 Welcome back to our lecture series Math 3120, Transition to Advanced Mathematics for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Miseldine. In lecture seven, we talk about two main topics, that of power sets and that of bi-conditional statements. We'll do this one in the next video. The current video, we're gonna talk about what is a power set? So previously in, well, in the last couple of videos, we've been talking about basic set theory, what is a set, what are the operations of a set, like unions, intersections, Cartesian products. These are examples of how we can build a new set using other sets. So like with a union, we combine two sets together to make its union. Similar statements can be said for intersections and Cartesian products. The power set is yet another way to build a new set from another set, but it's done using subsets. So if A is set, this could be any set under the sun, a finite set, an infinite set, the empty set doesn't matter. If you have any set whatsoever, P of A, which is gonna be the set of all subsets of A is called the power set of A, which is what the P stands for, P of A. Now some authors use the notation two to the A, and this is foreshadowing the formula we're gonna talk about in just a moment. This notation, it's definitely not universal. I don't think this one's used as common as perhaps this one. So that's the one we're gonna stick with in our lecture series. P of A denotes the power set. And again, the power set is the set of all subsets of the set itself. So in particular, the empty set is an element of the power set because the empty set is a subset of A and therefore is an element of the power set. Similarly, the set A itself is a subset of A and therefore it is an element of the power set of A. And in general, if X is some subset of A, then that means that X is an element of the power set. This is an important thing to remember that regardless of whatever the elements of the set A are, this power set of A contains as its elements sets. It's a set of sets. And those sets inside of the power set will be the subsets of A, all of the subsets of A. Now, it turns out that when we considered initially the idea of subsets, we were actually computing power sets at the same time. For example, previously in our lecture series, we had an example called example 133 in which case we listed all of the subsets of the set capital A, which contains three elements, lower case A, B and C. We showed in this example that the set A contained exactly eight subsets. And this includes the empty set, the three singletons, A, B, C, the three pairs, A, B, A, C, B, C, and then the whole set itself. Those were the eight sets. And we organized those sets and argued why those are the only subsets of A right here. Now, if you put all eight of these sets together, this is just what the power set is. The power set is the set of these eight subsets right there. So by listing all of the subsets of A, we were actually describing the power set just without the vocabulary yet. Now, I should also mention in this example, 133, we didn't just stick with this. We actually use the arguments on this set of cardinality three to argue what the list of all subsets of a general finite set are going to be. We considered how there's these decision binary trees and how based upon the path you take on the binary tree will determine which subsets you create. And that subsets of a finite set are in correspondence, one-to-one correspondence with these decision paths on a binary tree. And so in general, we argued that the number of subsets of a set of cardinality N is gonna be a power of two, two to the N. And so an immediate corollary of the proof we did in that example is the following. Remember corollary is a theorem whose proof is immediate from something you've already done in which case when we counted subsets of finite sets and recognizing the power set is just the set of subsets, doesn't have, the subsets don't have to be finite. The power set is just the set of subsets, we then immediately get that if A is a finite set whose cardinality is N, then the cardinality of the power set is gonna be two to the N. And that's gonna work for all finite sets and their respective power sets. Now, I do wanna mention that this takes care of how you work with power sets of a finite set. And this number gets big really quickly, of course, it's an exponential growth there. And so I'm not generally gonna ask someone to be like, oh, here's a set of cardinality 100, list all of the subsets, that's gonna be a huge set but we can count them, of course. It turns out that for infinite sets, things get a little bit more weird, things are a little bit more unruly in that sense. So consider, for example, one of the simplest of all infinite sets, the set of natural numbers, okay? It turns out that if you take some subset of the natural numbers, and of course, let's mention that the natural numbers is an infinite set, right? And like I said earlier, this is sort of like the simplest of all infinite sets. For reasons there'll be more clear on in our lecture series. But if you take some subset of the natural numbers, I claim that with this subset, you can create a binary sequence. And so what do we mean by a binary sequence? A binary sequence would be something like zero, zero, one, one, one, zero, zero, zero, one, one, et cetera. It's just a sequence of numbers, zeroes and ones. Turns out that you can create a binary sequence. A binary sequence you can think of is just like, oh, you have some initial term, we're gonna call it X naught. Zero is included in the natural numbers here. You have an X1, an X2, an X3, an X4, et cetera. Okay, so you have some binary sequence going on right here. And maybe some people write sequences with these little curly braces. I'm actually gonna drop them off in retrospect here because I wouldn't want us to think of this as a set because the order in which you list the terms in the sequence does determine the sequence. The order matters here. So a binary sequence really is like an infinite list that we've talked about previously here. Now it turns out that there's this one-to-one correspondence between subsets of the natural numbers and binary sequences. And this one-to-one correspondence that is every subset has a binary sequence unique to it and every binary sequence gives us unique subset. This has to do with this counting argument we used before, right? When we had a set with three elements in it, we were like, oh, there's three decisions that have to be made. The first decision is that, is A in the subset or not? The second decision is B in the subset or not. And the third decision was is C in the subset or not, okay? And so as the set A gets larger and larger and larger, while staying finite, then you just get more and more and more decisions. Now when you consider the natural numbers, which there is now infinitely many, you then have infinite decisions that have to be made. And you can make them in order, right? We'll start with the smallest natural number zero. We decide is zero in there, is zero not in there. Then we decide is one in there, is two in there, is there three in there? And so each subset is like an infinite list of decisions. And that's the same thing as this binary sequence. The terms in this sequence XI, so XI is the ith bit in this sequence, it can either be zero or one, we'll set it to be one exactly when I belongs to the subset X there and we'll put zero and I doesn't belong to that. So you have this bit that determines is I part of the subset or not? That's what goes over here and this is how that corresponds, it's been established, okay? So what this tells us is that the subsets of the natural numbers are in correspondence with these binary sequences, but there's infinitely many binary sequences. So it turns out by this observation that the power set of the natural numbers is likewise an infinite set, all right? So if the natural numbers are infinite, which they are, then the power sets can be infinite as well. And this actually happens in general that if you take the cardinality, if you take a set whose cardinality is infinite, then its power set will likewise be infinite as well. And the basic reasoning is the following. It turns out that the cardinality of a set is always less than or equal to the cardinality of the power set. And the reason for this is basically if you consider the set of singletons, right? Like you take a singleton. So if some element X is inside of our set A, then the singleton will be an element of the power set, like so. And so the singletons are members of the power set and each element of A naturally is associated to its singleton there. So basically the elements of the power set is at least as big as the set itself because of the singletons. But I say the word at least because it could be bigger, right? There's other subsets beyond that. In the finite case, we see there's a lot more elements, two to the N, many elements there. And so two to the N is generally gonna be a lot bigger than N itself. But what about an infinite case? It turns out that because of this counting argument right here, the power set of the natural numbers has the same cardinality as the set of binary sequences. And we're gonna see much later on in this lecture series that this is actually a bigger infinity than this infinity right here. And this might be really unruly for people the first time they hear it that different, there are such things as different infinities. Some infinities can be bigger than others. And this is an example of this. But what I can finish this video with is the following statement. A power set always is bigger than the original set itself. Even in the infinite case.