 Welcome back to our lecture series Math 31-20, Transition to Advanced Mathematics for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Angela Missildine. Now before we begin lecture 23, I wanted to give a quick summary of where we are in our lecture series and then give us some highlights of where we're going to be going to. So just so you're aware that like with our lecture series as usual, we have three categories we always talk about in the lecture. We have our mathematical topics we talk about, we have our logical topics that we talk about, and we also have our communication topics that we talk about. Now amongst our mathematical topics, we began our lecture series with the conversation of set theory, we transitioned to commited torques, which was computing the cardinality of finite sets. And then we've had a recent discussion about the set of integers being a very important set that we've been talking about. With regard to our logic, we've started with things like Boolean algebra, understanding the different operations and meanings of statements and the arithmetic involved with logic. Then we started learning proof patterns. So this includes things like direct proof, contra positive proof by contradiction. And then with regard to our integer unit, this was really a hybrid unit where the mathematical topics and the logical topics were actually in harmony. The whole focus of that third unit was mathematical induction. So now as we move on to a fourth unit in our mathematical category, we're going to then talk about what we refer to in mathematics as a relation. Now I want to give you some connections of where this is going to be leading to us. Relations which we'll define in this video will ultimately lead to our fifth and final unit in this lecture series that functions well able to mathematically define properly what a function is and then look at some applications of that. For example, how do we work with sets with infinite cardinality? Cognitive torques was all about computing finite cardinalities of sets. But it turns out infinite cardinalities have some subtleties that will be necessary to consider as we transition in mathematics. And relations is going to be the first step in that direction. In the next couple lectures, we'll develop the notions of relation and then we'll transition to functions as our final unit, like I said. Now like we saw in unit three, our topics on integers, the mathematical topics and the proof topics actually would hand in hand. And this actually kind of shows that the deeper and deeper one gets into in mathematics, you really can't separate the logic from the mathematics anymore. We're going to see that over and over and over again as we introduce relations in lecture 23 and lecture 24 and beyond as we've talked about relations, it's going to be constantly we're proving stuff about relations over and over and over again. And we're going to be using the proof patterns that we've now developed. The logic that we've developed in this lecture series is now able to come to bear and help us understand these more complicated mathematical topics like relations and functions. And so while we will continue for a while to have some lectures, helping to improve our ability to write proofs, communication wise, logically wise, be aware that for the rest of the series, our goal is to prove things about the mathematical topics we're introducing and all of the logic we developed is to get us to this moment where we can start to analyze our mathematical objects through the lens of logic. So with that said, what is a relation? Well, in plain English, we have a notion of what a relation is. Two people related, like in a family point of view, two people related, maybe their parent child, our cousins or, you know, third cousins once removed, whatever spouses, these are all types of like family relationships. It means there's a connection between the two people in that situation. Mathematically, we want to classify what is a relation and it turns out we can do this with Cartesian products. So we say that a relation R between two sets A and B is actually a subset of their Cartesian product written in another way. A relation is the subset of the Cartesian product A cross B. And so if we take some element of A, the first set, and we take some element B from the second set, we say that A is related to B with respect to this relationship R and we'll denote this as A R B. So the R is then used the R of the set, which is the relation is then used as a symbol connecting A to B. A is related to B with respect to this relation R. And we say A is related to be exactly when A comma B is an element of the subset. So if the ordered pair belongs to the subset of the Cartesian product, then we say that A is related to B. And this will be denoted as A R B. Now conversely, if A and B are not related, that would indicate that the element A comma B, the pair, is not belonging to the set R. And this will typically be denoted as A is not R B. So we draw a slash through it, typically in this direction here, we draw a slash through the symbol R to show that A and B are not related to each other. Now, while we can talk about relations in general between two sets, A and B, typically when we talk about relations, we actually only have one set in mind. So the set A and B are actually the one and the same thing. And so instead of saying that R is a relation from A to B, we actually might talk about a relation on A itself. Now, when we talk about functions in the future, those will be relations where the set A and B are not typically the same. But at least for the next couple lectures, we're going to be putting a lot of focus on when the relation is just on a single set. So we're looking for a subset of A cross A in that situation, which is often denoted as just A squared for short. Now, as the name suggests, relation, right? It's a mathematical relation captures the notion of relationship between quantities. And it turns out we've seen relations all the time in mathematics. We just may not have thought about it in that way before. Like, for example, we could say that one is less than five. That is a mathematical relation. We might also say that three is not less than two. With every relation, you also get its negation, okay? One is less than five, but three is not less than two. When we work with equations, like X plus one is equal to three, it turns out that the equal symbol is itself a relation. It's connecting this quantity with that quantity. We say the two things are equal if it's the same quantity. But apparently, they might look different based upon the expression and play here. Like I said, with any relation, there is its negation. We could say that X plus five doesn't equal three. Just put a slash to the relation and that's perfectly fine. With regard to sets, the subset symbol, of course, is a relation. We say that this set is a subset of that set. And of course, we can also do its negation, right? But we also get lots of variations here. We might say something like, well, A is a subset of B but not equal to B. Again, lots of relations with regard to set containment. We can also do this with element containment. For example, one half is an element of the rational numbers. But let's say something like, oh, the square root of two is not a rational number. We've proven that. Again, this element, this in, like I'm an element in that set, that is itself a relational symbol. And you're indicating a relationship. There's a relationship between this number and the set. And this would indicate, oh, you're an element of that set. There's a relationship between this number and that set, but it's the opposite. It's the negation. This relationship is that, oh, the square root of two is not an element of that set. And so we've seen lots of these things. Let's take divisibility. Six divides 24. But on the other hand, five does not divide 24. Again, these are examples of relations that we've introduced in our lecture series here. The divisibility symbol is a relationship between integers. And again, it doesn't stop there. We can keep on going, right? I mean, we had these inequalities beforehand. One is less than five. Three is not less than two. We, of course, can do things like less than or equal to, greater than, greater than or equal to. Things like x squared is greater than equal to y or three is not greater than or equal to pi. Things like that. The approximate symbol, right? Pi is approximately 3.14. There is a relationship there, but it's a different relationship to equality. Pi is not equal to 3.14. But there was a relationship of nearness. They're close to each other. We can do a lot of things. Like the square root of two, we might say, is not approximately 10. Maybe we think that's too far away. And so in mathematics, mathematics is rich with relationships. We're talking about them all the time. And so these are just some common relationships that we have seen in previous mathematics also in this lecture series as well. So let's look at some examples of relationships in this more formal set-theoretic setting that we defined on the previous slide. Let's take for our set the set of integers for a moment. And let's consider a relationship on the integers and integers. So just a relationship on the integers. We'll call it R. We typically refer to a relationship for R. It's a very clever mnemonic device there, R for relation. So we'll say that R is the set of ordered pairs inside of z cross z satisfying the relationship that x minus y is a natural number. So x minus y is either equal to zero or it's a positive integer in that case. So let's look at some examples of this. So notice that by definition of this set, the element 3 comma 2 belongs to the set R. And that's because 3 minus 2, which is equal to 1, is a natural number. Now, since the ordered pair 3 comma 2 belongs to the set R, that means that 3 is related to 2 with respect to R. So we'd say 3R2. And we're using the symbol R here because we're talking about a generic relation, even though we have a specific one in mind here. But this is similar, like saying like 3 is greater than 2 or 3 is equal to 2, which of course, that's a false statement. But we just put the relational symbol between the two quantities that are related or not related. And so even if we don't have a specific symbol here, we can make up a new symbol on the fly, call it R. We use the same pattern there, a number related to a number. And that relation, of course, could be the negation of the relation, of course. We'll see some examples of that. Another example, we say that with respect to this relation R, 17 is related to 5. Why is that? Well, it's because 17 minus 5 is equal to 12. That is a natural number. And by definition, that means the ordered pair 17 comma 5 belongs to the relation. So if the ordered pair belongs to the set, that means the two elements in the relation is related. Now be aware that as we've defined relationships using ordered pairs, the order does in fact matter. If you switch the order, you can actually get a false statement. For example, when it comes to 5 minus 17, this is now equal to negative 12. And that is not a natural number since it's a negative number. And so this tells us that 5 is not related to 17 because R is related to 17 if and only if 5 comma 17 belongs to the set. But in this case, it does not. So if the ordered pair belongs to the set, then you're related. If the ordered pair doesn't belong to the set, then you're not related. And these are equivalent notions by the definition of a relationship. But as we've defined this one, be aware that if you switch the order of the two elements, x and y, you don't necessarily get, you don't get the relatable. So in this case, you have that 17 is related to 5, but 5 is not related to 17. If we go back to our examples of families, we could say something like, we could say something like, you are the child of your parent, but like, let's just make, let's take some fictitious people here. We'll take Bob and we'll take Susie here. And so we can say that Bob is the son of his mother, Susie here. And so that's the relationship. You know, Susie is Bob's parent, but you can't reverse that around. While Susie is Bob's parent, Bob is not Susie's parent. I mean, biologically, that's impossible to do. Absence of weird sci-fi time traveling movies or something like that. So in that regard, it doesn't work. You can't reverse the relationship around. Susie is the parent of Bob, but Bob is not the parent of Susie. And as such, some relationships cannot be turned around. As we've defined this relationship R in consideration here, you can't turn it around. So while 17 relates to 5, 5 does not relate to 17. Another example is 2 and 5 are not related because 2 minus 5 is negative 3, not natural number. The ordered pair 2, 5 does not belong to the relation. Therefore, 2 is not related to 5, even though 5 is related to 2. Okay. Another example, negative 2 is not related to 2 because the ordered pair negative 2, 2 doesn't belong to R. If I take negative 2 and I subtract from it 2, I get negative 4. It's not a natural number. Okay. But I could say that I could say something like negative 2, negative 2 does belong to the relation. And that means that negative 2 is relatable to negative 2. And in fact, I could say that for all integers n, you actually have that n is related to n. That actually is a thing you could prove. And this is something we'll look more into, of course, into the next lecture. So we won't focus on that right now. Now I do want to make mention that this relationship R that we're talking about, this wasn't just drawn from the ether whatsoever. This symbol, this R, this relationship R is none other than just the greater than or equal to symbol with respect to the integers, right? Integers have the property. This is true for all real numbers, in fact. Two real numbers have the property that X is greater than or equal to Y, if and only if X minus Y is non-negative. That's what that means. Now, of course, this might seem like circular reasoning. So you could write that as like, oh, okay. X minus Y is not negative like so. That's what greater than or equal to means here. Less than or equal to is also defined similarly. So this relationship that we define using R is actually a relation we already are familiar with. That does happen sometimes. Let's look at another example. And this time focusing on the real numbers, we're going to take S to be the subset of R cross R, where we take all order pairs of the form X comma X. So this means that the order pair 3, 3 is inside of S. So E is related to E with respect to this relation S. Same thing, five is related to five with respect to this relation, because five comma five would be inside the set. Conversely, two comma five doesn't belong to the relation. It doesn't belong to the set. That's because two doesn't equal five. And therefore, S is not, excuse me, two is not related to five with respect to this relation. Now, this symbol right here is kind of given in a way, right? S is none other than the equal sign. And I know this kind of looks mathematically weird. S is equal to equals. But I'm describing the relationship, right? So this is a set. You can think of these things as sets, right? We can actually view, again, as strange as you might think of this, we can view equals as a subset of R cross R, because it's a relationship. It's a relation on the set of R cross R. It's exactly this relation here. So S sets S and equals are equal to each other. I know it seems kind of bizarre, but we're discussing relations in a formal rigorous abstract manner. And so this type of statement does come into play. So this statement here is just saying that this relation we introduced is none other than the equal relation on the set R cross R. I should say on the set R. All right. So now we've looked at some examples of relations. I want to look at a couple more before we end this video here. But one way of trying to understand a relation is to try to visualize this. This is not just true for relations, but in general, when you take an abstract concept like relations or sets, we often try to look for illustrations that can help us visualize the abstract concept. Like as we've talked about sets in the past, we use things like Venn diagrams to help us understand things like sets, unions, intersections, et cetera. We can do a similar thing for relation. And so if you have a relation R, it's a relation from the set A, from A to B there, because again, the ordered pairs, the direction does matter. So sometimes you emphasize from A to B. We can visualize a relation, particularly when it's the finite relation, using what's called a directed graph or a digraph for short. Now in advanced mathematics, there is a concept called a graph. And that typically is used to describe what we refer to as an undirected graph. And when I talk about graph, I don't mean like, here's the graph of y equals x squared. And we have a very different meaning in mind here. In which case, what we're describing right now is formally a directed graph or a digraph. But if you do call it a graph, it's not exactly incorrect. But one must be careful in the combinatorial study of graph theory, there can be some, there's different meanings in players. So I don't want to get too much into that right now. So we're going to introduce this notion of a digraph. Now associated to every relation is a digraph. So this is the relation digraph, for which a digraph consists of two sets. There's one set, which we refer to as the vertex set. And we have one set referred to as the edge set. Well, what is the vertex set? Well, for a the in general, this vertex set can be any set whatsoever. But when you're constructing the digraph associated to a relation, the vertex set will be the union of the two sets A and B, which of course, if A and B are the same set, then you just get the vertex set is A. And the elements of the vertex set we refer to as vertices, vertices being the plural of vertex there. We think of them actually as points in the plane. So when we draw a graph, we often will draw it by drawing little circles, little points. So our vertex set could be something like this. And you could arrange it however you want for convenience. Next, what we're going to do is then we consider this edge set. Sometimes it's called the arc set. But edge sets more commonly used here in this context, a link also as a possibility. The edge set for a general digraph is going to be a collection of arrows that connect the vertices together. So you get something like this. So that arrow points to that one. This arrow points to that one. This one points to that one. And be aware that we draw arrows, but they don't have to be drawn straight for convenience. You could draw them however exotic as you want. That's perfectly fine. You have the possibility of an edge pointing back to itself. So you have a loop in that situation. Sometimes when you draw a digraph, you allow the possibilities for a multi edge there, multiple edges. You could also have them going in both directions. They are directed there, but you can go in both directions. Now, if they go in both directions, sometimes you just drop the arrowheads entirely and just draw a single link right there. So this is for a pretty crazy graph that we've drawn on the screen right now. Now, with regard to relations, the edge set of a relation is actually the relation itself. It's a set of ordered pairs, and they are ordered pairs called edges. Like I said, some people call them arcs. Some people call them links. These are all synonyms in this place. And they're ordered pairs, but we can think of them as arrows with direction. So if this point was A and this vertex was B, then we would draw an arrow from A to B whenever A is related to B, which of course happens if and only if the ordered pair A comma B belongs to the relation. So that if A is related to B, if the ordered pair A comma B belongs to the relation, then we draw an arrow from the vertex A to the vertex B, respecting that direction. And like I said before, if it goes in both directions, you can drop the arrowhead because then there's no ambiguity in that situation. This can be a helpful visual tool to understand relations. So let's look at a few examples of finite sets and then draw their relational digraph there. So for the remaining examples of this video, consider the finite set A equals 1, 2, 3, 4, 5. And so I'm going to consider five different relations on top of this set A right here. The first one we're going to call are, and the elements of the relation are the following, 1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 1, 4, 4, 4, 3, 4, 2, 4, 1, 5, 1, 5, 2, 5, 3, 5, 4 and 5, 5. And this of course gives us a relation on A squared right here. And so how do we interpret this? Well, if you look at the first one, 1, 1, this means that the element 1 is related to 1. And it also means with regard to our diagram, we're going to draw an we're going to draw an arrow from 1 back onto itself. This is what we called a loop from before. If you look at the second element there, 2, 1, this means that 2 is related to 1. And we would draw in our digraph an arrow from 2 towards 1, showing that relationship that you can see right there. 2, 2 indicates another loop, which we see right here. 3, 3 is also a loop, like so. If you look at 3, 2, again, this means there's an arrow from 3 to 2. 3, 1 means there's an arrow from 3 to 1, like so. And if we continue to do that for all of them, there's a loop 4, 4. There's an arrow 4, 3. There's an arrow 4, 2. There's an arrow 4, 1. For 5, there's a loop from 5 to 5. There's an arrow from 5 to 4, 5 to 3, 5 to 2, and 5 to 1. Although I think I wrote those in different orders, that doesn't really matter. So this right here is a very, very interesting looking relation. And so notice what we have here. If I look at the diagram, there is a relation from 3 to 2 there. Look at the direction there, of course. So 3 is related to 2. But on the other hand, if we were to consider something like, okay, is 1 relatable to 4? When you look at the arrow here, the arrow goes the wrong way. There's 4 is related to 1, but 1 is not related to 4. So it turns out you don't have a relation between those things there. Now, if you view your set A as a subset of the integers, which makes sense. It's 5 integers, 1, 2, 3, 4, 5. This symbol, again, is none other than just r. Greater than or equal to symbol that we were looking at earlier. And so this relation, if you were to just restrict it to the 5 elements, 1, 2, 3, 4, 5, greater than or equal to would look something like this. And you'll notice that the arrow is always pointing from the bigger number to the smaller number. So like if you look at number 1 right here, all arrows point to 1. There's no arrows exiting 1, because with this set 1 through 5, 1 is the smallest number. It's not bigger than anyone. So if you were to follow a path along this thing, you get stuck at 1, because 1 is the smallest there. Conversely, if you look at 5, 5 is an example in this set where none of the arrows are pointing to 5 other than the loop itself, of course. If you were to follow an arrow out of 5, you can't get back to it, because it's the biggest 1. And so no one is related to 5 other than 5 itself, because 5 is bigger than all the other ones. Let's look at another example. Let's look at this example S right here, for which our elements are the following. 1 goes to 2, 1 is related to 3, 1 is related to 4, and 1 is related to 5. We also have that 2 is related to 3, 2 is related to 4, and 2 is related to 5. We have that 3 is related to 4, and 3 is related to 5, and we have that 4 is related to 5. So notice some of the things we have here. We have that 3 relates, I guess our relation is S now, 3 relates to 5 because of the arrow that we see right here, but conversely, 5 does not relate to 3, because the arrow doesn't go the other direction. Now again, this little diagram here, we still have our 5 points. There's no loops this time, and the arrows are actually pointing the exact opposite direction. That's because in this situation, our symbol S is now the relation that we're going to do less than, the less than relation. 1 is less than 2, 2 is less than 5, 2 is less than 3, 3 is less than 5, but notice it's not less than or equal to, right? Because if it were less than or equal to, our diagram would need to have all these loops put into that. You would need that the element is relatable to itself, which we don't see that in this diagram. We don't see it in the set, and so we don't get less than or equal to, but we do get less than. So it's kind of like the sort of like the opposite relation to what we had before. Alright, let's look at this one here, T. This one's an interesting example here. The relation as we have illustrated is 1 is related to 1, 3 is related to 3, 5 is related to 5, 2 is related to 2, 4 is related to 4. So in this case, everyone's related to themselves. You see these 5 loops in the diagram to suggest that these things are related to one another. That is, they're related to each other. But what relationships do we have? 1 is related to 3, 3 is related to 1. So in this case, we do have arrows going in both directions. We haven't seen that in the previous examples. Like I said, sometimes you'll just draw this as an arrow, a line with no arrows whatsoever is what I meant to say. But for our examples, we'll keep it as a digraph. So there's arrows going in both directions. We have that 3 is related to 5, but we also have that 5 is related to 3. So again, you see arrows going in both directions. We also have that, let's see, what do we say? 1 related to 3, 3 related to 1. We have that 3 is related to 5, 5 is related to 3. And then we also have that 1 is related to 5 and 5 is related to 1. That was my mistake once. See arrows going back in directions there as well, both directions. Now, you see that in this case, though, if you look at 1, there is no relationship between 1 and 2. But conversely, there's also no relationship between 2 and 1. They're not connected to each other whatsoever inside of this graph. 2 is related to itself, and it's related to 4. That's all there is. On the other hand, you have 4, which is related to 2, and 4 is related to 4. And so while I drew this picture this way, using the same 5 pentagonal shapes that I did before, the same high points, one could actually redraw this picture in the following way. You have sort of like a triangle where you have, again, double arrows. I'm just going to write just lines in that situation. You have something like this, and there's the loops, of course. So it'd be like 1, 3, and 5. And then you can actually move the picture entirely like the 2 and 4, like so. And so you can actually separate them, and you see that there really is like the separation of the graph. It's sort of like two subgraphs that have come together. This is an interesting example, because this example we're looking at right now is our first formal analysis of what we will eventually call an equivalence relation. An equivalence relation requires that the relation be reflexive, which basically means at this moment that there are loops at every single point on the graph. It has to also be symmetric, which symmetric means that every arrow is actually double headed. It goes in both directions. And then thirdly, there's a property called transitivity for which if you're on one, you can actually loop around, right? You can go, if you're connected, you can reach each other if you're on the same branch. But this one has these different components that aren't necessarily connected. It's not required for a equivalence relation. We'll define this formally in a future lecture, but these properties that are being reflexive, being symmetric, being transitive, these are topics that we will discuss in the very next lecture, lecture 24 in our series here. Let's look at just a few more examples and then we'll end this video on relations, just meant to be an introduction. Now, one thing that'll be interesting to say here is that if you have two relations like S and T that we had before, so the S relation we had before, remember that was the less than symbol, our symbol T, we didn't say anything about that yet, but that symbol T actually had to do with parity. Notice that with the, I think I misspelled the word parity there, JK on that one, parity, there you go. When it comes to parity, we had in our relation that one through five were related because they're all odd numbers and two and four were related to each other because they're both even numbers. So we have those two different relations that have meanings, less than and parity. We can actually combine them together. If you have two relations, you can take the intersection of the relation. What would that mean? The intersection of the relation would suggest that you want the relation where this relation is satisfied and this relation is satisfied. After all, to be a relation, you just have to be a subset of a squared. And therefore, if you take the intersection of two subsets of a squared, you get another subset of a squared, it gives you a relation. So this would be the relation that incorporates the notion of and the number is less than and they have the same parity. So this is going to give us a smaller relation. We have only a few, a few relations now. You have one three because one is less than three and they're both odd. You have one five, one is less than five and they're both odd. You'll have three five, which again, three is less than five and they're both, and they're both odd. And then lastly, also have two four, two of two is less than four and they're both even numbers. So notice our, our digraph is much more reduced this time. All of the loops we had before are gone because a number is not less than itself and our two directions are now gone. It only points towards the bigger number now. So we actually, we don't have the same structure that we had before with regard to our relations. Okay. One, I should also mention one could do a similar thing with unions. Like you could take S union T. What would that mean? This would be like, oh, you want to be less than or you want to have the same parity. Oh boy, why can't I smell this word today? Parity. In which case you could, I'll let, I'll let the viewer consider what would that, what would that relation look like? What would the S union T look like? Because after all, as S and T are subsets, their union is also a subset of A squared. So it is also a relation. We could talk about the compliments, right, of a relation. We could talk about set differences and all of those, all of those mechanics we can do with sets you can do with relations as well because they themselves are subsets. All right. So let's look at one last example here. Let's, we're now on, we're now on U going through the alphabet here, U. This time U has the relation that it's one is related to three, three is related to three, five is related to two, two is related to five and one is related to two. For which if you illustrate that in a diagram, if you get the following thing, there are some oddities to mention about this one. So first, why does three have a loop and no one else has a loop? We haven't seen that before. When loops were present, they seem to actually be universally present in our diagrams. This one only has a loop at three, nowhere else. Okay. Some things we have seen before is like everyone exits from one, everyone enters into five. Okay. You've seen that before. So you could call this like a source and this like a sink because once the water goes out of the sink, you can't take it out anymore. But look at four, like no one is related to four, not even four itself is related to four. Now four was part of the set A, but there's no relations between four. It seems kind of odd, but that's, that is a perfectly acceptable relation. Now, when you look at this one, this relation U, it kind of feels like, well, I can't really think of any familiar relation that could connect here, like less thans equal to a parity, all those ones we could describe with using words, but this one doesn't really have a familiar description. You could argue this is like a random relation, but be aware to be a relation, you just have to be a subset of the ordered pair. So this is a relation just as good as all the other ones. We might not have the appropriate context to the relation, but it is still a relation nonetheless. And so as a closer remark, I want to make mention here that if you take the power set of A cross B, this is the set of all relations from the set A to the set B. Every subset of A cross B is a relation. So if you take the power set, all the subsets, you get all the relations. And there are two notable relations I should make mention. There is the relation A cross B itself. What if everyone is related to everything? That's, that is a relation. Is it a very interesting relation? Maybe it depends, but that is a relation. And on the other extreme, the empty set, the empty set is a subset of A cross B. It gives you the empty relation, the relation where no one is related to each other. So like in this case, four is not related to anyone. The empty relation would be that no one is related to anyone. It's an extreme, but it is a relation, right? Oftentimes we're going to look for some happy medium, right? We're looking for a subset between A cross B and empty set. There are some relations, but we don't necessarily want all the relations because if everyone's related, then, then maybe there's no, that doesn't mean anything, right? If the relationship is everything, then there, there, maybe there's no relationship at all, right? The distinction is kind of is the relevant thing in that situation. All right. And so this gives us some examples of relations. Hopefully you enjoyed these examples. In the next lecture, as already mentioned, we'll actually start studying properties of relations, things like transitive, reflective, symmetric, anti-symmetric, just to name some of the possibilities.