The Power Set

The name power set has a nice ring to it. The power set is indeed a very powerful thing. Only recently in my real analysis class did I finally realize this. Even though I knew what a power set was, for some reason the connection to different subjects didn’t click until this quarter started. I’m going to use it to talk about infinitely many infinities after a brief overview of what a power set is.

But first, I’m going to introduce some notation and language to hopefully decomplicate things. Instead of saying “things” for the stuff in sets, I will start using the proper name which is elements. We say a set is made up of elements. I also want to use a notation for number of elements in a set. The notation is $|S|$, which is read as “the cardinality of S” in fancy math or “the number of elements in S” for sane people.

The power set of a set S is the set of all subsets of S. It sounds a bit complicated, but examples will ease the idea into your brain. First, the term subset should be explained with an example. Say that I have a set
S $= \{a,b,c\}$,
then an example of a subset would be the set $\{a,b\}$ because it is contained in S.
A subset of a general set S, is a set that is contained in S. To be completely contained in another set means that all of it’s elements are inside of the other set. Another example would be the set $\{b,c\}$. Because both of it’s elements, b and c, are elements in S.

The power set is the set of ALL possible subsets of the original given set! It will be a set of sets so the notation is going to contain a lot of curly brackets. Again for an example let
S $= \{a,b,c\}$
Let’s find some stuff that would be in the power set.
As I said before, $\{a,b\},\{b,c\},$ are in the power set, we also have $\{a,c\}$ and of course all the single elements $\{a\} ,\{b\} ,\{c\}$.
Note that each of these are sets, not just elements. $a$ is an element of S, while $\{a\}$ is the set containing the element a. So $\{a\}$ is a subset of S, while a is an element of S. Since the power set is made up of subsets, the elements of a power set are subsets of the original set.
So $\{a\}$ is an element of the power set, while $\{a\}$ is a subset of S, while a is an element in S. Got all that? Good.

Lets write all of these together in one set, with the notation $P(S)$, which read in English is “the power set of S.”

$P(S) = \{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\}$.
I purposely left some information out. This is not the entire power set, we are missing two things that mathematicians sort of just agreed on. The first is that any set is automatically it’s own subset because everything is contained in itself. So I also need to put $\{a,b,c\}$ into the power set list I have.
The other thing I’m missing is the mystical object known as the empty set . The empty set is the set of nothing and is denoted by $\emptyset$. Because nothing is contained in everything, the empty set is a subset of every set and thus is always inside of the power set (the set of ALL subsets.)
So now what we have is

$P(S) = \{\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\},\emptyset \}$

…Cool. So what is the big deal with this idea? Well first we have to think about properties that it has. One thing interesting thing is that the number of elements (which again are subsets of the original set) in the power set is always easy to find, and is based on the number of elements in the original set.
Using the earlier notation introduced for number of elements in a set, we want to find $|P(S)|$.

Well that’s not too bad, we can just count up the number of elements in P(S). There are 8 elements in P(S) because there are the 3 sets with only one element,
$\{a\} \text{ and } \{b\} \text{ and } \{c\}$
there are 3 sets of two elements,
$\{a,b\} \text{ and } \{a,c\} \text{ and } \{b,c\}$
there is the entire set,
$\{a,b,c\}$
and there is the empty set.
$\emptyset$
Then we have : $3+3+1+1 = 8$

Let’s do the cases now where $S = \{a\}$ and then $S = \{a,b\}$. Truthfully, the letters do not matter I am really concerned about how many elements are in S. So I want to see what happens to
$|P(S)|$ when I make $|S| = 1 \text{ or } |S|=2$

When $S = \{a\}$ then $P(S) = \{\{a\},\emptyset \}$ so $|P(S)| = 2$
When $S = \{a,b\}$ then $P(S) = \{\{a\},\{b\},\{a,b\},\emptyset \}$ so $|P(S)|=4$
The general pattern emerging is that when $|S| = n$ then $|P(S)|= 2^{n}$
To check just remember that
$2^{1} = 2$
$2^{2} = 4$
$2^{3} = 8$
I may do the proof of this formula another day because this entry is becoming pretty long. I’d rather explain the importance of the power set for mathematics and what happens with infinity. Knowing what the power set is, how do we get the power set of something like
$\mathbb{N} = \{0,1,2,3,...\}$
We need the set of all subsets, but this set is infinite so it is also going to have infinite subsets, like the even numbers for example. It would also contain the set of odd numbers and a whole bunch of other sets.

Too many to count actually (see what I’m getting at?) In general what happens when you take a power set of an infinite set is that the size of the power set will be a bigger infinity than the original set.
But there is nothing stopping you from taking the power set of another power set! This would look like
$P(P(S))$. So what happens when you keep taking more and more power sets of an infinite set? Well you just get bigger and bigger infinities! You can literally get infinitely many infinities, each one bigger than the previous one!

It’s almost impossible to grasp. People can grasp the first infinity moderately well, and even the first uncountable infinity, but wrapping your head around the size of a bigger infinity than that is crazy.
These last few things are going to use words most people don’t know so feel free to stop reading here if you aren’t a math person.

For some reason when I first saw the definition of a topology as a collection of subsets of X blah blah blah, it didn’t register to me that this was just a subset of the power set. I came across that definition many more times since then and even in the grad level Topology course I didn’t make that connection. Only after an algebra was defined for me as a subset of the power set blah blah blah, did I finally realize it for topologies as well.

It’s interesting to me that the foundation of topology and modern analysis is built upon subsets of a power set with certain properties. It helped me a lot to think of topologies and algebras as subsets of a power set with properties added on, I can’t really explain it. Yes, that is the same thing thing that their definitions say, but I never saw it that way until recently.

Why is the notion of structure defined in terms of the power set? Why does a subset of the power set having specific properties give us nice structures?