 Now that we've defined sets and some of the useful relationships we can find between them, we are in a position to talk about cardinalities again. And so by defining cardinality as a set property, this allows us to compare cardinalities by looking at the underlying set properties. And so we have, first off, our old definition, which we've already used, which is that two sets, A and B, have the same cardinality, if and only if A is similar to B. A is equivalent to B. Well, if they're not equal, one of them is greater, one of them is less, and so I might take a look at N, the cardinality of A, less than or equal to the cardinality of B, if and only if, two things, A is similar to C, where C is a subset of B. Well, if the cardinalities aren't the same, if we think about cardinality as the number of things in a set, then one of them is larger. And so we get that with the second idea. The cardinality of the set A is less than or equal to the cardinality of the set B, if and only if A is similar to C, where C is a subset of B. And note here, we're allowing this subset to be the entire set. It is a possibility we're not requiring it be a proper subset, which is why we're getting the less than or equal to. Now, this may seem, this formulation may be a little bit strange at first, because we have this new phrase, if and only if, and this ominous sounding phrase, is actually a good thing to hear, because what it means is that if you have one side, if you have either end here or here, if you have either end, you get the other one. What we have is a two for one deal. If you have this, you get all of this for free. And if you have this, you get this for free. So, if I have the cardinality of A being equal to the cardinality of B, I know that A is similar to B, and vice versa. Likewise for if I know the cardinality of A is less than or equal to the cardinality of B, I know that I have a similarity to a subset. Well, let's see if we can do anything with this. Well, let's take a look at a problem here. We'll take C being the set of things where what I'm talking about is a city in the United States. M being a set of things where what I'm talking about is a U.S. city whose name includes the word new. And let's see if we can prove or disprove the cardinality of M is less than or equal to the cardinality of C. Now, that's a prove or disprove. I'm not yet committing myself to whether this is actually a true statement or not. Now, if I persist in thinking about cardinality of M, cardinality of C is the number of things in these two sets, I have to know the number of cities in the U.S., and then I also have to know the number of cities in the U.S. whose name includes the word new. So, I can go through an atlas and count the number of cities in the U.S., and then I can count the number of cities whose names include the word new, and a couple hours later I'll be able to prove or disprove this statement. On the other hand, I might do something a little bit more easy, and because I know the meaning of this concept, cardinality of M less than or equal to cardinality of C, I might actually be able to do something more directly and a lot easier. Now, in order to do that, I need to pull up what the definition, what the meaning of cardinality of C less than or equal to cardinality of M is, and there's my if and only if definition, cardinality of one is less than or equal to the other, if and only if M is similar to N, which is a subset of C. And here's a useful fact of life. Data is never relevant. Oh, wait, no, sorry, wrong fact. Data is always relevant. Information is your friend. And so one of the things we can start with is we can begin by making any observations whatsoever that will help us find a relationship between our two sets, C and our set M. And it's helpful to start up by thinking about, well, what are things that are in these sets? So let's take a look at that. So let's see elements of C. Well, these are cities in the United States. And so let me think, these are places like, I don't know, Boston, New York City, Nacogduchess, and a bunch of other places. And our set M, these are US cities whose name includes the word new. And these are places like, well, New York City, New Windsor, New Rochelle, and a whole bunch of other places. And so this helps us get an idea of what these two sets look like. And it's worth making an observation that every element of M, everything that's a US city whose name includes the word new, is a city in the US. So every element of M is an element of C. And so that says that M is a subset of C. And that emerges directly from our definition of what a subset is. And we want to formalize that, which means we want to write it down. Since every US city with the word new in the name is a city in the US, our set M is a subset of C. Now, our definition of cardinality tells us that the cardinality of A is less than or equal to the cardinality of B, if and only if A is equivalent to C, where C is a subset of B. But any set is equivalent to itself. So M is equivalent to itself. M is a subset of C. And that's all I need to fill out my if and only if definition of what it means for cardinality of M to be less than or equal to the cardinality of C. Again, remember, if you have one side, and here we have the one side of the if and only if, we get the other side for free. Now, one final note here. This is a proof problem, and the portion that you prove, the proof is everything in green. The answer is not complete unless you have everything that is written in green.