 Welcome back to our lecture series, Math 3120, Transition to Advanced Mathematics for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Miseldine. In this first video for lecture 24, we're going to talk about some important properties of relations. Now remember, we introduced the set theoretic definition of relations in lecture 23. Just as a quick summary here, we say that a symbol R is a relation on two sets A and B. If R is in fact a subset of their Cartesian product. We like to think of it as the element A relates to the element B, if and only if their ordered pair belongs to the relation itself. There's a lot of benefits from the set theoretic definition of relations, some of which we're going to see in this video as well. Now before we get into these important properties of relations, let's mention a little bit, just make sure we understand exactly what is what we're talking about here. When we have a relation, we might say something like generically, we'd say like X is related to Y or if you want more specific examples, you might have things like five is less than seven or three is equal to seven, something like that. Now be aware that these relations are in fact statements. They are either true or they're false. That's the only value that you can give to a relation here. These are relational expressions I should say. The relation itself is a set, but with a relation, you make these expressions like the following five is less than seven. This is a true statement. Three is equal to seven. This is a false statement. Then the original one X is related to Y. Currently, this is an open statement because we don't know what X, Y, or the relation are or themselves. But every relational expression is a statement. It's either true or false. Now on the other hand, if you were to take like an algebraic expression as opposed to these relational expressions, you get things like three plus four, you get two times nine, you get one divided by two. In each of these situations over here, these aren't statements, these are quantities. They're numbers and therefore they could be a real number, an integer, a rational number, complex number, depends on the number set that you have there. Now, the way that we write an algebraic expression is very similar to how we do a relational expression. You're gonna have some number, you're gonna have some symbol, and then you're gonna have some other number, right? You put those together. Now with these algebraic expressions, your symbol is this operation. You're combining the numbers together, this operational symbol over here. And so when you combine two numbers with an operation, you get back another number. On the other hand, over here, when you have your relational expressions, you connect together two quantities, which honestly any object in mathematics, you could call it a quantity, right? I mean, what's a number after all? A number is just toys that mathematicians like to play with, right? So over here with operations, excuse me, you connect two quantities together and you produce another quantity. With relations, you connect two quantities together via the relation like these ones over here, and then you don't necessarily produce the quantity back, you actually do this true-false over here. So with operations, we compute things, but with relations, we prove things, which admittedly since the value you attach to a statement is a true or false, it's a Boolean value, you can make the argument that proofs are just calculations for statements, much like algebra does calculations with operations, but I somewhat digress. It's important to realize that relational expressions are statements, they're either true or false, and therefore we often have to prove things related to relations in this situation. So in this video, what I want to do is describe some very important properties that relations can have and then start to discuss how one can prove various properties here. Because while we can look at individual statements like two are five, it would be better to think of things generically, like if I take a generic element X and a generic element Y, what can I say about the relationship, the relation as a whole and not necessarily individual or pairs? And so there are some very important properties of relations that I'm gonna list in this video that these are considered all over the place. So these are the five top properties that we're interested in when you discuss a relation. So in all of these examples, we are gonna consider relations from a set back onto itself. So R is a relation on the set X here. We say that R is reflexive if for all elements X inside the set, it holds that X is related to X. So it's reflexive if every element is related to itself. Now it turns out the antonym of a reflexive relation is an irreflexive relation. We say that R is irreflexive if for every element X of the set, it holds that X is not related to X. So what are some examples of reflexive and irreflexive relations here? For the reflexive relations, we have things like equality. This is like the poster child of a reflexive operation relation, excuse me, because every element is equal to itself. We do also have that less than or equal to is a reflexive operation relation, excuse me, greater than or equal to, but admittedly that's because they contain the equality relation inside of them, of course. You can take set containment, set containment. So a set is a subset of itself. So equality does happen, it's reflexive. You have divisibility in that situation. A number does divide itself. You can also take the approximate symbol, if you're like pi is approximately pi. That is true, right? That's a really good approximation, but generally you don't have equality, but it does allow for it. And in fact, in general, a relation will be reflexive if and only if you actually have the equality relation as a subset of that. So if the relation contains all of the equality, which of course what I mean in that situation is you have the relation X comma X for all X inside of X. If this is a subset of the relation that makes it reflexive, okay? Now in the previous video we had about relations, we did introduce this idea of a relational digraph. It's an illustration of the relation itself. And this is really great when you have like a finite set. We've looked at some examples of relations on sets with only five elements. We just called them one, two, three, four, five. And how do you detect reflexivity on the diagram, on the digraph there? It comes down to if every vertex has a loop, then the relation has to be reflexive. So for example, you look at this one right here. This is an example we considered beforehand. Notice how every element has a loop. These arrows you then interpret as your relationships. So like for example, there's an arrow between one and five because one is related to five and also five is related to one. And so the arrow pointing from one back to itself suggests that one is related to one. And this arrow here says that two is related to two, et cetera, et cetera. When you look at the digraph, a reflexive relation will have a loop at every single vertex. And therefore you see this one right here. This is not a reflexive relation because there is not a loop at every single vertex. This actually leads to the second topic we were talking about right now. What does it mean to be irreflexive? Irreflexive means that equality never happens, basically. In other words, an element is never related to itself. Now we've seen examples of this. Of course, you have not equal to is definitely an irreflexive relation. It's like the poster child of irreflective or irreflective relations there. But you also get things like less than or greater than. These are also irreflexive because equality is not allowed in that situation. If you take the negation of the set symbol, it turns out that is also irreflexive. Now, so this one means you're not a subset of and since a set is a subset of itself, the negation would be irreflexive. But some people also introduce this symbol right here where you take the subset symbol but you put a line through the equal part of it saying that you have to get a proper subset that would also be irreflexive. We could also mention that if you take not to visibility, not approximately, these are all also irreflexive things here. It turns out that if you take any relation, you could always introduce its negation by putting a slash through it. And it turns out that if you have a reflexive relation, its negation will always be irreflexive. So honestly, most of these symbols here are just the negations of the things we had above. Like you have not equal, you have not subset, not divide, not approximate. When it comes to our inequalities here, they actually got switched around. If you take the negation of less than or equal to, then you're gonna basically get the opposite here. So those guys get switched around. So if you ever negate a reflexive relation, you get an irreflexive relation. That basically gives you each and every one of them. So what about our symbol of element containment inside of a set, right? Can it be a reflexive operation, a relation, I should say? So if you take something like A is inside of A, this is actually forbidden in the axioms of set theory because we want to avoid Russell's paradox. So this never happens, which means this does happen. The not element, or not in symbol is actually a reflexive relation, which then means its negation is irreflexive. So it's important to include that in that list right there. With regard to the relational digraphs, notice that irreflexivity is like the opposite of reflexivity. So if having a loop at every single point makes you reflexive, then the absence of a loop at every single point makes you irreflexive. So this example right here of the pentagon is an irreflexive relation. You also have, for example, this one right here. This is an example of an irreflexive relation because there's no loops on any vertex whatsoever. This one right here, I couldn't quite fit it above. This is an example of a reflexive relation because we can see the loops at every single spot on the list there. I want to point your attention to this one here as well. So this one is neither reflexive nor is it irreflexive. It's not reflexive because we don't have loops at every single vertex. So it's not reflexive, but it's also not irreflexive because it does actually allow for loops. So notice here, it happens that two is not related to two in this example, so it's not reflexive, but we also have that three is related to three. So this reflexivity versus irreflexivity, it's not like an either or. If you're one, you can't be the other. If you're reflexive, you can't be irreflexive. And if you're irreflexive, then you can't be reflexive. Those are going to be opposites of each other. I should mention there is one example of a relation which is both reflexive and irreflexive. And this is basically the only one there is. You take X yourself itself to be the empty set, which means the only relation on the empty set is the empty set itself. This is an example of a reflexive and irreflexive relation. It's the only such one because to be reflexive, you have to take every element of inside of X and it has to be related to itself. But since X doesn't have any elements, it's the empty set, this condition is vacuously true for the empty relation. But for irreflexivity, it works the other way around too, that for all elements, you can't be related to each other. So if there's nothing, then it's not related to each other. So you're good, right? So this is an example of something being vacuously true in that situation because to be reflexive, let's say you're trying to show like, oh, is it reflexive or not? Well, if it weren't reflexive, that means there has to be an element not related to itself. No such element exists because it's the empty set, so it's reflexive. For irreflexivity, in that situation, to be not irreflexive, there have to be an element for which it's related to itself, no such element exists. So this is an irreflexive relation. So this is the only one that there's possible here. Now, when we were discussing the reflexive property and the irreflexive property previously, I introduced the notion of a negation of a relation. We did that, of course, in the previous lecture, but we made it a little bit more formal. I put the definition on the screen there, although I didn't really draw attention to it because we've already really talked about it. Given any relation, there is its negation. So the relation there has the condition that A is related to B if and only if A, B is inside the set. Well, what does it mean for A to be not related to B? That actually would mean that A, B is not in R. In other words, the negation of a relation is just, of course, the complement of R when you view it as a subset of A cross B. So given any relation, we can define a new relation, its negation. Another important relation that we can construct from a given relation is its inverse. So if R is a relation from A to B, then the inverse of A, which is denoted as R to the negative one superscript, this is going to be a relation from B to A. So notice the direction switches around. That is, R inverse is a subset of B cross A, and this is the relation such that you have the element B, A exactly when A, B is an element of R. So in other words, we have that B is related by the inverse to A if and only if A is related to B. That's the relationship there. This happens if and only if, of course, A, B is inside of R. So we have this inverse relation as well. And then, of course, you can take the negation of the inverse. You can combine those together if you want to. We weren't going to talk about that one so much, but the inverse is an important relation. In particular, when A equals B, all four of these things are relations on the exact same set. So R is a relation on A, the negation is a relation on A, the inverse is a relation on A, and the negation of the inverse is a relation on A. We mentioned when we talked about reflexivity that there's a relationship between the properties between a reflective relation and its negation, which necessarily has to be irreflective. So one property of one can influence the property of another. And this is why we want to introduce this inverse property because for the next one, the next property we're going to talk about, which is known as symmetry, turns out there's a relationship between the inverse of a relation and the relation itself when symmetry gets involved there. So what is symmetry here? So we say that R, the relation R is symmetric if for all points X and Y inside the set X, if X is related to Y, then Y is related to X. The opposite of symmetry here is what we call anti-symmetry, which is a little bit different than the irreflexivity we saw before. Anti-symmetry actually means that if you have elements X and Y, and if it holds that X is related to Y and Y is related to X, then it must have been the case that X equals Y. So basically for symmetry, if you have a relation in one direction, you have a relation in the other direction. For anti-symmetry, that never happens with the exception of equality. So some examples of symmetric relations that we have introduced previously. So equality is a symmetric relation. So if three equals two, then two equals three, which of course I know that's not true, but if you had like X equals three, X equals X, sure that could be true. That kind of led me to my next one, non-equality. In this case, if you're not equal, that is a symmetric relation. I'll say that one, two is not equal to three, that implies three is not equal to two. It's a symmetric relationship, approximately equal to. Is symmetric not approximately equal to? Turns out there's a connection between symmetry of a relation with the symmetry of its negation. Notice these are negations of each other, both symmetric. We'll see in just a second that for every symmetric relation, it's inverse, excuse me, it's negation. It's also symmetric. Okay. Now when it comes to a relational digraph, how do you see symmetry? Symmetry is parent when it's present when you can see arrows going in both directions. So every arrow is reversible. So when you look at this example here, the arrows only go in one direction. You can't turn around. So this is not a symmetric relation. You look at this next one right here, same thing. The arrows only go in one direction. It's not symmetric. But if we look at this third option here, this one, you'll notice that every time there's an arrow, it's a double arrow. So this is an example of a symmetric relationship here. We'll look at some more examples in just a moment. Let's then talk about anti-symmetry, then sort of the opposite notion of symmetry. So you're anti-symmetric that if whenever you have a relationship going in both directions, so whenever you have a double arrow, it turns out it actually was a loop. That is only equality. It's the only way you can be symmetric in that situation. So examples of relations that we've seen before that are anti-symmetric, less than is an anti-symmetric relationship here. Greater than, less than or equal to, greater than or equal to. Now I should mention that these are all anti-symmetric. Anti-symmetry doesn't say that you have to be reflexive. Anti-symmetry says that if you have symmetry, it only happens when you're equality. Equality is not required. So equality is allowed here, but equality is forbidden here, but all together these are anti-symmetric because for this one and for this one, there's never a case where you have A is less than B and B is less than A. That never happens. Less than or equal to is possible, but never for less than or greater thans. Okay, so those are anti-symmetric relationships. Of course, there is the set containment symbol. That's going to be anti-symmetric. And so if you look at the examples we've seen so far, I kind of already suggested this, that for an anti-symmetry digraph, it'll be anti-symmetric if there's never bi-directional arrows. The arrows never go in two directions. Loops, that's okay. A loop is okay. You don't have to have loops though. So when we look at these ones again, so this example is an example of a anti-symmetric digraph here. The relation is anti-symmetrical because the arrows never go in two directions. The only potential situation is a loop. The loops don't count against anti-symmetry. This one is also anti-symmetric because the arrows never go in two directions. This one, like we already mentioned, is in fact not anti-symmetric because the arrows go in two directions. This one is an anti-symmetric diagram because again, the arrows never go in two directions. And then lastly, this one right here, this is an example of one that is neither symmetric nor anti-symmetric. It's not symmetric because there do exist arrows that go in only one direction, such as one is related to two, but two is not related to one. But it's also not an anti-symmetric one because you do have by directions here. You do have that two is related to five and five is related to two. So not an anti-symmetric diagram, not a symmetric one. So you can have relations which are symmetric. You can have relations which are anti-symmetric. You can have relations that are neither symmetric nor anti-symmetric. You can have relations that are both symmetric and anti-symmetric. So we consider that for reflexive and irreflexive relations there for which I said only the empty relation can do that. Well, if you take the equality relation, so x, the only things related to each other are x and x is related to x, that's it. So if you take the equality relation, which again, this is a subset of x cross x. If you take any relation that's a subset of that, this is going to be both symmetric and it's going to be anti-symmetric. And it turns out you could show that the only symmetric anti-symmetric relations are going to be exactly those which are subsets of equality. Now I alluded to this fact earlier and I wanted to make it explicit now. With regard to a relation, if it's symmetric and then it says we can actually say something about its negation. If a relation is symmetric, then its negation is also symmetric. This is what I meant earlier, that the properties of one relation can influence the properties of its relatives. So for example, the negation is also going to be symmetric. How does one prove symmetry? Symmetry is itself a conditional statement which is universally quantified. So for all elements x and y, if x is related to y, then y is also related to x. So we want to prove that universal conditional statement there. So what we're going to do is we're going to take two generic elements, x and y, and suppose that the relation satisfied. Now I'm not talking about the relation r, I'm talking about the relation not r. So let's suppose that x is not related to y. Now I need to then prove that this implies that y is not related to x. I'm going to prove this by contradiction. So I'm going to take the conclusion I want and I'm going to negate it. So suppose that y is not related to x is that y is related to x. So suppose to the contrary that y is not related to x. Now if y is related to x, sorry, suppose to the contrary that y is not not related to x. So y is related to x. Well, this is now involving the original relation r, which is a symmetric relation. So if y is related to x, that's what we're assuming, that means that x is related to y. But that then contradicts our original one there. We get a contradiction. That's the opposite of what we assumed. So therefore, this is now false. We get that y is not related to x, which is then just this one turned around. So that actually tells us that the negation is a symmetric relation. So if a relation is symmetric, it's negation is also symmetric. This is why since equality is symmetric, not equals is also symmetric. Approximately we said was symmetric, which means not approximately is also symmetric. Now we introduced this inverse. How are they related? Well, it turns out that a relation is symmetric if and only if it's equal to its inverse. So the inverse reverses the order of things. Well, if you reverse all of the order pairs and you back the exact same thing, that's exactly what it means for a relation to be symmetric. So far we've introduced reflexive and then it's opposite irreflexive. We've introduced symmetric and it's opposite anti-symmetric. There's one last property of relations we wanna introduce in this video and this is what we call a transitive relation. We don't have a opposite notion of transitivity. At least we're not gonna introduce one in this video here. Again, if r is a relation on x, we say that r is transitive if for all elements x, y, and z inside of the set x, whenever x is related to y and y is related to z, it holds that x is related to z. And so let's see some examples of transitive relations. Well, equality is a transitive relation. Less than is transitive, greater than is transitive. Less than or equal to is transitive, greater than or equal to is transitive. What else? We have set containment. It is transitive. We have divisibility. That is a transitive relation. I wanna come back to this divisibility one for just a second. Actually, I wanna focus on this one for a second, but actually because with regard to symmetry, there's something important to mention here. Is this a symmetric relationship? Well, it depends on the set that you're considering. If you think of divisibility equipped with the integers, so if you take integers with divisibility, it turns out this is not symmetric. Well, it's not symmetric. You actually never get symmetry here. I meant to say anti-symmetry. It's not anti-symmetric. It's not symmetric because like two divides four, but four doesn't divide two. So it's not symmetric. But is it anti-symmetric? It's also not anti-symmetric because you get two divides negative two. You get that negative two divides two, but you don't have that two equals negative two. So with regard to the integers, divisibility is not anti-symmetric. It's not symmetric. But if you change the set to the natural numbers, so you've looked at the natural numbers with respect to divisibility, it's still not symmetric, but it is now anti-symmetric. In that situation, if A divides B and if B divides A, then we actually have that A equals B. And for this reason, because divisibility is anti-symmetric and we restrict ourselves to the positive integers and zero, this is oftentimes why when we talk about divisibility, we restrict our attention to natural numbers or just positive numbers if we don't want to include zero either. Because we want anti-symmetry for this relation, we just had to introduce those terms yet. So divisibility is an anti-symmetric relation when you're on the natural numbers, but not with the integers. So the set in play does matter. It makes a big difference there. I should also mention with regard to this symbol, not that symbol. I don't want to make it, you think that it's transitive there, but with regard to symmetry, right? Is it possible that A is an element of B and B is an element of A? With sets, again, we don't allow this thing to happen. No, no, no, no, no, no. In this case, you cannot have A as an element of B and B as an element of A. So that never happens with sets whatsoever. And therefore that actually makes this set symbol, this actually makes it anti-symmetric. Because the hypothesis of the conditional never is satisfied, it's actually vacuously true that this is an anti-symmetric property there. It's not symmetric, though, for sure. So those are some of the things I should have said on the previous slide. Anyways, let's get back to transitivity. If whenever R is related to Y and Y is related to Z, it must also hold that X is related to Z. So equality less than, greater than, less than or equal to greater than equal to set containment divisibility. These are all examples of transitive relations that we have seen previously in our lecture series. With regard to the relational digraph, you'll notice that a relation is transitive if whenever there's a path from one point to another point, that actually implies there's a direct arrow. So look at this graph right here, which represents a relation. There is a path from, say, five to one. Notice you can take this path from five to one. Since there's a direct path, that's a good sign there. There is a path from, say, five to two, but there's also a direct path in that situation. And so a graph is transitive exactly when there's always a direct link whenever a path exists. So this is a transitive relation. This is also a transitive relation. When you look at this one, this one is also transitive. Whenever two points are connected, there is a link that actually connects both of them there. Let's look at a few other ones. This is an example of a transitive relation. Again, whenever there is a connection between two points, there's always a direct path between them. Is this one transitive? Okay. You'll notice that four is all by itself. So it's disconnected from the rest of the group. That's okay. That's perfectly okay. To be transitive, what has to happen is that if there is a connection, there's a direct connection. So notice in this situation that there is a path from one to five, but there is no direct path here. So this is an example of a non-transitive relation. So it is possible for relations to be non-transitive. All right. Some other ones I should talk about. Let's talk about like approximately. It's approximately a transitive relation. Well, what does approximately actually mean? If two numbers are approximately equal to each other, it means they're close to each other. And so you can have it like, you can have a number X1 that's approximately X2. You can have a number that's like X2 that's approximately X3. You can have a number that's approximately, X3 that's approximately X4. But the thing is approximately doesn't mean like, it just means they're close to each other. So if one and two are close and two and three are close and three and four are close, eventually there could reach a point where you have like somewhere down this sequence, you have like X in minus one is approximately X in, but you have that X1 is not approximately equal to X in. Like, cause the thing is like, this means they're close and these are close. But when you put them together, even though like X2 is close to one and close to three, it could be that X1 and X3 are not close enough to be considered approximate, right? And so approximately, honestly, the approximate relation is not really well-defined. What does it mean? Like why is pi approximately 3.14? Like what does that actually mean? It's a little ill-defined, but if you take that idea of closeness, we can make this rigorous and calculus time on epsilon delta. But there could be a point that just because these two are close and these two are close, it doesn't mean that these two are close to each other. Okay? So this approximate symbol is, it's symmetric, it's reflexive, but it's not transitive. Okay? And so these topics we're introducing, it's important to realize they all offer something new that you can have a relation that's not transitive but is symmetric-reflexive. You can have something that's reflexive and transitive but not symmetric, like less than is such an example. It's less than or equal to it's reflexive, but transitive but not symmetric. It's actually anti-symmetric. But you can take equal, for example, equals reflexive and transitive, but not anti-symmetric. It's actually symmetric in that situation. So you can construct examples of relations that satisfy some of these conditions, but not all of the conditions, which is why we've introduced all five of them. I also wanna mention that this symbol, this set element is also an example of a non-transitive relation. So you could have things like A is inside of B and B is inside of C, but we definitely should not have that A is inside of C. That'd be kind of weird. Maybe, maybe not, it depends. It actually can happen sometimes, but let me show you it's not transitive. Let's take, for example, we'll take the set A to be the set that contains the natural numbers, the integers, and the rational numbers, something like this. And so notice that zero is an element of the natural numbers. And the natural numbers is an element of A, but we have that zero is not an element of A. So again, this shows that it's not transitive. And so these five properties are gonna be properties of relations we explore in the future. They're all independent of each other. A relation can have some of the properties, but not other ones. And they're not necessarily mutually exclusive. There are some times where even opposite looking properties can be cohabitating, like you can be reflexive and irreflexive in one special case. You can be symmetric and anti-symmetric in some special cases and such. Definitely you can have some of these properties together. And we're gonna talk about relations in the future that have some of these properties and then develop the theories around those.