Power Set
OK? Got that? Maybe an example will help...
All The Subsets
When we have a set {a,b,c}:
 Then a subset of it could be {a} or {b}, or {a,c}, and so on,
 And {a,b,c} is also a subset of {a,b,c} (yes it is a subset, but not a "proper subset")
 And the empty set {} is also a subset of {a,b,c}
In fact, when we list all the subsets of S={a,b,c} we get the Power Set of {a,b,c}:
P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
Think of it as all the different ways we can select the items (the order of the items doesn't matter), including selecting none, or all.
How Many Subsets
Easy! If the original set has n members, then the Power Set will have 2^{n} members
Example: in the {a,b,c} example above, there are three members (a,b and c).
So, the Power Set should have 2^{3} = 8, which it does!
Notation
The number of members of a set is often written as S, so we can write:
Example: for the set S={1,2,3,4,5} how many members will the power set have?
Well, S has 5 members, so:
P(S) = 2^{n} = 2^{5} = 32
You will see in a minute why the number of members is a power of 2
It's Binary!
And here is the most amazing thing. To create the Power Set, just write down the sequence of binary numbers (using n digits), and then let "1" mean "put the matching member into this subset".
OK, it is easier to show with an example:
abc  Subset  

0  000  { } 
1  001  {c} 
2  010  {b} 
3  011  {b,c} 
4  100  {a} 
5  101  {a,c} 
6  110  {a,b} 
7  111  {a,b,c} 
Well, they are not in a pretty order, but they are all there.
Another Example
Let's eat! We have four flavors of icecream: banana, chocolate, lemon, and strawberry. How many different ways can we have them? Let's use letters for the flavors: {b, c, l, s}. Example selections include:

bcls  Subset  

0  0000  {} 
1  0001  {s} 
2  0010  {l} 
3  0011  {l,s} 
...  ... etc ..  ... etc ... 
12  1100  {b,c} 
13  1101  {b,c,s} 
14  1110  {b,c,l} 
15  1111  {b,c,l,s} 
And the result is (more neatly arranged):
P = { {}, {b}, {c}, {l}, {s}, {b,c}, {b,l}, {b,s}, {c,l}, {c,s}, {l,s}, {b,c,l}, {b,c,s},
{b,l,s}, {c,l,s}, {b,c,l,s} }
SymmetryIn the table above, did you notice that the first subset is empty and the last has every member? But did you also notice that the second subset has "s", and the second last subset has everything except "s"? 

In fact when we mirror that table about the middle we see there is a kind of symmetry. This is because the binary numbers that we used to help us get all combinations have a beautiful and elegant pattern. 
A Prime Example
The Power Set can be useful in unexpected areas. I wanted to find out the factors (not just the prime factors, but all factors) of a number.
I could test all possible numbers: to find the factors of, say 330, I could check 2,3,4,5,6,7,8,9,10 ... etc. Well, there are ways to improve that, but it still takes a long time for large numbers (in my tests, the computer just sat there for ages).
But I could already find the prime factors very quickly, so couldn't I somehow combine the prime factors to make all factors?
Let me see, 330 = 2×3×5×11 (using prime factor tool).
So, all the factors of 330 are:
 2,3,5, and 11,
 2×3, 2×5 and 2×11 as well, and
 2×3×5 and 2×3×11 ... ? Aha! I need a Power Set!
So, my oriignal set is {2,3,5,11} and the power set can be worked out using:
2,3,5,11  Subset  Factor  

0  0000  { }  1 
1  0001  {11}  11 
2  0010  {5}  5 
3  0011  {5,11}  5 × 11 = 55 
4  0100  {3}  3 
5  0101  {3,11}  3 × 11 = 33 
... etc ...  ... etc ...  ... etc ...  
15  1111  {2,3,5,11}  2 × 3 × 5 × 11 = 330 
And the result? The factors of 330 are 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330, and 1, 2, 3, etc as well (using the All Factors Tool).
Automated
I couldn't resist making this available to you in an automated way.
So, when you need a power set, try Power Set Maker.