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

It corresponds with a finite set in mathematics.

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

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



Task

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



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

 {{}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {1,4}, {2,4}, {1,2,4}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}.
For a set which contains n elements, the corresponding power set has 2n elements, including the edge cases of empty set.


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


 





P




{\displaystyle {\mathcal {P}}}

(



∅


{\displaystyle \varnothing }

) = { 



∅


{\displaystyle \varnothing }

 }

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


 





P




{\displaystyle {\mathcal {P}}}

({



∅


{\displaystyle \varnothing }

}) = { 



∅


{\displaystyle \varnothing }

, { 



∅


{\displaystyle \varnothing }

 } }


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




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")
        }
    }
}