So What's the Deal With Power Sets?
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A Power Set...
 Is the set of all possible subsets
 Includes the empty set (X = 0 members in the subset)
 Doesn't care about order
 (i.e., order doesn't matter)
 Is the sum of the combinations of all subsets
A Power Set...

Is not a permutation. Order does not matter in a power set.
 (In fact, for mathematicians working in this area, calling something a set specifically means the collection is unordered. Ordered collections are called lists or strings.)
 Is not a combination. Combinations concern themselves with one specific value of X. Power sets consider all values of X. So the number of items in the power set is the sum of all individual combinations.
An Example
My mother bought me a scarf, a pair of mittens, and a hat to wear this winter.
I can wear
 none of it (0 items),
 1 item,
 2 items,
 or all 3 items.
How many different ways can I wear or not wear these 3 items? That is, how many different subsets are there?
Let's Count the Ways
 Subsets of size X = 0 (i.e., I wear nothing):
 [ ] = 1 subset
 Subsets of size X = 1:
 [hat] [mittens] [scarf] = 3 subsets
 Subsets of size X = 2:
 [hat mittens] [hat scarf] [mittens scarf] = 3 subsets
 Subsets of size X = 3:
 [hat mittens scarf] = 1 subset
So there are 1 + 3 + 3 + 1 = 8 possible subsets
Let's Count the Ways Using
Subsets of 0:
Subsets of 1:
Subsets of 2:
Subsets of 3:
So there are 1 + 3 + 3 + 1 = 8 possible subsets
Like I said, the number of items in the power set is the sum of all combinations, X = 0 to n.
Let's Count the Ways
Using Binary Coding
Let's code an item 1 if I wear it and 0 if I don't wear it (i.e., 1 if the item is in the subset, 0 if it's not).
The pattern is: [hat mittens scarf].
 [ ] = [0 0 0]
 [hat] = [1 0 0]
 [mittens] = [0 1 0]
 [scarf] = [0 0 1]
 [hat mittens] = [1 1 0]
 [hat scarf] = [1 0 1]
 [mittens scarf] = [0 1 1]
 [hat mittens scarf] = [1 1 1]
So you can see, to find the number of possible subsets, you have to read the largest subset (the subset with all 1s in it) as a binary number. In this case, that's 111, which is 7 in decimal form. Adding the empty set, [ ], makes 8.
Let's Count the Ways Using the Quickest Method
The simplest way to count subsets in a power set is to form the largest binary number (111), plus 1 for the empty set, as an exponent with base 2:
where n is the number of items in the mother set.
That's 2 possible values (0 or 1) for hat, 2 for mittens, and 2 for scarf. So there are 2 possibilities of hat AND 2 more possibilities of mittens AND 2 more possibilities of scarf.
Combined, that's 2 x 2 x 2 = 8 total possibilites.
OK, One More Example
I want to take my Sunday school class on a trip to hear AC/DC in concert. There are 6 students in the class. I may have 0 come to the concert, 1, 2, 3, 4, 5, or all 6.
How many different subsets are possible?
So there are 64 possible combinations of students that could go to the concert.
Fin
1. Probability  Counting Power Sets
By smilinjoe
1. Probability  Counting Power Sets
The set of all possible subsets one could form from the mother set.
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