# Finding The Power Set Of A Given Set

Bashyboy

{∅,{∅}}

## The Attempt at a Solution

My answer is {∅, {{∅}}, {∅, {∅}}}
but the actual answer is: {∅,{∅},{{∅}},{∅,{∅}}}
I don't understand how the second element, {∅}, appears...

Mentor
your set { a, b } so the power set has all proper subsets including the null set and the set itself:

{ nullset, {a}, {b}, {a,b} }

and in your example then the { a } corresponds to the { nullset } and the { b } corresponds to { { nullset } }

Bashyboy
Okay, thank you.

Bashyboy
Actually, I have another question concerning power sets. My book says, "If a set has n elements, then its power set has 2^n elements." Why does this formula work?

Mentor
okay the best way to thing of this is using a binary string of digits where each digit of the string represents an element of the set so I can encode the various subsets as strings of binary digis as follows:

000000 represents the nullset { }
100000 represents the first element {a}
010000 the 2nd { b }
001000 the 3rd... { c }
111000 a subset containing {a,b,c}
...
111111 the set itself { a,b,c,d,e,f }

so in creating a subset then

a is or isnt in the subset: 2 choices
b is or isnt in the subset : x 2 =4 choices
...
f is or isn't in the subset x 2 = 64 choices

or more succinctly:

a x b x c x d x e x f
2 x 2 x 2 x 2 x 2 x 2 = 2^6 = 64 possible subsets

and therefore the powerset contains 64 elements.