) ok() bool {
    for i, e0 := range s {
        for _, e1 := range s[i+1:] {
            if e0.Eq(e1) {
                return false
            }
        }
    }
    return true
}

// elem.Eq
func (s set) Eq(e elem) bool {
    t, ok := e.(set)
    if !ok {
        return false
    }
    if len(s) != len(t) {
        return false
    }
    for _, se := range s {
        if !t.has(se) {
            return false
        }
    }
    return true
}

// elem.String
func (s set) String() string {
    if len(s) == 0 {
        return "∅"
    }
    var buf strings.Builder
    buf.WriteRune('{')
    for i, e := range s {
        if i > 0 {
            buf.WriteRune(',')
        }
        buf.WriteString(e.String())
    }
    buf.WriteRune('}')
    return buf.String()
}

// method required for task
func (s set) powerSet() set {
    r := set{set{}}
    for _, es := range s {
        var u set
        for _, er := range r {
            er := er.(set)
            u = append(u, append(er[:len(er):len(er)], es))
        }
        r = append(r, u...)
    }
    return r
}

func main() {
    var s set
    for _, i := range []Int{1, 2, 2, 3, 4, 4, 4} {
        s.add(i)
    }
    fmt.Println("      s:", s, "length:", len(s))
    ps := s.powerSet()
    fmt.Println("   𝑷(s):", ps, "length:", len(ps))

    fmt.Println("\n(extra credit)")
    var empty set
    fmt.Println("  empty:", empty, "len:", len(empty))
    ps = empty.powerSet()
    fmt.Println("   𝑷(∅):", ps, "len:", len(ps))
    ps = ps.powerSet()
    fmt.Println("𝑷(𝑷(∅)):", ps, "len:", len(ps))

    fmt.Println("\n(regression test for earlier bug)")
    s = set{Int(1), Int(2), Int(3), Int(4), Int(5)}
    fmt.Println("      s:", s, "length:", len(s), "ok:", s.ok())
    ps = s.powerSet()
    fmt.Println("   𝑷(s):", "length:", len(ps), "ok:", ps.ok())
    for _, e := range ps {
        if !e.(set).ok() {
            panic("invalid set in ps")
        }
    }
}

Output:

      s: {1,2,3,4} length: 4
   𝑷(s): {∅,{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}} length: 16

(extra credit)
  empty: ∅ len: 0
   𝑷(∅): {∅} len: 1
𝑷(𝑷(∅)): {∅,{∅}} len: 2

(regression test for earlier bug)
      s: {1,2,3,4,5} length: 5 ok: true
   𝑷(s): length: 32 ok: true

Groovy

Builds on the Combinations solution. Sets are not a "natural" collection type in Groovy. Lists are much more richly supported. Thus, this solution is liberally sprinkled with coercion from Set to List and from List to Set.

def powerSetRec(head, tail) {
    if (!tail) return [head]
    powerSetRec(head, tail.tail()) + powerSetRec(head + [tail.head()], tail.tail())
}

def powerSet(set) { powerSetRec([], set as List) as Set}

Test program:

def vocalists = [ 'C', 'S', 'N', 'Y' ] as Set
println vocalists
println powerSet(vocalists)

Output:

[C, S, N, Y]
[[], [Y], [N], [N, Y], [S], [S, Y], [S, N], [S, N, Y], [C], [C, Y], [C, N], [C, N, Y], [C, S], [C, S, Y], [C, S, N], [C, S, N, Y]]

Haskell

import Data.Set
import Control.Monad

powerset :: Ord a => Set a -> Set (Set a)
powerset = fromList . fmap fromList . listPowerset . toList

listPowerset :: [a] -> [[a]]
listPowerset = filterM (const [True, False])

listPowerset describes the result as all possible (using the list monad) filterings (using filterM) of the input list, regardless (using const) of each item's value. powerset simply converts the input and output from lists to sets.

Alternate Solution

powerset [] = [[]]
powerset (head:tail) = acc ++ map (head:) acc where acc = powerset tail

or

powerSet :: [a] -> [[a]]
powerSet = foldr (\x acc -> acc ++ map (x:) acc) [[]]

which could also be understood, in point-free terms, as:

powerSet :: [a] -> [[a]]
powerSet = foldr ((mappend <*>) . fmap . (:)) (pure [])

Examples:

*Main> listPowerset [1,2,3]
[[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]
*Main> powerset (Data.Set.fromList [1,2,3])
{{},{1},{1,2},{1,2,3},{1,3},{2},{2,3},{3}}

Works with: GHC version 6.10

Prelude> import Data.List
Prelude Data.List> subsequences [1,2,3]
[[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]

Alternate solution

A method using only set operations and set mapping is also possible. Ideally, Set would be defined as a Monad, but that's impossible given the constraint that the type of inputs to Set.map (and a few other functions) be ordered.

import qualified Data.Set as Set
type Set=Set.Set
unionAll :: (Ord a) => Set (Set a) -> Set a
unionAll = Set.fold Set.union Set.empty

--slift is the analogue of liftA2 for sets.
slift :: (Ord a, Ord b, Ord c) => (a->b->c) -> Set a -> Set b -> Set c
slift f s0 s1 = unionAll (Set.map (\e->Set.map (f e) s1) s0)

--a -> {{},{a}}
makeSet :: (Ord a) => a -> Set (Set a)
makeSet = (Set.insert Set.empty) . Set.singleton.Set.singleton

powerSet :: (Ord a) => Set a -> Set (Set a)
powerSet = (Set.fold (slift Set.union) (Set.singleton Set.empty)) . Set.map makeSet

Usage:

Prelude Data.Set> powerSet fromList [1,2,3]
fromList [fromList [], fromList [1], fromList [1,2], fromList [1,2,3], fromList [1,3], fromList [2], fromList [2,3], fromList [3]]

Icon and Unicon

The two examples below show the similarities and differences between constructing an explicit representation of the solution, i.e. a set containing the powerset, and one using generators. The basic recursive algorithm is the same in each case, but wherever the first stores part of the result away, the second uses 'suspend' to immediately pass the result back to the caller. The caller may then decide to store the results in a set, a list, or dispose of each one as it appears.

Set building

The following version returns a set containing the powerset:

procedure power_set (s)
  result := set ()
  if *s = 0 
    then insert (result, set ()) # empty set
    else {
      head := set(?s) # take a random element
      # and find powerset of remaining part of set
      tail_pset := power_set (x -- head)
      result ++:= tail_pset # add powerset of remainder to results
      every ps := !tail_pset do # and add head to each powerset from the remainder
        insert (result, ps ++ head)
    }
  return result
end

To test the above procedure:

procedure main ()
  every s := !power_set (set(1,2,3,4)) do { # requires '!' to generate items in the result set
    writes ("[ ")
    every writes (!s || " ")
    write ("]")
  }
end

Output:

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

Generator

An alternative version, which generates each item in the power set in turn:

procedure power_set (s)
  if *s = 0 
    then suspend set ()
    else {
      head := set(?s)
      every ps := power_set (s -- head) do {
        suspend ps
        suspend ps ++ head
      }
    }
end

procedure main ()
  every s := power_set (set(1,2,3,4)) do { # power_set's values are generated by 'every'
    writes ("[ ")
    every writes (!s || " ")
    write ("]")
  }
end

J

There are a number of ways to generate a power set in J. Here's one:

ps =: #~ 2 #:@i.@^ #

For example:

   ps 'ACE'
   
E  
C  
CE 
A  
AE 
AC 
ACE

In the typical use, this operation makes sense on collections of unique elements.

   ~.1 2 3 2 1
1 2 3
   #ps 1 2 3 2 1
32
   #ps ~.1 2 3 2 1
8

In other words, the power set of a 5 element set has 32 sets where the power set of a 3 element set has 8 sets. Thus if elements of the original "set" were not unique then sets of the power "set" will also not be unique sets.

Java

Works with: Java version 1.5+

Recursion

This implementation sorts each subset, but not the whole list of subsets (which would require a custom comparator). It also destroys the original set.

public static ArrayList<String> getpowerset(int a[],int n,ArrayList<String> ps)
    {
        if(n<0)
        {
            return null;
        }
        if(n==0)
        {
            if(ps==null)
                ps=new ArrayList<String>();
            ps.add(" ");
            return ps;
        }
        ps=getpowerset(a, n-1, ps);
        ArrayList<String> tmp=new ArrayList<String>();
        for(String s:ps)
        {
            if(s.equals(" "))
                tmp.add(""+a[n-1]);
            else
                tmp.add(s+a[n-1]);
        }
        ps.addAll(tmp);
        return ps;
    }

Iterative

The iterative implementation of the above idea. Each subset is in the order that the element appears in the input list. This implementation preserves the input.

public static <T> List<List<T>> powerset(Collection<T> list) {
  List<List<T>> ps = new ArrayList<List<T>>();
  ps.add(new ArrayList<T>());   // add the empty set

  // for every item in the original list
  for (T item : list) {
    List<List<T>> newPs = new ArrayList<List<T>>();

    for (List<T> subset : ps) {
      // copy all of the current powerset's subsets
      newPs.add(subset);

      // plus the subsets appended with the current item
      List<T> newSubset = new ArrayList<T>(subset);
      newSubset.add(item);
      newPs.add(newSubset);
    }

    // powerset is now powerset of list.subList(0, list.indexOf(item)+1)
    ps = newPs;
  }
  return ps;
}

Binary String

This implementation works on idea that each element in the original set can either be in the power set or not in it. With n elements in the original set, each combination can be represented by a binary string of length n. To get all possible combinations, all you need is a counter from 0 to 2ⁿ - 1. If the k^th bit in the binary string is 1, the k^th element of the original set is in this combination.

public static <T extends Comparable<? super T>> LinkedList<LinkedList<T>> BinPowSet(
		LinkedList<T> A){
	LinkedList<LinkedList<T>> ans= new LinkedList<LinkedList<T>>();
	int ansSize = (int)Math.pow(2, A.size());
	for(int i= 0;i< ansSize;++i){
		String bin= Integer.toBinaryString(i); //convert to binary
		while(bin.length() < A.size()) bin = "0" + bin; //pad with 0's
		LinkedList<T> thisComb = new LinkedList<T>(); //place to put one combination
		for(int j= 0;j< A.size();++j){
			if(bin.charAt(j) == '1')thisComb.add(A.get(j));
		}
		Collections.sort(thisComb); //sort it for easy checking
		ans.add(thisComb); //put this set in the answer list
	}
	return ans;
}

JavaScript

ES5

Iteration

Uses a JSON stringifier from http://www.json.org/js.html

Works with: SpiderMonkey

function powerset(ary) {
    var ps = [[]];
    for (var i=0; i < ary.length; i++) {
        for (var j = 0, len = ps.length; j < len; j++) {
            ps.push(ps[j].concat(ary[i]));
        }
    }
    return ps;
}

var res = powerset([1,2,3,4]);

load('json2.js');
print(JSON.stringify(res));

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]]

Functional composition

Translation of: Haskell

(function () {

   // translating:  powerset = foldr (\x acc -> acc ++ map (x:) acc) [[]]

    function powerset(xs) {
        return xs.reduceRight(function (a, x) {
            return a.concat(a.map(function (y) {
                return [x].concat(y);
            }));
        }, [[]]);
    }


    // TEST
    return {
        '[1,2,3] ->': powerset([1, 2, 3]),
        'empty set ->': powerset([]),
        'set which contains only the empty set ->': powerset([[]])
    }

})();

Output:

{
 "[1,2,3] ->":[[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]],
 "empty set ->":[[]],
 "set which contains only the empty set ->":[[], [[]]]
}

ES6

(() => {
    'use strict';

    // powerset :: [a] -> [[a]]
    const powerset = xs =>
        xs.reduceRight((a, x) => [...a, ...a.map(y => [x, ...y])], [
            []
        ]);


    // TEST
    return {
        '[1,2,3] ->': powerset([1, 2, 3]),
        'empty set ->': powerset([]),
        'set which contains only the empty set ->': powerset([
            []
        ])
    };
})()

Output:

{"[1,2,3] ->":[[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]], 
"empty set ->":[[]], 
"set which contains only the empty set ->":[[], [[]]]}

jq

def powerset:
  reduce .[] as $i ([[]];
     reduce .[] as $r (.; . + [$r + [$i]]));

Example:

[range(0;10)]|powerset|length
# => 1024

Extra credit:

# The power set of the empty set:
  [] | powerset
  # => [[]]

# The power set of the set which contains only the empty set:
  [ [] ] | powerset
  # => [[],[[]]]

Recursive version

def powerset:
  if length == 0 then [[]]
  else .[0] as $first
    | (.[1:] | powerset) 
    | map([$first] + . ) + .
  end;

Example:

[1,2,3]|powerset
# => [[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]

Julia

function powerset(x::Vector{T})::Vector{Vector{T}} where T
    result = Vector{T}[[]]
    for elem in x, j in eachindex(result)
        push!(result, [result[j] ; elem])
    end
    result
end

Output:

julia> show(powerset([1,2,3]))
[Int64[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]

Non-Mutating Solution

using Base.Iterators

function bitmask(u, max_size)
    res = BitArray(undef, max_size)
    res.chunks[1] = u%UInt64
    res
end

function powerset(input_collection::Vector{T})::Vector{Vector{T}} where T
    num_elements = length(input_collection)
    bitmask_map(x) = Iterators.map(y -> bitmask(y, num_elements), x)
    getindex_map(x) = Iterators.map(y -> input_collection[y], x)

    UnitRange(0, (2^num_elements)-1) |>
        bitmask_map |>
        getindex_map |>
        collect
end

Output:

julia> show(powerset([1,2,3]))
[Int64[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]

K

   ps:{x@&:'+2_vs!_2^#x}

Usage:

   ps "ABC"
(""
 ,"C"
 ,"B"
 "BC"
 ,"A"
 "AC"
 "AB"
 "ABC")

Kotlin

// purely functional & lazy version, leveraging recursion and Sequences (a.k.a. streams)
fun <T> Set<T>.subsets(): Sequence<Set<T>> =
    when (size) {
        0 -> sequenceOf(emptySet())
        else -> {
            val head = first()
            val tail = this - head
            tail.subsets() + tail.subsets().map { setOf(head) + it }
        }
    }

// if recursion is an issue, you may change it this way:

fun <T> Set<T>.subsets(): Sequence<Set<T>> = sequence {
    when (size) {
        0 -> yield(emptySet<T>())
        else -> {
            val head = first()
            val tail = this@subsets - head
            yieldAll(tail.subsets())
            for (subset in tail.subsets()) {
                yield(setOf(head) + subset)
            }
        }
    }
}

Output:

Power set of setOf(1, 2, 3, 4) comprises:
[]
[4]
[3]
[3, 4]
[2]
[2, 4]
[2, 3]
[2, 3, 4]
[1]
[1, 4]
[1, 3]
[1, 3, 4]
[1, 2]
[1, 2, 4]
[1, 2, 3]
[1, 2, 3, 4]

Power set of emptySet<Any>() comprises:
[]

Power set of setOf(emptySet<Any>()) comprises:
[]
[[]]

Lambdatalk

{def powerset

{def powerset.r
 {lambda {:ary :ps :i}
  {if {= :i {A.length :ary}}
   then :ps
   else {powerset.r :ary                 
                    {powerset.rr :ary :ps {A.length :ps} :i 0}
                    {+ :i 1}} }}}

{def powerset.rr
 {lambda {:ary :ps :len :i :j}
  {if {= :j :len}
   then :ps
   else {powerset.rr :ary 
                     {A.addlast! {A.concat {A.get :j :ps}
                                           {A.new {A.get :i :ary}}}
                                 :ps}
                     :len
                     :i 
                     {+ :j 1}} }}}

 {lambda {:ary}
  {A.new {powerset.r :ary {A.new {A.new}} 0}}}} 

-> powerset

{powerset {A.new 1 2 3 4}}
-> [[],[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]]]

Logo

to powerset :set
  if empty? :set [output [[]]]
  localmake "rest powerset butfirst :set
  output sentence  map [sentence first :set ?] :rest  :rest
end

show powerset [1 2 3]
[[1 2 3] [1 2] [1 3] [1] [2 3] [2] [3] []]

Logtalk

:- object(set).

    :- public(powerset/2).

    powerset(Set, PowerSet) :-
        reverse(Set, RSet),
        powerset_1(RSet, [[]], PowerSet).

    powerset_1([], PowerSet, PowerSet).
    powerset_1([X| Xs], Yss0, Yss) :-
        powerset_2(Yss0, X, Yss1),
        powerset_1(Xs, Yss1, Yss).

    powerset_2([], _, []).
    powerset_2([Zs| Zss], X, [Zs, [X| Zs]| Yss]) :-
        powerset_2(Zss, X, Yss).

    reverse(List, Reversed) :-
        reverse(List, [], Reversed).

    reverse([], Reversed, Reversed).
    reverse([Head| Tail], List, Reversed) :-
        reverse(Tail, [Head| List], Reversed).

:- end_object.

Usage example:

| ?- set::powerset([1, 2, 3, 4], PowerSet).

PowerSet = [[],[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]]
yes

Lua

--returns the powerset of s, out of order.
function powerset(s, start)
  start = start or 1
  if(start > #s) then return {{}} end
  local ret = powerset(s, start + 1)
  for i = 1, #ret do
    ret[#ret + 1] = {s[start], unpack(ret[i])}
  end
  return ret
end

--non-recurse implementation
function powerset(s)
   local t = {{}}
   for i = 1, #s do
      for j = 1, #t do
         t[#t+1] = {s[i],unpack(t[j])}
      end
   end
   return t
end

--alternative, copied from the Python implementation
function powerset2(s)
  local ret = {{}}
  for i = 1, #s do
    local k = #ret
    for j = 1, k do
      ret[k + j] = {s[i], unpack(ret[j])}
    end
  end
  return ret
end

M4

define(`for',
  `ifelse($#, 0, ``$0'',
          eval($2 <= $3), 1,
          `pushdef(`$1', `$2')$4`'popdef(
             `$1')$0(`$1', incr($2), $3, `$4')')')dnl
define(`nth',
  `ifelse($1, 1, $2,
          `nth(decr($1), shift(shift($@)))')')dnl
define(`range',
  `for(`x', eval($1 + 2), eval($2 + 2),
       `nth(x, $@)`'ifelse(x, eval($2+2), `', `,')')')dnl
define(`powerpart',
  `{range(2, incr($1), $@)}`'ifelse(incr($1), $#, `',
     `for(`x', eval($1+2), $#,
        `,powerpart(incr($1), ifelse(
           eval(2 <= ($1 + 1)), 1,
           `range(2,incr($1), $@), ')`'nth(x, $@)`'ifelse(
              eval((x + 1) <= $#),1,`,range(incr(x), $#, $@)'))')')')dnl
define(`powerset',
  `{powerpart(0, substr(`$1', 1, eval(len(`$1') - 2)))}')dnl
dnl
powerset(`{a,b,c}')

Output:

{{},{a},{a,b},{a,b,c},{a,c},{b},{b,c},{c}}

Maple

combinat:-powerset({1,2,3,4});

Output:

{{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, 

    {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}

Mathematica/Wolfram Language

Built-in function that either gives all possible subsets, subsets with at most n elements, subsets with exactly n elements or subsets containing between n and m elements. Example of all subsets:

Subsets[{a, b, c}]

gives:

{{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

Subsets[list, {n, Infinity}] gives all the subsets that have n elements or more.

Subsets[list, n] gives all the subsets that have at most n elements.

Subsets[list, {n}] gives all the subsets that have exactly n elements.

Subsets[list, {m,n}] gives all the subsets that have between m and n elements.

MATLAB

Sets are not an explicit data type in MATLAB, but cell arrays can be used for the same purpose. In fact, cell arrays have the benefit of containing any kind of data structure. So, this powerset function will work on a set of any type of data structure, without the need to overload any operators.

function pset = powerset(theSet)

    pset = cell(size(theSet)); %Preallocate memory

    %Generate all numbers from 0 to 2^(num elements of the set)-1
    for i = ( 0:(2^numel(theSet))-1 )
       
        %Convert i into binary, convert each digit in binary to a boolean
        %and store that array of booleans
        indicies = logical(bitget( i,(1:numel(theSet)) )); 
        
        %Use the array of booleans to extract the members of the original
        %set, and store the set containing these members in the powerset
        pset(i+1) = {theSet(indicies)};
       
    end
    
end

Sample Usage: Powerset of the set of the empty set.

powerset({{}})

ans = 

     {}    {1x1 cell} %This is the same as { {},{{}} }

Powerset of { {1,2},3 }.

powerset({{1,2},3})

ans = 

    {1x0 cell}    {1x1 cell}    {1x1 cell}    {1x2 cell} %This is the same as { {},{{1,2}},{3},{{1,2},3} }

Maxima

powerset({1, 2, 3, 4});
/* {{}, {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}} */

Nim

import sets, hashes
 
proc hash(x: HashSet[int]): Hash =
  var h = 0
  for i in x: h = h !& hash(i)
  result = !$h
 
proc powerset[T](inset: HashSet[T]): HashSet[HashSet[T]] =
  result.incl(initHashSet[T]())  # Initialized with empty set.
  for val in inset:
    let previous = result
    for aSet in previous:
      var newSet = aSet
      newSet.incl(val)
      result.incl(newSet)
 
echo powerset([1,2,3,4].toHashSet())

Output:

{{4, 3, 1}, {3, 2, 1}, {3}, {3, 1}, {2}, {4, 3, 2, 1}, {}, {4, 2}, {4, 2, 1}, {4, 3, 2}, {1}, {3, 2}, {4, 3}, {4}, {4, 1}, {2, 1}}

Objective-C

#import <Foundation/Foundation.h>

+ (NSArray *)powerSetForArray:(NSArray *)array {
	UInt32 subsetCount = 1 << array.count;
	NSMutableArray *subsets = [NSMutableArray arrayWithCapacity:subsetCount];
	for(int subsetIndex = 0; subsetIndex < subsetCount; subsetIndex++) {
		NSMutableArray *subset = [[NSMutableArray alloc] init];
		for (int itemIndex = 0; itemIndex < array.count; itemIndex++) {
			if((subsetIndex >> itemIndex) & 0x1) {
				[subset addObject:array[itemIndex]];
			}
		}		
		[subsets addObject:subset];
	}
	return subsets;
}

OCaml

The standard library already implements a proper Set datatype. As the base type is unspecified, the powerset must be parameterized as a module. Also, the library is lacking a map operation, which we have to implement first.

module PowerSet(S: Set.S) =
struct

  include Set.Make (S)

  let map f s =
    let work x r = add (f x) r in
    fold work s empty
  ;;

  let powerset s = 
    let base = singleton (S.empty) in
    let work x r = union r (map (S.add x) r) in 
    S.fold work s base
  ;;

end;; (* PowerSet *)

version for lists:

let subsets xs = List.fold_right (fun x rest -> rest @ List.map (fun ys -> x::ys) rest) xs [[]]

OPL

{string} s={"A","B","C","D"};
range r=1.. ftoi(pow(2,card(s)));
{string} s2 [k in r] = {i | i in s: ((k div (ftoi(pow(2,(ord(s,i))))) mod 2) == 1)};

execute
{
 writeln(s2);
}

which gives

[{} {"A"} {"B"} {"A" "B"} {"C"} {"A" "C"} {"B" "C"} {"A" "B" "C"} {"D"} {"A"
         "D"} {"B" "D"} {"A" "B" "D"} {"C" "D"} {"A" "C" "D"} {"B" "C" "D"}
         {"A" "B" "C" "D"}]

Oz

Oz has a library for finite set constraints. Creating a power set is a trivial application of that:

declare
  %% Given a set as a list, returns its powerset (again as a list)
  fun {Powerset Set}
     proc {Describe Root}
        %% Describe sets by lower bound (nil) and upper bound (Set)
        Root = {FS.var.bounds nil Set}
        %% enumerate all possible sets
        {FS.distribute naive [Root]}
     end
     AllSets = {SearchAll Describe}
  in
     %% convert to list representation
     {Map AllSets FS.reflect.lowerBoundList}
  end
in
  {Inspect {Powerset [1 2 3 4]}}

A more convential implementation without finite set constaints:

fun {Powerset2 Set}
   case Set of nil then [nil]
   [] H|T thens
      Acc = {Powerset2 T}
   in
      {Append Acc {Map Acc fun {$ A} H|A end}}
   end
end

PARI/GP

vector(1<<#S,i,vecextract(S,i-1))

Works with: PARI/GP version 2.10.0+

The forsubset iterator was added in version 2.10.0 to efficiently iterate over combinations and power sets.

S=["a","b","c"]
forsubset(#S,s,print1(vecextract(S,s)"  "))

Output:

[]  ["a"]  ["b"]  ["c"]  ["a", "b"]  ["a", "c"]  ["b", "c"]  ["a", "b", "c"]

Perl

Perl does not have a built-in set data-type. However, you can...

Module: Algorithm::Combinatorics

This module has an iterator over the power set. Note that it does not enforce that the input array is a set (no duplication). If each subset is processed immediately, this has an advantage of very low memory use.

use Algorithm::Combinatorics "subsets";
my @S = ("a","b","c");
my @PS;
my $iter = subsets(\@S);
while (my $p = $iter->next) {
  push @PS, "[@$p]"
}
say join("  ",@PS);

Output:

[a b c]  [b c]  [a c]  [c]  [a b]  [b]  [a]  []

Module: ntheory

Library: ntheory

The simplest solution is to use the one argument version of the combination iterator, which iterates over the power set.

use ntheory "forcomb";
my @S = qw/a b c/;
forcomb { print "[@S[@_]]  " } scalar(@S);
print "\n";

Output:

[]  [a]  [b]  [c]  [a b]  [a c]  [b c]  [a b c]

Using the two argument version of the iterator gives a solution similar to the Raku and Python array versions.

use ntheory "forcomb";
my @S = qw/a b c/;
for $k (0..@S) {
  # Iterate over each $#S+1,$k combination.
  forcomb { print "[@S[@_]]  " } @S,$k;
}
print "\n";

Output:

[]  [a]  [b]  [c]  [a b]  [a c]  [b c]  [a b c]

Similar to the Pari/GP solution, one can also use vecextract with an integer mask to select elements. Note that it does not enforce that the input array is a set (no duplication). This also has low memory if each subset is processed immediately and the range is applied with a loop rather than a map. A solution using vecreduce could be done identical to the array reduce solution shown later.

use ntheory "vecextract";
my @S = qw/a b c/;
my @PS = map { "[".join(" ",vecextract(\@S,$_))."]" } 0..2**scalar(@S)-1;
say join("  ",@PS);

Output:

[]  [a]  [b]  [a b]  [c]  [a c]  [b c]  [a b c]

Module: Set::Object

The CPAN module Set::Object provides a set implementation for sets of arbitrary objects, for which a powerset function could be defined and used like so:

use Set::Object qw(set);

sub powerset {
    my $p = Set::Object->new( set() );
    foreach my $i (shift->elements) {
        $p->insert( map { set($_->elements, $i) } $p->elements );
    }
    return $p;
}

my $set = set(1, 2, 3);
my $powerset = powerset($set);

print $powerset->as_string, "\n";

Output:

Set::Object(Set::Object() Set::Object(1 2 3) Set::Object(1 2) Set::Object(1 3) Set::Object(1) Set::Object(2 3) Set::Object(2) Set::Object(3))

Simple custom hash-based set type

It's also easy to define a custom type for sets of strings or numbers, using a hash as the underlying representation (like the task description suggests):

package Set {
    sub new       { bless { map {$_ => undef} @_[1..$#_] }, shift; }
    sub elements  { sort keys %{shift()} }
    sub as_string { 'Set(' . join(' ', sort keys %{shift()}) . ')' }
    # ...more set methods could be defined here...
}

(Note: For a ready-to-use module that uses this approach, and comes with all the standard set methods that you would expect, see the CPAN module Set::Tiny)

The limitation of this approach is that only primitive strings/numbers are allowed as hash keys in Perl, so a Set of Set's cannot be represented, and the return value of our powerset function will thus have to be a list of sets rather than being a Set object itself.

We could implement the function as an imperative foreach loop similar to the Set::Object based solution above, but using list folding (with the help of Perl's List::Util core module) seems a little more elegant in this case:

use List::Util qw(reduce);

sub powerset {
    @{( reduce { [@$a, map { Set->new($_->elements, $b) } @$a ] }
               [Set->new()], shift->elements )};
}

my $set = Set->new(1, 2, 3);
my @subsets = powerset($set);

print $_->as_string, "\n" for @subsets;

Output:

Set()
Set(1)
Set(2)
Set(1 2)
Set(3)
Set(1 3)
Set(2 3)
Set(1 2 3)

Arrays

If you don't actually need a proper set data-type that guarantees uniqueness of its elements, the simplest approach is to use arrays to store "sets" of items, in which case the implementation of the powerset function becomes quite short.

Recursive solution:

sub powerset {
    @_ ? map { $_, [$_[0], @$_] } powerset(@_[1..$#_]) : [];
}

List folding solution:

use List::Util qw(reduce);

sub powerset {
    @{( reduce { [@$a, map([@$_, $b], @$a)] } [[]], @_ )}
}

Usage & output:

my @set = (1, 2, 3);
my @powerset = powerset(@set);

sub set_to_string {
    "{" . join(", ", map { ref $_ ? set_to_string(@$_) : $_ } @_) . "}"
}

print set_to_string(@powerset), "\n";

Output:

{{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}

Lazy evaluation

If the initial set is quite large, constructing it's powerset all at once can consume lots of memory.

If you want to iterate through all of the elements of the powerset of a set, and don't mind each element being generated immediately before you process it, and being thrown away immediately after you're done with it, you can use vastly less memory. This is similar to the earlier solutions using the Algorithm::Combinatorics and ntheory modules.

The following algorithm uses one bit of memory for every element of the original set (technically it uses several bytes per element with current versions of Perl). This is essentially doing a vecextract operation by hand.

use strict;
use warnings;
sub powerset :prototype(&@) {
    my $callback = shift;
    my $bitmask = '';
    my $bytes = @_/8;
    {
       my @indices = grep vec($bitmask, $_, 1), 0..$#_;
       $callback->( @_[@indices] );
       ++vec($bitmask, $_, 8) and last for 0 .. $bytes;
       redo if @indices != @_;
    }
}

print "powerset of empty set:\n";
powerset { print "[@_]\n" };
print "powerset of set {1,2,3,4}:\n";
powerset { print "[@_]\n" } 1..4;
my $i = 0;
powerset { ++$i } 1..9;
print "The powerset of a nine element set contains $i elements.\n";

Output:

powerset of empty set:
[]
powerset of set {1,2,3,4}:
[]
[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]  
The powerset of a nine element set contains 512 elements.

The technique shown above will work with arbitrarily large sets, and uses a trivial amount of memory.

Phix

sequence powerset
integer step = 1
 
function pst(object key, object /*data*/, object /*user_data*/)
    integer k = 1
    while k<length(powerset) do
        k += step
        for j=1 to step do
            powerset[k] = append(powerset[k],key)
            k += 1
        end for
    end while
    step *= 2
    return 1
end function
 
function power_set(integer d)
    powerset = repeat({},power(2,dict_size(d)))
    step = 1
    traverse_dict(routine_id("pst"),0,d)
    return powerset
end function
 
integer d1234 = new_dict({{1,0},{2,0},{3,0},{4,0}})
?power_set(d1234)
integer d0 = new_dict()
?power_set(d0)
setd({},0,d0)
?power_set(d0)

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}}
{{}}
{{},{{}}}

alternative

Adapted from the one I used on Ascending_primes#powerset.

with javascript_semantics
function power_set(sequence s)
    sequence powerset = {{}}, subset = {{{},0}}
    while length(subset) do
        sequence next = {}
        for i=1 to length(subset) do
            {sequence sub, integer k} = subset[i]
            for j=k+1 to length(s) do
                sequence ni = append(deep_copy(sub),s[j])
                next = append(next,{ni,j})
                powerset = append(powerset,ni)
            end for
        end for
        subset = next
    end while
    assert(length(powerset)=power(2,length(s)))
    return powerset
end function
 
?power_set({1,2,3,4})
?power_set({4,3,2,1})
?power_set({})
?power_set({{}})

Output:

Guaranteed to be in length order, and index order within each length.

{{},{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}}
{{},{4},{3},{2},{1},{4,3},{4,2},{4,1},{3,2},{3,1},{2,1},{4,3,2},{4,3,1},{4,2,1},{3,2,1},{4,3,2,1}}
{{}}
{{},{{}}}

PHP

<?php
function get_subset($binary, $arr) {
  // based on true/false values in $binary array, include/exclude
  // values from $arr
  $subset = array();
  foreach (range(0, count($arr)-1) as $i) {
    if ($binary[$i]) {
      $subset[] = $arr[count($arr) - $i - 1];
    } 
  }
  return $subset;
}

function print_array($arr) {
  if (count($arr) > 0) {
    echo join(" ", $arr);
  } else {
    echo "(empty)";
  }
  echo '<br>';
}

function print_power_sets($arr) {
  echo "POWER SET of [" . join(", ", $arr) . "]<br>";
  foreach (power_set($arr) as $subset) {
    print_array($subset);
  }
}
  
function power_set($arr) {  
  $binary = array();
  foreach (range(1, count($arr)) as $i) {
    $binary[] = false;
  }
  $n = count($arr);
  $powerset = array();
  
  while (count($binary) <= count($arr)) {
    $powerset[] = get_subset($binary, $arr);
    $i = 0;
    while (true) {
      if ($binary[$i]) {
        $binary[$i] = false;
        $i += 1;
      } else {
        $binary[$i] = true;
        break;
      }
    }
    $binary[$i] = true;
  }
  
  return $powerset;
}
 
print_power_sets(array());
print_power_sets(array('singleton'));
print_power_sets(array('dog', 'c', 'b', 'a'));
?>

Output:

POWER SET of []
POWER SET of [singleton]
(empty)
singleton
POWER SET of [dog, c, b, a]
(empty)
a
b
a b
c
a c
b c
a b c
dog
a dog
b dog
a b dog
c dog
a c dog
b c dog
a b c dog

PicoLisp

(de powerset (Lst)
   (ifn Lst
      (cons)
      (let L (powerset (cdr Lst))
         (conc
            (mapcar '((X) (cons (car Lst) X)) L)
            L ) ) ) )

PL/I

Translation of: REXX

*process source attributes xref or(!);
 /*--------------------------------------------------------------------
 * 06.01.2014 Walter Pachl  translated from REXX
 *-------------------------------------------------------------------*/
 powerset: Proc Options(main);
 Dcl (hbound,index,left,substr) Builtin;
 Dcl sysprint Print;
 Dcl s(4) Char(5) Var Init('one','two','three','four');
 Dcl ps   Char(1000) Var;
 Dcl (n,chunk,p) Bin Fixed(31);
 n=hbound(s);                      /* number of items in the list.   */
 ps='{} ';                         /* start with a null power set.   */
 Do chunk=1 To n;                  /* loop through the ...     .     */
   ps=ps!!combn(chunk);            /* a CHUNK at a time.             */
   End;
 Do While(ps>'');
   p=index(ps,' ');
   Put Edit(left(ps,p-1))(Skip,a);
   ps=substr(ps,p+1);
   End;

 combn: Proc(y) Returns(Char(1000) Var);
 /*--------------------------------------------------------------------
 * returns the list of subsets with y elements of set s
 *-------------------------------------------------------------------*/
 Dcl (y,base,bbase,ym,p,j,d,u) Bin Fixed(31);
 Dcl (z,l) Char(1000) Var Init('');
 Dcl a(20) Bin Fixed(31) Init((20)0);
 Dcl i Bin Fixed(31);
 base=hbound(s)+1;
 bbase=base-y;
 ym=y-1;
 Do p=1 To y;
   a(p)=p;
   End;
 Do j=1 By 1;
   l='';
   Do d=1 To y;
     u=a(d);
     l=l!!','!!s(u);
     End;
   z=z!!'{'!!substr(l,2)!!'} ';
   a(y)=a(y)+1;
   If a(y)=base Then
     If combu(ym) Then
       Leave;
   End;
 /* Put Edit('combn',y,z)(Skip,a,f(2),x(1),a); */
 Return(z);

 combu: Proc(d) Recursive Returns(Bin Fixed(31));
 Dcl (d,u) Bin Fixed(31);
 If d=0 Then
   Return(1);
 p=a(d);
 Do u=d To y;
   a(u)=p+1;
   If a(u)=bbase+u Then
     Return(combu(u-1));
   p=a(u);
   End;
 Return(0);
 End;
 End;

 End;

Output:

{}
{one}
{two}
{three}
{four}
{one,two}
{one,three}
{one,four}
{two,three}
{two,four}
{three,four}
{one,two,three}
{one,two,four}
{one,three,four}
{two,three,four}
{one,two,three,four}

PowerShell

function power-set ($array) {
    if($array) {
        $n = $array.Count
        function state($set, $i){  
            if($i -gt -1) {
                state $set ($i-1)
                state ($set+@($array[$i])) ($i-1)   
            } else {
                "$($set | sort)"
            }
        }
        $set = state @() ($n-1)
        $power = 0..($set.Count-1) | foreach{@(0)}
        $i = 0
        $set | sort | foreach{$power[$i++] = $_.Split()}
        $power | sort {$_.Count}
    } else {@()}

}
$OFS = " "
$setA = power-set  @(1,2,3,4)
"number of sets in setA: $($setA.Count)"
"sets in setA:"
$OFS = ", "
$setA | foreach{"{"+"$_"+"}"} 
$setB = @()
"number of sets in setB: $($setB.Count)"
"sets in setB:"
$setB | foreach{"{"+"$_"+"}"} 
$setC = @(@(), @(@()))
"number of sets in setC: $($setC.Count)"
"sets in setC:"
$setC | foreach{"{"+"$_"+"}"} 
$OFS = " "

Output:

number of sets in setA: 16
sets in setA:
{}
{1}
{2}
{3}
{4}
{1, 2}
{1, 3}
{1, 4}
{2, 3}
{2, 4}
{3, 4}
{1, 2, 3}
{1, 2, 4}
{1, 3, 4}
{2, 3, 4}
{1, 2, 3, 4}
number of sets in setB: 0
sets in setB:
number of sets in setC: 2
sets in setC:
{}
{}

Prolog

Logical (cut-free) Definition

The predicate powerset(X,Y) defined here can be read as "Y is the powerset of X", it being understood that lists are used to represent sets.

The predicate subseq(X,Y) is true if and only if the list X is a subsequence of the list Y.

The definitions here are elementary, logical (cut-free), and efficient (within the class of comparably generic implementations).

powerset(X,Y) :- bagof( S, subseq(S,X), Y).

subseq( [], []).
subseq( [], [_|_]).
subseq( [X|Xs], [X|Ys] ) :- subseq(Xs, Ys).
subseq( [X|Xs], [_|Ys] ) :- append(_, [X|Zs], Ys), subseq(Xs, Zs).

Output:

?- powerset([1,2,3], X).
X = [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]].

% Symbolic:
?- powerset( [X,Y], S).
S = [[], [X], [X, Y], [Y]].

% In reverse:
?- powerset( [X,Y], [[], [1], [1, 2], [2]] ).
X = 1,
Y = 2.

Single-Functor Definition

power_set( [], [[]]).
power_set( [X|Xs], PS) :-
  power_set(Xs, PS1),
  maplist( append([X]), PS1, PS2 ), % i.e. prepend X to each PS1
  append(PS1, PS2, PS).

Output:

?- power_set([1,2,3,4,5,6,7,8], X), length(X,N), writeln(N).
256

Constraint Handling Rules

CHR is a programming language created by Professor Thom Frühwirth.
Works with SWI-Prolog and module chr written by Tom Schrijvers and Jan Wielemaker.

:- use_module(library(chr)).

:- chr_constraint chr_power_set/2, chr_power_set/1, clean/0.

clean @ clean \ chr_power_set(_) <=> true.
clean @ clean <=> true.

only_one @ chr_power_set(A) \ chr_power_set(A) <=> true.


creation @ chr_power_set([H | T], A) <=>
           append(A, [H], B),
	   chr_power_set(T, A),
           chr_power_set(T, B),
	   chr_power_set(B).


empty_element @ chr_power_set([], _) <=> chr_power_set([]).

Output:

 ?- chr_power_set([1,2,3,4], []), findall(L, find_chr_constraint(chr_power_set(L)), LL), clean.
LL = [[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],[]] .

PureBasic

This code is for console mode.

If OpenConsole()
  Define argc=CountProgramParameters()
  If argc>=(SizeOf(Integer)*8) Or argc<1
    PrintN("Set out of range.")
    End 1
  Else
    Define i, j, text$
    Define.q bset=1<<argc
    Print("{")
    For i=0 To bset-1   ; check all binary combinations
      If Not i: text$=  "{"
      Else    : text$=", {"
      EndIf
      k=0
      For j=0 To argc-1  ; step through each bit   
        If i&(1<<j)
          If k: text$+", ": EndIf         ; pad the output 
          text$+ProgramParameter(j): k+1  ; append each matching bit 
        EndIf
      Next j
      Print(text$+"}")
    Next i
    PrintN("}")
  EndIf
EndIf

Output:

C:\Users\PureBasic_User\Desktop>"Power Set.exe" 1 2 3 4
{{}, {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}}

Python

def list_powerset(lst):
    # the power set of the empty set has one element, the empty set
    result = [[]]
    for x in lst:
        # for every additional element in our set
        # the power set consists of the subsets that don't
        # contain this element (just take the previous power set)
        # plus the subsets that do contain the element (use list
        # comprehension to add [x] onto everything in the
        # previous power set)
        result.extend([subset + [x] for subset in result])
    return result

# the above function in one statement
def list_powerset2(lst):
    return reduce(lambda result, x: result + [subset + [x] for subset in result],
                  lst, [[]])

def powerset(s):
    return frozenset(map(frozenset, list_powerset(list(s))))

list_powerset computes the power set of a list of distinct elements. powerset simply converts the input and output from lists to sets. We use the frozenset type here for immutable sets, because unlike mutable sets, it can be put into other sets.

Example:

>>> list_powerset([1,2,3])
[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]
>>> powerset(frozenset([1,2,3]))
frozenset([frozenset([3]), frozenset([1, 2]), frozenset([]), frozenset([2, 3]), frozenset([1]), frozenset([1, 3]), frozenset([1, 2, 3]), frozenset([2])])

Further Explanation

If you take out the requirement to produce sets and produce list versions of each powerset element, then add a print to trace the execution, you get this simplified version of the program above where it is easier to trace the inner workings

def powersetlist(s):
    r = [[]]
    for e in s:
        print "r: %-55r e: %r" % (r,e)
        r += [x+[e] for x in r]
    return r

s= [0,1,2,3]    
print "\npowersetlist(%r) =\n  %r" % (s, powersetlist(s))

Output:

r: [[]]                                                    e: 0
r: [[], [0]]                                               e: 1
r: [[], [0], [1], [0, 1]]                                  e: 2
r: [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]  e: 3

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

Binary Count method

If you list the members of the set and include them according to if the corresponding bit position of a binary count is true then you generate the powerset. (Note that only frozensets can be members of a set in the second function)

def powersequence(val):
    ''' Generate a 'powerset' for sequence types that are indexable by integers.
        Uses a binary count to enumerate the members and returns a list

        Examples:
            >>> powersequence('STR')   # String
            ['', 'S', 'T', 'ST', 'R', 'SR', 'TR', 'STR']
            >>> powersequence([0,1,2]) # List
            [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]
            >>> powersequence((3,4,5)) # Tuple
            [(), (3,), (4,), (3, 4), (5,), (3, 5), (4, 5), (3, 4, 5)]
            >>> 
    '''
    vtype = type(val); vlen = len(val); vrange = range(vlen)
    return [ reduce( lambda x,y: x+y, (val[i:i+1] for i in vrange if 2**i & n), vtype())
             for n in range(2**vlen) ]

def powerset(s):
    ''' Generate the powerset of s

        Example:
            >>> powerset(set([6,7,8]))
            set([frozenset([7]), frozenset([8, 6, 7]), frozenset([6]), frozenset([6, 7]), frozenset([]), frozenset([8]), frozenset([8, 7]), frozenset([8, 6])])
    '''
    return set( frozenset(x) for x in powersequence(list(s)) )

Recursive Alternative

This is an (inefficient) recursive version that almost reflects the recursive definition of a power set as explained in http://en.wikipedia.org/wiki/Power_set#Algorithms. It does not create a sorted output.

def p(l):
    if not l: return [[]]
    return p(l[1:]) + [[l[0]] + x for x in p(l[1:])]

Python: Standard documentation

Pythons documentation has a method that produces the groupings, but not as sets:

>>> from pprint import pprint as pp
>>> from itertools import chain, combinations
>>> 
>>> 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))

>>> pp(set(powerset({1,2,3,4})))
{(),
 (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,)}
>>>

Qi

Translation of: Scheme

(define powerset
  [] -> [[]]
  [A|As] -> (append (map (cons A) (powerset As))
                    (powerset As)))

Quackery

Quackery is, when seen from a certain perspective, an assembly language that recognises only three types, "operators", which correspond to op-codes, "numbers" i.e. bignums, and "nests" which are ordered sequences of zero of more operator, bignums and nests. Everything else is a matter of interpretation.

Integers can be used as a set of natural numbers, with (in binary) 0 corresponding to the empty set, 1 corresponding to the set of the natural number 1, 10 corresponding to the set of the natural number 2, 11 corresponding to the set of the natural numbers 1 and 2, and so on. With this sort of set, enumerating the powerset of the numbers 0 to 4, for example simply consists of enumerating the numbers 0 to 15 inclusive. Operations on this sort of set, such as union and intersection, correspond to bitwise logic operators.

The other way of implementing sets is with nests, with each item in a nest corresponding to an item in the set. This is computationally slower and more complex to code, but has the advantage that it permits sets of sets, which are required for this task.

  [ stack ]                              is (ps).stack
  [ stack ]                              is (ps).items
  [ stack ]                              is (ps).result
 
  [ 1 - (ps).items put
    0 (ps).stack put
    [] (ps).result put
    [ (ps).result take
      (ps).stack behead 
      drop nested join
      (ps).result put
      (ps).stack take
      dup (ps).items share
      = iff
          [ drop
            (ps).stack size 1 > iff
              [ 1 (ps).stack tally ] ]
            else
              [ dup (ps).stack put
                1+ (ps).stack put ]
             (ps).stack size 1 = until ]
    (ps).items release
    (ps).result take ]                   is (ps)     (   n -->   )

  [ dup size dip
      [ witheach
          [ over swap peek swap ] ]
      nip pack ]                         is arrange  ( [ [ --> [ )

  [ dup [] = iff
      nested done
    dup size (ps) 
    ' [ [ ] ] swap join
    [] unrot witheach 
      [ dip dup arrange 
        nested 
        rot swap join swap ]
    drop ]                               is powerset (   [ --> [ )

   ' [ [ 1 2 3 4 ] [ ] [ [ ] ] ]
   witheach 
     [ say "The powerset of "
       dup echo cr 
       powerset witheach [ echo cr ] 
       cr ]

Output:

The powerset of [ 1 2 3 4 ]
[ ]
[ 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 ]

The powerset of [ ]
[ ]

The powerset of [ [ ] ]
[ ]
[ [ ] ]

R

Non-recursive version

The conceptual basis for this algorithm is the following:

for each element in the set:
	for each subset constructed so far:
		new subset = (subset + element)

This method is much faster than a recursive method, though the speed is still O(2^n).

powerset <- function(set){
	ps <- list()
	ps[[1]] <- numeric()						#Start with the empty set.
	for(element in set){						#For each element in the set, take all subsets
		temp <- vector(mode="list",length=length(ps))		#currently in "ps" and create new subsets (in "temp")
		for(subset in 1:length(ps)){				#by adding "element" to each of them.
			temp[[subset]] = c(ps[[subset]],element)
		}
		ps <- c(ps,temp)						#Add the additional subsets ("temp") to "ps".
	}
	ps
}

powerset(1:4)

The list "temp" is a compromise between the speed costs of doing arithmetic and of creating new lists (since R lists are immutable, appending to a list means actually creating a new list object). Thus, "temp" collects new subsets that are later added to the power set. This improves the speed by 4x compared to extending the list "ps" at every step.

Recursive version

Library: sets

The sets package includes a recursive method to calculate the power set. However, this method takes ~100 times longer than the non-recursive method above.

library(sets)

An example with a vector.

v <- (1:3)^2
sv <- as.set(v)
2^sv

{{}, {1}, {4}, {9}, {1, 4}, {1, 9}, {4, 9}, {1, 4, 9}}

An example with a list.

l <- list(a=1, b="qwerty", c=list(d=TRUE, e=1:3))
sl <- as.set(l)
2^sl

{{}, {1}, {"qwerty"}, {<<list(2)>>}, {1, <<list(2)>>}, {"qwerty",
 1}, {"qwerty", <<list(2)>>}, {"qwerty", 1, <<list(2)>>}}

Racket

;;; Direct translation of 'functional' ruby method
(define (powerset s)
  (for/fold ([outer-set (set(set))]) ([element s])
    (set-union outer-set 
               (list->set (set-map outer-set
                                   (λ(inner-set) (set-add inner-set element)))))))

Raku

(formerly Perl 6)

Works with: rakudo version 2014-02-25

sub powerset(Set $s) { $s.combinations.map(*.Set).Set }
say powerset set <a b c d>;

Output:

set(set(), set(a), set(b), set(c), set(d), set(a, b), set(a, c), set(a, d), set(b, c), set(b, d), set(c, d), set(a, b, c), set(a, b, d), set(a, c, d), set(b, c, d), set(a, b, c, d))

If you don't care about the actual Set type, the .combinations method by itself may be good enough for you:

.say for <a b c d>.combinations

Output:

 
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

Rascal

import Set;

public set[set[&T]] PowerSet(set[&T] s) = power(s);

Output:

rascal>PowerSet({1,2,3,4})
set[set[int]]: {
  {4,3},
  {4,2,1},
  {4,3,1},
  {4,2},
  {4,3,2},
  {4,1},
  {4,3,2,1},
  {4},
  {3},
  {2,1},
  {3,1},
  {2},
  {3,2},
  {1},
  {3,2,1},
  {}
}

REXX

/*REXX program  displays a  power set;  items may be  anything  (but can't have blanks).*/
Parse Arg text                                   /*allow the user specify optional set. */
If text='' Then                                  /*Not specified?  Then use the default.*/
  text='one two three four'
n=words(text)
psi=0
Do k=0 To n               /* loops from 0 to n elements of a set      */
  cc=comb(n,k)            /* returns the combinations of 1 through k  */
  Do while pos('/',cc)>0        /* as long as there is a combination  */
    Parse Var cc elist '/' cc   /* get i from comb's result string    */
    psl=''                      /* initialize the list of words       */
    psi=psi+1                   /* index of this set                  */
    Do While elist<>''          /* loop through elements              */
      parse var elist e elist   /* get an element (a digit)           */
      psl=psl','word(text,e)    /* add corresponding test word to set */
      End
    psl=substr(psl,2)           /* get rid of leading comma           */
    Say right(psi,2) '{'psl'}'  /* show this element of the power set */
    End
  End
Exit
comb: Procedure
/***********************************************************************
* Returns the combinations of size digits out of things digits
* e.g. comb(4,2) -> ' 1 2/1 3/1 4/2 3/2 4/3 4/'
                      1 2/  1 3/  1 4/  2 3/  2 4/  3 4 /
***********************************************************************/
Parse Arg things,size
n=2**things-1
list=''
Do u=1 To n
  co=combinations(u)
  If co>'' Then
    list=list '/' combinations(u)
  End
Return substr(space(list),2) '/'    /* remove leading / */

combinations: Procedure Expose things size
  Parse Arg u
  nc=0
  bu=x2b(d2x(u))
  bu1=space(translate(bu,' ',0),0)
  If length(bu1)=size Then Do
    ub=reverse(bu)
    res=''
    Do i=1 To things
      c=i
      If substr(ub,i,1)=1 Then res=res i
      End
    Return res
    End
  Else
    Return ''

output when using the default input:

 1 {}
 2 {one}
 3 {two}
 4 {three}
 5 {four}
 6 {one,two}
 7 {one,three}
 8 {one,four}
 9 {two,three}
10 {two,four}
11 {three,four}
12 {one,two,three}
13 {one,two,four}
14 {one,three,four}
15 {two,three,four}
16 {one,two,three,four}

Ring

# Project : Power set

list = ["1", "2", "3", "4"]
see powerset(list)
 
func powerset(list)
        s = "{"
        for i = 1 to (2 << len(list)) - 1 step 2
             s = s + "{"
             for j = 1 to len(list) 
                  if i & (1 << j)
                     s = s + list[j] + ","
                  ok
             next
             if right(s,1) = ","
                s = left(s,len(s)-1)
             ok
             s = s + "},"
        next
        return left(s,len(s)-1) + "}"

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}}

RPL

Works with: Halcyon Calc version 4.2.8

RPL code

Comment

≪ 
  IF DUP SIZE 
  THEN LAST OVER SWAP GET → last
  ≪ LIST→ 1 - SWAP DROP →LIST POWST 
     1 OVER SIZE FOR j 
        DUP j GET last + 1 →LIST + NEXT
  ≫ 
  ELSE 1 →LIST END
≫ 'POWST' STO

POWST ( { set } -- { power set } ) 
if set is not empty
then store last item
     get power set of { set } - last item
     for all sets of { set } - last item power set
     add last item to set, then set to power set

else return { { } }

{ 1 2 3 4 } POWST
{ } POWST

Output:

2: { { } { 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 } }
1: { { } }

Ruby

# Based on http://johncarrino.net/blog/2006/08/11/powerset-in-ruby/ 
# See the link if you want a shorter version. 
# This was intended to show the reader how the method works. 
class Array
  # Adds a power_set method to every array, i.e.: [1, 2].power_set
  def power_set
    
    # Injects into a blank array of arrays.
    # acc is what we're injecting into
    # you is each element of the array
    inject([[]]) do |acc, you|
      ret = []             # Set up a new array to add into
      acc.each do |i|      # For each array in the injected array,
        ret << i           # Add itself into the new array
        ret << i + [you]   # Merge the array with a new array of the current element
      end
      ret       # Return the array we're looking at to inject more.
    end
    
  end
  
  # A more functional and even clearer variant.
  def func_power_set
    inject([[]]) { |ps,item|    # for each item in the Array
      ps +                      # take the powerset up to now and add
      ps.map { |e| e + [item] } # it again, with the item appended to each element
    }
  end
end

#A direct translation of the "power array" version above
require 'set'
class Set
  def powerset 
    inject(Set[Set[]]) do |ps, item| 
      ps.union ps.map {|e| e.union (Set.new [item])}
    end
  end
end

p [1,2,3,4].power_set
p %w(one two three).func_power_set

p Set[1,2,3].powerset

Output:

[[], [4], [3], [3, 4], [2], [2, 4], [2, 3], [2, 3, 4], [1], [1, 4], [1, 3], [1, 3, 4], [1, 2], [1, 2, 4], [1, 2, 3], [1, 2, 3, 4]]
[[], ["one"], ["two"], ["one", "two"], ["three"], ["one", "three"], ["two", "three"], ["one", "two", "three"]]
#<Set: {#<Set: {}>, #<Set: {1}>, #<Set: {2}>, #<Set: {1, 2}>, #<Set: {3}>, #<Set: {1, 3}>, #<Set: {2, 3}>, #<Set: {1, 2, 3}>}>

Rust

This implementation consumes the input set, requires that the type T has a full order a.k.a implements the Ord trait and that T is clonable.

use std::collections::BTreeSet;

fn powerset<T: Ord + Clone>(mut set: BTreeSet<T>) -> BTreeSet<BTreeSet<T>> {
    if set.is_empty() {
        let mut powerset = BTreeSet::new();
        powerset.insert(set);
        return powerset;
    }
    // Access the first value. This could be replaced with `set.pop_first().unwrap()`
    // But this is an unstable feature 
    let entry = set.iter().nth(0).unwrap().clone(); 
    set.remove(&entry);
    let mut powerset = powerset(set);
    for mut set in powerset.clone().into_iter() {
        set.insert(entry.clone());
        powerset.insert(set);
    }
    powerset
}

fn main() {
    let set = (1..5).collect();
    let set = powerset(set);
    println!("{:?}", set);

    let set = ["a", "b", "c", "d"].iter().collect();
    let set = powerset(set);
    println!("{:?}", set);
}

Output:

{{}, {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}}
{{}, {"a"}, {"a", "b"}, {"a", "b", "c"}, {"a", "b", "c", "d"}, {"a", "b", "d"}, {"a", "c"}, {"a", "c", "d"}, {"a", "d"}, {"b"}, {"b", "c"}, {"b", "c", "d"}, {"b", "d"}, {"c"}, {"c", "d"}, {"d"}}

SAS

options mprint mlogic symbolgen source source2;

%macro SubSets (FieldCount = );
data _NULL_;
	Fields = &FieldCount;
	SubSets = 2**Fields;
	call symput ("NumSubSets", SubSets);
run;

%put &NumSubSets;

data inital;
	%do j = 1 %to &FieldCount;
		F&j. = 1;
	%end;
run;

data SubSets;
	set inital;
	RowCount =_n_;
	call symput("SetCount",RowCount);
run;

%put SetCount ;

%do %while (&SetCount < &NumSubSets);

data loop;
	%do j=1 %to &FieldCount;
		if rand('GAUSSIAN') > rand('GAUSSIAN') then F&j. = 1;
	%end;

data SubSets_  ;
set SubSets loop;
run;

proc sort data=SubSets_  nodupkey;
	by F1 - F&FieldCount.;
run;

data Subsets;
	set SubSets_;
	RowCount =_n_;
run;

proc sql noprint;
	select max(RowCount) into :SetCount
	from SubSets;
quit;
run; 

%end;
%Mend SubSets;

You can then call the macro as:

%SubSets(FieldCount = 5);

The output will be the dataset SUBSETS and will have a 5 columns F1, F2, F3, F4, F5 and 32 columns, one with each combination of 1 and missing values.

Output:

Obs	F1	F2	F3	F4	F5	RowCount
1	.	.	.	.	.	1
2	.	.	.	.	1	2
3	.	.	.	1	.	3
4	.	.	.	1	1	4
5	.	.	1	.	.	5
6	.	.	1	.	1	6
7	.	.	1	1	.	7
8	.	.	1	1	1	8
9	.	1	.	.	.	9
10	.	1	.	.	1	10
11	.	1	.	1	.	11
12	.	1	.	1	1	12
13	.	1	1	.	.	13
14	.	1	1	.	1	14
15	.	1	1	1	.	15
16	.	1	1	1	1	16
17	1	.	.	.	.	17
18	1	.	.	.	1	18
19	1	.	.	1	.	19
20	1	.	.	1	1	20
21	1	.	1	.	.	21
22	1	.	1	.	1	22
23	1	.	1	1	.	23
24	1	.	1	1	1	24
25	1	1	.	.	.	25
26	1	1	.	.	1	26
27	1	1	.	1	.	27
28	1	1	.	1	1	28
29	1	1	1	.	.	29
30	1	1	1	.	1	30
31	1	1	1	1	.	31
32	1	1	1	1	1	32

Scala

import scala.compat.Platform.currentTime

object Powerset extends App {
  def powerset[A](s: Set[A]) = s.foldLeft(Set(Set.empty[A])) { case (ss, el) => ss ++ ss.map(_ + el)}

  assert(powerset(Set(1, 2, 3, 4)) == Set(Set.empty, Set(1), Set(2), Set(3), Set(4), Set(1, 2), Set(1, 3), Set(1, 4),
    Set(2, 3), Set(2, 4), Set(3, 4), Set(1, 2, 3), Set(1, 3, 4), Set(1, 2, 4), Set(2, 3, 4), Set(1, 2, 3, 4)))
  println(s"Successfully completed without errors. [total ${currentTime - executionStart} ms]")
}

Another option that produces lazy sequence of the sets:

def powerset[A](s: Set[A]) = (0 to s.size).map(s.toSeq.combinations(_)).reduce(_ ++ _).map(_.toSet)

A tail-recursive version:

def powerset[A](s: Set[A]) = {
  def powerset_rec(acc: List[Set[A]], remaining: List[A]): List[Set[A]] = remaining match {
    case Nil => acc
    case head :: tail => powerset_rec(acc ++ acc.map(_ + head), tail)
  }
  powerset_rec(List(Set.empty[A]), s.toList)
}

Scheme

Translation of: Common Lisp

(define (power-set set)
  (if (null? set)
      '(())
      (let ((rest (power-set (cdr set))))
        (append (map (lambda (element) (cons (car set) element))
                     rest)
                rest))))

(display (power-set (list 1 2 3)))
(newline)

(display (power-set (list "A" "C" "E")))
(newline)

Output:

((1 2 3) (1 2) (1 3) (1) (2 3) (2) (3) ())
((A C E) (A C) (A E) (A) (C E) (C) (E) ())

Call/cc generation:

(define (power-set lst)
  (define (iter yield)
    (let recur ((a '()) (b lst))
      (if (null? b) (set! yield
		      (call-with-current-continuation
			(lambda (resume)
			  (set! iter resume)
			  (yield a))))
	(begin (recur (append a (list (car b))) (cdr b))
	       (recur a (cdr b)))))

    ;; signal end of generation
    (yield 'end-of-seq))

  (lambda () (call-with-current-continuation iter)))

(define x (power-set '(1 2 3)))
(let loop ((a (x)))
  (if (eq? a 'end-of-seq) #f
    (begin
      (display a)
      (newline)
      (loop (x)))))

Output:

(1 2)
(1 3)
(1)
(2 3)
(2)
(3)
()

Iterative:

(define (power_set_iter set)
  (let loop ((res '(())) (s set))
    (if (empty? s)
        res
        (loop (append (map (lambda (i) (cons (car s) i)) res) res) (cdr s)))))

Output:

'((e d c b a)
  (e d c b)
  (e d c a)
  (e d c)
  (e d b a)
  (e d b)
  (e d a)
  (e d)
  (e c b a)
  (e c b)
  (e c a)
  (e c)
  (e b a)
  (e b)
  (e a)
  (e)
  (d c b a)
  (d c b)
  (d c a)
  (d c)
  (d b a)
  (d b)
  (d a)
  (d)
  (c b a)
  (c b)
  (c a)
  (c)
  (b a)
  (b)
  (a)
  ())

Seed7

$ include "seed7_05.s7i";
 
const func array bitset: powerSet (in bitset: baseSet) is func
  result
    var array bitset: pwrSet is [] (bitset.value);
  local
    var integer: element is 0;
    var integer: index is 0;
    var bitset: aSet is bitset.value;
  begin
    for element range baseSet do
      for key index range pwrSet do
        aSet := pwrSet[index];
        if element not in aSet then
          incl(aSet, element);
          pwrSet &:= aSet;
        end if;
      end for;
    end for;
  end func;

const proc: main is func
  local
    var bitset: aSet is bitset.value;
  begin
    for aSet range powerSet({1, 2, 3, 4}) do
      writeln(aSet);
    end for;
  end func;

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}

SETL

Pfour := pow({1, 2, 3, 4});
Pempty := pow({});
PPempty := pow(Pempty);

print(Pfour);
print(Pempty);
print(PPempty);

Output:

{{} {1} {2} {3} {4} {1 2} {1 3} {1 4} {2 3} {2 4} {3 4} {1 2 3} {1 2 4} {1 3 4} {2 3 4} {1 2 3 4}}
{{}}
{{} {{}}}

Sidef

var arr = %w(a b c)
for i in (0 .. arr.len) {
    say arr.combinations(i)
}

Output:

[[]]
[["a"], ["b"], ["c"]]
[["a", "b"], ["a", "c"], ["b", "c"]]
[["a", "b", "c"]]

Simula

SIMSET
BEGIN

    LINK CLASS LOF_INT(N); INTEGER N;;

    LINK CLASS LOF_LOF_INT(H); REF(HEAD) H;;

    REF(HEAD) PROCEDURE MAP(P_LI, P_LLI);
        REF(HEAD) P_LI;
        REF(HEAD) P_LLI;
    BEGIN
        REF(HEAD) V_RESULT;
        V_RESULT :- NEW HEAD;
        IF NOT P_LLI.EMPTY THEN BEGIN
            REF(LOF_LOF_INT) V_LLI;
            V_LLI :- P_LLI.FIRST QUA LOF_LOF_INT;
            WHILE V_LLI =/= NONE DO BEGIN
                REF(HEAD) V_NEWLIST;
                V_NEWLIST :- NEW HEAD;
                ! ADD THE SAME 1ST ELEMENT TO EVERY NEWLIST ;
                NEW LOF_INT(P_LI.FIRST QUA LOF_INT.N).INTO(V_NEWLIST);
                IF NOT V_LLI.H.EMPTY THEN BEGIN
                    REF(LOF_INT) V_LI;
                    V_LI :- V_LLI.H.FIRST QUA LOF_INT;
                    WHILE V_LI =/= NONE DO BEGIN
                        NEW LOF_INT(V_LI.N).INTO(V_NEWLIST);
                        V_LI :- V_LI.SUC;
                    END;
                END;
                NEW LOF_LOF_INT(V_NEWLIST).INTO(V_RESULT);
                V_LLI :- V_LLI.SUC;
            END;
        END;
        MAP :- V_RESULT;
    END MAP;

    REF(HEAD) PROCEDURE SUBSETS(P_LI);
        REF(HEAD) P_LI;
    BEGIN
        REF(HEAD) V_RESULT;
        IF P_LI.EMPTY THEN BEGIN
            V_RESULT :- NEW HEAD;
            NEW LOF_LOF_INT(NEW HEAD).INTO(V_RESULT);
        END ELSE BEGIN
            REF(HEAD) V_SUBSET, V_MAP;
            REF(LOF_INT) V_LI;
            V_SUBSET :- NEW HEAD;
            V_LI :- P_LI.FIRST QUA LOF_INT;
            ! SKIP OVER 1ST ELEMENT ;
            IF V_LI =/= NONE THEN V_LI :- V_LI.SUC;
            WHILE V_LI =/= NONE DO BEGIN
                NEW LOF_INT(V_LI.N).INTO(V_SUBSET);
                V_LI :- V_LI.SUC;
            END;
            V_RESULT :- SUBSETS(V_SUBSET);
            V_MAP :- MAP(P_LI, V_RESULT);
            IF NOT V_MAP.EMPTY THEN BEGIN
                REF(LOF_LOF_INT) V_LLI;
                V_LLI :- V_MAP.FIRST QUA LOF_LOF_INT;
                WHILE V_LLI =/= NONE DO BEGIN
                    NEW LOF_LOF_INT(V_LLI.H).INTO(V_RESULT);
                    V_LLI :- V_LLI.SUC;
                END;
            END;
        END;
        SUBSETS :- V_RESULT;
    END SUBSETS;

    PROCEDURE PRINT_LIST(P_LI); REF(HEAD) P_LI;
    BEGIN
        OUTTEXT("[");
        IF NOT P_LI.EMPTY THEN BEGIN
            INTEGER I;
            REF(LOF_INT) V_LI;
            I := 0;
            V_LI :- P_LI.FIRST QUA LOF_INT;
            WHILE V_LI =/= NONE DO BEGIN
                IF I > 0 THEN OUTTEXT(",");
                OUTINT(V_LI.N, 0);
                V_LI :- V_LI.SUC;
                I := I+1;
            END;
        END;
        OUTTEXT("]");
    END PRINT_LIST;

    PROCEDURE PRINT_LIST_LIST(P_LLI); REF(HEAD) P_LLI;
    BEGIN
        OUTTEXT("[");
        IF NOT P_LLI.EMPTY THEN BEGIN
            INTEGER I;
            REF(LOF_LOF_INT) V_LLI;
            I := 0;
            V_LLI :- P_LLI.FIRST QUA LOF_LOF_INT;
            WHILE V_LLI =/= NONE DO BEGIN
                IF I > 0 THEN BEGIN
                    OUTTEXT(",");
                !   OUTIMAGE;
                END;
                PRINT_LIST(V_LLI.H);
                V_LLI :- V_LLI.SUC;
                I := I+1;
            END;
        END;
        OUTTEXT("]");
        OUTIMAGE;
    END PRINT_LIST_LIST;

    INTEGER N;
    REF(HEAD) V_RANGE;
    REF(HEAD) V_LISTS;

    V_RANGE :- NEW HEAD;
    V_LISTS :- SUBSETS(V_RANGE);
    PRINT_LIST_LIST(V_LISTS);
    OUTIMAGE;
    FOR N := 1 STEP 1 UNTIL 4 DO BEGIN
        NEW LOF_INT(N).INTO(V_RANGE);
        V_LISTS :- SUBSETS(V_RANGE);
        PRINT_LIST_LIST(V_LISTS);
        OUTIMAGE;
    END;
END.

Output:

[[]]

[[],[1]]

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

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

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

Smalltalk

Works with: GNU Smalltalk

Code from Bonzini's blog

Collection extend [
    power [
        ^(0 to: (1 bitShift: self size) - 1) readStream collect: [ :each || i |
            i := 0.
            self select: [ :elem | (each bitAt: (i := i + 1)) = 1 ] ]
    ]
].

#(1 2 4) power do: [ :each |
    each asArray printNl ].

#( 'A' 'C' 'E' ) power do: [ :each |
    each asArray printNl ].

Standard ML

version for lists:

fun subsets xs = foldr (fn (x, rest) => rest @ map (fn ys => x::ys) rest) [[]] xs

Swift

Works with: Swift version Revision 4 - tested with Xcode 9.2 playground

func powersetFrom<T>(_ elements: Set<T>) -> Set<Set<T>> {
  guard elements.count > 0 else {
    return [[]]
  }
  var powerset: Set<Set<T>> = [[]]
  for element in elements {
    for subset in powerset {
      powerset.insert(subset.union([element]))
    }
  }
  return powerset
}

// Example:
powersetFrom([1, 2, 4])

Output:

{
  {2, 4}
  {4, 1}
  {4},
  {2, 4, 1}
  {2, 1}
  Set([])
  {1}
  {2}
}

//Example:
powersetFrom(["a", "b", "d"])

Output:

{
  {"b", "d"}
  {"b"}
  {"d"},
  {"a"}
  {"b", "d", "a"}
  Set([])
  {"d", "a"}
  {"b", "a"}
}

Tcl

proc subsets {l} {
    set res [list [list]]
    foreach e $l {
        foreach subset $res {lappend res [lappend subset $e]}
    }
    return $res
}
puts [subsets {a b c d}]

Output:

{} a b {a b} c {a c} {b c} {a b c} d {a d} {b d} {a b d} {c d} {a c d} {b c d} {a b c d}

Binary Count Method

proc powersetb set {
   set res {}
   for {set i 0} {$i < 2**[llength $set]} {incr i} {
      set pos -1
      set pset {}
      foreach el $set {
          if {$i & 1<<[incr pos]} {lappend pset $el}
      }
      lappend res $pset
   }
   return $res
}

TXR

The power set function can be written concisely like this:

(defun power-set (s)
  (mappend* (op comb s) (range 0 (length s))))

This generates the lists of combinations of all possible lengths, from 0 to the length of s and catenates them. The comb function generates a lazy list, so it is appropriate to use mappend* (the lazy version of mappend) to keep the behavior lazy.

A complete program which takes command line arguments and prints the power set in comma-separated brace notation:

@(do (defun power-set (s)
       (mappend* (op comb s) (range 0 (length s)))))
@(bind pset @(power-set *args*))
@(output)
@  (repeat)
{@(rep)@pset, @(last)@pset@(empty)@(end)}
@  (end)
@(end)

Output:

$ txr rosetta/power-set.txr  1 2 3
{1, 2, 3}
{1, 2}
{1, 3}
{1}
{2, 3}
{2}
{3}
{}

The above power-set function generalizes to strings and vectors.

@(do (defun power-set (s)
       (mappend* (op comb s) (range 0 (length s))))
     (prinl (power-set "abc"))
     (prinl (power-set "b"))
     (prinl (power-set ""))
     (prinl (power-set #(1 2 3))))

Output:

$ txr power-set-generic.txr
("" "a" "b" "c" "ab" "ac" "bc" "abc")
("" "b")
("")
(#() #(1) #(2) #(3) #(1 2) #(1 3) #(2 3) #(1 2 3))

UNIX Shell

From here

p() { [ $# -eq 0 ] && echo || (shift; p "$@") | while read r ; do echo -e "$1 $r\n$r"; done }

Usage

|p `cat` | sort | uniq                                                                        
A
C
E
^D

UnixPipes

| cat A
a
b
c

| cat A |\
   xargs -n 1 ksh -c 'echo \{`cat A`\}' |\
   xargs |\
   sed -e 's; ;,;g' \
       -e 's;^;echo ;g' \
       -e 's;\},;}\\ ;g' |\
   ksh |unfold `wc -l A` |\
   xargs -n1 -I{} ksh -c 'echo {} |\
        unfold 1 |sort -u |xargs' |sort -u

a
a b
a b c
a c
b
b c
c

Ursala

Sets are a built in type constructor in Ursala, represented as lexically sorted lists with duplicates removed. The powerset function is a standard library function, but could be defined as shown below.

powerset = ~&NiC+ ~&i&& ~&at^?\~&aNC ~&ahPfatPRXlNrCDrT

test program:

#cast %sSS

test = powerset {'a','b','c','d'}

Output:

{
   {},
   {'a'},
   {'a','b'},
   {'a','b','c'},
   {'a','b','c','d'},
   {'a','b','d'},
   {'a','c'},
   {'a','c','d'},
   {'a','d'},
   {'b'},
   {'b','c'},
   {'b','c','d'},
   {'b','d'},
   {'c'},
   {'c','d'},
   {'d'}}

V

V has a built in called powerlist

[A C E] powerlist
=[[A C E] [A C] [A E] [A] [C E] [C] [E] []]

its implementation in std.v is (like joy)

[powerlist
   [null?]
   [unitlist]
   [uncons]
   [dup swapd [cons] map popd swoncat]
    linrec].

VBA

Option Base 1
Private Function power_set(ByRef st As Collection) As Collection
    Dim subset As Collection, pwset As New Collection
    For i = 0 To 2 ^ st.Count - 1
        Set subset = New Collection
        For j = 1 To st.Count
            If i And 2 ^ (j - 1) Then subset.Add st(j)
        Next j
        pwset.Add subset
    Next i
    Set power_set = pwset
End Function
Private Function print_set(ByRef st As Collection) As String
    'assume st is a collection of collections, holding integer variables
    Dim s() As String, t() As String
    ReDim s(st.Count)
    'Debug.Print "{";
    For i = 1 To st.Count
        If st(i).Count > 0 Then
            ReDim t(st(i).Count)
            For j = 1 To st(i).Count
                Select Case TypeName(st(i)(j))
                    Case "Integer": t(j) = CStr(st(i)(j))
                    Case "Collection": t(j) = "{}" 'assumes empty
                End Select
            Next j
            s(i) = "{" & Join(t, ", ") & "}"
        Else
            s(i) = "{}"
        End If
    Next i
    print_set = "{" & Join(s, ", ") & "}"
End Function
Public Sub rc()
    Dim rcset As New Collection, result As Collection
    For i = 1 To 4
        rcset.Add i
    Next i
    Debug.Print print_set(power_set(rcset))
    Set rcset = New Collection
    Debug.Print print_set(power_set(rcset))
    Dim emptyset As New Collection
    rcset.Add emptyset
    Debug.Print print_set(power_set(rcset))
    Debug.Print
End Sub

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}}
{{}}
{{}, {{}}}

VBScript

Function Dec2Bin(n)
	q = n
	Dec2Bin = ""
	Do Until q = 0
		Dec2Bin = CStr(q Mod 2) & Dec2Bin
		q = Int(q / 2)
	Loop
	Dec2Bin = Right("00000" & Dec2Bin,6)
End Function

Function PowerSet(s)
	arrS = Split(s,",")
	PowerSet = "{"
	For i = 0 To 2^(UBound(arrS)+1)-1
		If i = 0 Then
			PowerSet = PowerSet & "{},"
		Else
			binS = Dec2Bin(i)
			PowerSet = PowerSet & "{"
			c = 0
			For j = Len(binS) To 1 Step -1
				If CInt(Mid(binS,j,1)) = 1 Then
					PowerSet = PowerSet & arrS(c) & ","	
				End If
				c = c + 1
			Next
			PowerSet = Mid(PowerSet,1,Len(PowerSet)-1) & "},"
		End If
	Next
	PowerSet = Mid(PowerSet,1,Len(PowerSet)-1) & "}"
End Function

WScript.StdOut.Write 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}}

Wren

Library: Wren-perm

Although we have a module for sets, they are based on maps whose keys must be value types. This means that sets of sets are technically impossible because sets themselves are not value types.

We therefore use lists to represent sets which works fine here.

import "./perm" for Powerset

var sets  = [ [1, 2, 3, 4], [], [[]] ]
for (set in sets) {
    System.print("The power set of %(set) is:")
    System.print(Powerset.list(set))
    System.print()
}

Output:

The power set of [1, 2, 3, 4] is:
[[], [1], [2], [3], [4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]

The power set of [] is:
[[]]

The power set of [[]] is:
[[], [[]]]

XPL0

func PowSet(Set, Size);
int  Set, Size;
int  N, M, Mask, DoComma;
[ChOut(0, ^{);
for N:= 0 to 1<<Size -1 do
    [if N>0 then ChOut(0, ^,);
    ChOut(0, ^{);
    Mask:= 1;  DoComma:= false;
    for M:= 0 to Size-1 do
        [if Mask & N then
            [if DoComma then ChOut(0, ^,);
            IntOut(0, Set(M));
            DoComma:= true;
            ];
        Mask:= Mask << 1;
        ];
    ChOut(0, ^});
    ];
Text(0, "}^m^j");
];

[PowSet([2, 3, 5, 7], 4);
 PowSet([1], 1);
 PowSet(0, 0);
]

Output:

{{},{2},{3},{2,3},{5},{2,5},{3,5},{2,3,5},{7},{2,7},{3,7},{2,3,7},{5,7},{2,5,7},{3,5,7},{2,3,5,7}}
{{},{1}}
{{}}

zkl

Using a combinations function, build the power set from combinations of 1,2,... items.

fcn pwerSet(list){
  (0).pump(list.len(),List, Utils.Helpers.pickNFrom.fp1(list),
     T(Void.Write,Void.Write) ) .append(list)
}

foreach n in (5){
   ps:=pwerSet((1).pump(n,List)); ps.println(" Size = ",ps.len());
}

Output:

L(L()) Size = 1
L(L(),L(1)) Size = 2
L(L(),L(1),L(2),L(1,2)) Size = 4
L(L(),L(1),L(2),L(3),L(1,2),L(1,3),L(2,3),L(1,2,3)) Size = 8
L(L(),L(1),L(2),L(3),L(4),L(1,2),L(1,3),L(1,4),L(2,3),L(2,4),
   L(3,4),L(1,2,3),L(1,2,4),L(1,3,4),L(2,3,4),L(1,2,3,4)) Size = 16