 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 Missildine. This is the first video for lecture 25, for which let me remind you in the previous lecture 24, we introduced so-called properties of relations. We indicated five properties that a relation can have. Recall that a relation is a subset of the Cartesian product. It's a way of connecting two objects that belong to a set together. There were a couple of properties we introduced. We had talked about the reflexive property and its antonym, the irreflexive property. We talked about the symmetric property with its counterpart anti-symmetric property. And then we also talked about the transitive property. Now there was a reason why we picked these four properties. These are the four, five properties, excuse me. We talked about these five properties the most often because these are properties we often want relations to have. And it usually don't just want one or two. We usually want like a bunch of them, like three or some. Now some of them are for the most part mutually exclusive except for trivial cases. So we're not gonna get all five of them. But there are some examples of very important relations that exhibit many of those properties we had talked about. Now the one that gets the most attention in a course like Math 3120 is the notion of an equivalence relation for which we will talk about that in the very next lecture, number 26. But in this one, before we talk about equivalence relation, I actually wanna talk about its often forgotten stepbrother, the notion of a partial order. This sometimes gets neglected in a Math 3120 topic, lecture series for example. We're not gonna do it in our lecture series but like notice in the textbook that this lecture series is based upon the book of proof. There actually is no topic about partial order. So this is gonna be completely supplementary for this video. So if you have been following along with the book, do make sure to take amply good notes for this topic because again, this is completely coming out of, this is not in the book of proof I should say there. All right, so what is a partial order? So imagine we have a set X. First of all, a partial order is a relation on X. So that is our partial order we often are gonna draw a symbol that in some regard looks like the less than or equal to symbol. So many people will just use less than or equal to. You'll notice mine actually has a little bit more like curly Q to it. And that's to make it look like a less than or equal to symbol, but not actually the same thing. Now, if anyone's actually trying to create such a symbol in latex, I do wanna mention that the less than or equal to symbol you probably know is backslash le. But this symbol right here, its name is actually backslash P-R-E-Q, sorry, P-R-C-E-Q. The P-R-C would give you this without the line on the bottom. The E-Q puts the line on the bottom. The P-R-E-C is short for predecessor. So it's trying to suggest there's some type of relationship between an element and the next element. So that's what that symbol is in case you want to replicate it. But some other partial orders we're gonna see in a moment like the set containment symbol. I mean, it's even on the screen right here. It also very much resembles this, but as opposed to this one, which comes to a complete point, this is more rounded, this is a compromise, there is some curvature, you know, concavity is there, but it comes to a cusp nonetheless. Anyways, we're gonna try to use this symbol for a little bit as a generic partial order. And this is because it's by analog a generalization to the less than or equal to symbol. We've talked about that in just a second. So a relation is called a partial order if it satisfies the following three properties that a relation can sometimes satisfy. To be a partial order, you must have the reflexive property which remember the reflexive property says that each element X inside of the set is related to itself. And this holds for every element in the set. Every element is related to itself, it's reflexive. You look into the mirror, you see reflection, that's the reflexive property. Now for a partial order, it's also required that the anti-symmetric property holds for which reminder, what is the anti-symmetric property? The anti-symmetric property says that if X is related to Y and Y is related to X, then that can only happen when X and Y are equal to themselves, right? X and Y have to equal each other. Now we do allow for that because of the reflexive property, right? If X equals Y, then X of course is related to Y because X is related to X. Now the anti-symmetry says that the only time that the relation goes in both direction is when equality happens. So you can never reverse the relation without it being equal already. And then the third property that a partial order has is the transitive property, which as a reminder of the transitive property says that if X is related to Y and Y is related to Z, then that means that X is related to Z. And so any relation that satisfies these three conditions, reflexive, anti-symmetric and transitive, we call that a partial order. The naming of this will make a little bit more sense in just a moment. Now the set X for which the partial order is a relationship on, the set X equipped with a partial order makes the set into what we call a partially ordered set. Or some people refer to this as a post set for short, partially ordered set, post set. And so a partially ordered set is a set with a partial order. Sometimes we say a partial ordering on it. And the name, I mean, we'll get a little bit more into this in a moment. Why this called a partial order is because this relation is giving an ordering on it. It gives us some way of saying that an element is bigger than another element. And so the canonical example of a partial order is take the set of real numbers. If you take two elements inside the set of real numbers, A and B, and these could be positive, negative, zero, they could be rational, irrational, any real numbers here. We say that A is less than or equal to B if B minus A is a non-negative number. So that's what it means for A to be less than or equal to B, that when you take the difference, B minus A, you don't get something negative, you get something positive, or it could be zero. Like if A is equal to B, then B minus A will equal zero, that's okay, we do want this thing to be reflexive after all. If you take any number A, A minus A is equal to zero, which that is not negative, and therefore we get that A is less than or equal to A. It satisfies the reflexive property. What about anti-symmetry? Well, if you have anti-symmetry, if A is less than or equal to B, that means B minus A is not negative, okay? And likewise, if you have the other direction, B is less than or equal to A, that means A minus B is of course not negative, like so. And so what that then gives us here, which you'll notice that B minus A is just negative one times A minus B, and you can factor the negative out on the other side, B minus A, like so, for which then if you times both sides by negative one, you'll get that B minus A is less than or equal to zero, that is B minus A is not positive in this situation. Now you might wonder, like, is the definition of less than or equal to circular because I'm using this symbol to describe itself. This right here is just a shorthand for non-negative, and this right here is a shorthand for non-positive. I can describe a positive number without any reference to the linear ordering of the real line or anything like that. So don't worry, this is just shorthand. I'm not having a circular definition here. But my point is, to check anti-symmetry, if A is less than or equal to B, that gives you the B minus A is greater than or equal to zero. If B is less than or equal to A, that implies that B minus A is less than or equal to zero here, the only way these are both happening is because B minus A is equal to zero, which means that B equals A. So the anti-symmetry property holds as well. And then for the last one, for transitivity, for transitivity we would take A is less than or equal to B, we then also take A, sorry, B is less than or equal to C, and we wanna argue that A is less than C. So how do we do that? We'll notice that this right here tells us that B minus A is not negative. This one right here tells us that C minus B is not negative. When you add those together, if I take B, if I take C minus B and I add it to B minus A, well, that clearly simplifies to be C minus A. But if you take a non-negative number plus a non-negative number, that is still non-negative, and this then tells us that A is less than or equal to C. So we get that. So in fact, this is a transitive relation and therefore this is a partial order. And this is essentially where the name comes from. It's a partial order because we have an ordering. With the real numbers, this less than or equal to symbol tells us that you have one number less than or equal to the other number. That's where the ordering is coming from. Now, in the case of the real numbers, and you could also talk about the rational numbers, the integers or the natural numbers as well, as subsets of this poset, they themselves also become partially ordered sets using the exact same relation. Any subset of a poset is itself a poset with that same ordering. Now this right here with the real numbers is an example of what we call a total order or a linear order for which every element, there's a relationship between A or B. So if you take any two elements A, there is a relation between them. One is either less than or equal to the other or vice versa. Now in general, that's not a requirement. We'll see an example of that in just a second. But in general, that's not a requirement to be a partial order. There does not have to be a relation between elements of the same set. So given two elements X and Y, it doesn't have to be the case that X is less than or equal to Y or that Y is less than or equal to X. If that does happen, if there is a relationship between X and Y, we say the elements are comparable. This would happen if X is less than or equal to Y or Y is less than or equal to X. So let me give you an example of this. And this actually explains why we call this a partial order as opposed to a total order that happens on the real numbers. Because in general, only some of the elements are comparable and as such, we have them partially ordered. We need another axiom. If we want to guarantee a total ordering, these three properties here don't guarantee it by themselves. All right, so a canonical example of a partial order that is incomplete and hence why it's justified to call it partial is to look at the power set of any set. So X is any set whatsoever. It could be a finite set. It could be an infinite set. It could be an empty set for all I care. You take any set whatsoever, then P of X remember is the power set of X by definition, this was the set of all subsets of X. Then if you take the power set of X and you equip the relation of set containment, that is the subset symbol, this then forms a partial ordering on the power set and thus the power set can be thought of as a partially ordered set, okay? Now to prove that a relation is a partial order, you have to check the three axioms, the three properties that it needs to have. And typically people go in the order that I've already listed, reflexive, anti-symmetric, and transitive, but you can go in any order, it doesn't matter. People generally start with the reflexive property, it's the easiest one, it's the first one listed. Admittedly when it comes to a partial order, the anti-symmetric property is generally the hardest one to do, but it typically comes second. Transitive in my opinion is usually the medium one, but we usually put a third there. So let's check that. Why is the subset symbol, the subset relation reflexive? Well, if we want to show that a set is a subset of another, we take an arbitrary element of the left set and argue that it is inside of the right set. So if X is inside of A, then X is inside of A. That's a fairly straightforward statement, I think we all would agree that's a true statement. Now since X was an arbitrary set of A, this then shows that A is a subset of A. We already talked about this before, but here's a formal proof in case there was any doubt. A is a subset of A. This symbol is a reflexive relation, okay? Now let's go to anti-symmetry here. Now while this can be difficult in general for the subset symbol, this is actually fairly straightforward. If A is less than or equal to B, and B is less than or equal to A, so we assume the relation goes in both directions. This is then like the definition of what it means for two sets to be equal. If A is a subset of B, that means every element of A is an element of B. Now if B is a subset of A, that means every element of B is an element of A, and two sets are by definition equal to each other when they have the exact same elements. So every element of A is an element of B, every element of B is an element of A, so A and B have the exact same elements, that makes them equal. That's the definition of equality on sets. And in fact, this is how one usually proves that two sets are equal to each other. You show that they're subsets in both directions. So honestly, how we prove that sets is equal is essentially using the anti-symmetric property of this symbol here, right? So it is anti-symmetric. This one's fairly straightforward. It's really built into the definition of equality of sets. So that one wasn't so convoluted or anything like that. The third one, transitivity. We would, to show transitivity, we would assume two relations. A is related to B, and B is related to C. That would manifest itself in this situation as saying that A is a subset of B, and B is a subset of C. What we then need to show is that A is related to C. That is the first element here is related to the third element there, which means we need to show that A is a subset of C. Now, how do I show that A is a subset of C? I'm gonna take an arbitrary element of A and argue why it belongs to C. So take the element X right there. X, take X to be a generic element of A. Now, by assumption, since A is a subset of B, every element of X is an element of B. Therefore, X is an element of B. But by our other assumption, B is a subset of C. Every element of B is also an element of C. So since X is an element of B, that means that X is an element of C. And since X was a generic element of A, this shows that every element of A is an element of B, and that's what it means to be a subset. So we've now shown that the subset relation is reflexive, anti-symmetric, and transitive. That makes it into a partial order, and therefore the power set of any set is a poset, which is kind of fun, right? The power set is a poset, right? It's like an abbreviated power set. You might say poset, and someone might think it's the other poset, the partially ordered set, and you would be right still. That's kind of a fun little coincidence there. This is really, this example right here is sort of the canonical example of a partial order that's incomplete. Let's try to offer a concrete example. Let's take the set X, which contains three elements, A, B, and C, and let's introduce some other sets here. So we'll take Y to be the subset that only contains A. We'll take Z to be the subset that contains A and B. I need another one, but I ran out of letters, so I'll go back up the alphabet. We'll take W to be the set that contains B and C, like so. So these are all subsets of X, and thus members of the power set. Now, some relations are gonna be very straightforward here. So like Y is a subset of X, Y is a subset of Z, W is a subset of X, for example. These are relations that do hold inside of this power set, but it turns out there are some relations that don't hold. For example, Y is not a subset of W. Y contains the element A, W does not, and therefore Y is not a subset of W, but it also goes the other direction. Y, or excuse me, W is not a subset of Y. W contains the element B, Y does not. I mean, W also contains the element C and Y does not, but it turns out only one counter example is necessary here. Y is not a subset of W, but W is not a subset of Y. They are not comparable to each other. These are example of incomparable elements. Similarly, we have that Z is not a subset of W because Z contains A and W doesn't, but W is not a subset of Z because W contains C and Z does not. These give us two examples of incomparable elements that do exist inside of this partially ordered set, and this phenomenon here is exactly why we call it a partially ordered set. The relation does describe order. It does tell us when one element is larger than other, but because of incomparable elements, it's only partially ordered. Some elements we can't say are bigger or smaller than each other. Let's consider one more example of a very important partial order that we've discussed before, the divisibility relation. Of course, we never talked about it being a partial order before because we never had that vocabulary, but now let's actually argue that under the right conditions, divisibility forms a partial order. And so using divisibility, we can construct a poset. So take, for example, any positive integer N, then let X be the set of positive divisors of N. So for example, if we take N to equal 12, then we would take X to equal one, two, three, four, not five, six actually, N12. So these are the six positive divisors of 12. So consider this set right here, the divisors of 12, or the divisors of some positive integer N in that situation. If you equip the divisors of N with the divisibility relation, because it is a relation, this actually makes it into a partially ordered sets. Now, how do you show that something is a poset? You have to show that the relation is a partial order, which always means three things, reflexive, anti-symmetric, and transitive. And just go in that order, reflexive, anti-symmetric, transitive, like sing a little song there, reflexive, anti-symmetric, transitive, whatever you wanna do. I'm sure there's some mnemonic device you could do there. R, anti-symmetric, T, so rat, I guess. I don't know, think about a rat eating a relation. I don't know, use whatever mnemonic device you want. We got to check those three things, and we'll do that in that order. The proof of a partial order always has those three ingredients. Let's first check that that divisibility is a reflexive relation. Now to do that, we have to take an arbitrary element of X. Our set X is the set of positive divisors of N. So being an element of X is an equivalent thing to saying that D divides N. So just so you're aware, when I say that D divides N, now I'm just saying that it's a member of X, okay? So what I didn't need to do to show that it's reflexive, I need to show that D divides D. But to show that D divides D, I need D to equal something times D, and that has a natural candidate's one. So notice that D equals one times D, and that shows that D divides D. So we get that divisibility is a reflexive relation. Okay? I had mentioned earlier that anti-symmetry is generally the hardest one, and this will then manifest that prophecy I made. At least we'll give an example of such a thing. It turns out the divisibility anti-symmetry can be a little bit problematic. And this is actually the reason why we require things to be positive, because it turns out the proof of anti-symmetry would fail if we allowed for negative numbers here, but since we've only restricted our attention to positive integers, I think we'll be an anti-symmetric relation in that situation. All right, so to do that, we need to take two elements of our set. So take elements D and E, they both are positive divisors of N, that means we belong to the set X, and then let's suppose that D divides E and E divides D. So to prove anti-symmetry, since it's a conditional statement, there's an if and then a then. We assume the if part and then we prove the then part. We're gonna assume that D divides E and E divides D, and then we need to argue that D and E are equal to each other. That's what we're gonna go up. This is just basic direct proof right now. So because D divides E, that means there exists some integer A such that D times A is equal to E. So these ones are connected like so. Now, because E divides D, there exists some positive integer B such that EB is equal to D, like so. So you have the yellow relation that gives us that equation and the blue relation gives you the second equation like so. So, okay, let's then go forward with that. Notice that both of these equations involve the integers E and D. So if I substitute one into the other, like let's take this equation and then substitute the equation for E. If we carry that forward here, D is equal to EB, but D is itself equal, sorry, E echo is equal to DA times B, for which if you rewrite that, D is equal to D times AB. Now, there's a common factor of D here in this situation. And because this is an equation with integers, I can divide by D, but wait a second. If I wanna divide by D, I do need to make sure that D is not equal to zero. But remember, we're choosing positive divisors of N, D is not zero because zero is not positive. So you can divide both sides by D and you're gonna end up with one equals A times B. All right? Like we mentioned earlier, because D and E were positive integers, A and B themselves had to be positive integers to make those divisibility factorizations work. And as such, the only way you can have a product of positive integers equal to one is that A and B themselves have to equal one. But then come back to this equation right here, D equals EB. Well, D equals EB, but B is equal to one. So that means D equals one E, which then makes it equal to E. So because we had the relation going in both directions, it enforced that D and E were equal to each other. And this shows us that divisibility is anti-symmetric. And like I said earlier, it is requisite that we have positive integers. Because notice that negative two divides two and we have that two divides negative two, but two is not equal to negative two. If you allow divisibility to range over all integers, it's not anti-symmetric, but over the positive integers, you do get anti-symmetry in that situation. So that's an important, a little caveat I want to make mention there. Anti-symmetry was the hardest one. Transitivity is not so bad here. So let's take three arbitrary elements of X. So DEF, they're all elements of X, which means they divide N, they're divisors of N. And then we, again, transitive is and if then statement conditional. There's hypotheses, D divides E and E divides F. Then we have a conclusion, D divides F. So we will assume the hypothesis, we assume D divides E and E divides F. Then we conclude that D divides F, but we have to provide the justification in the meanwhile here, okay? Now since D divides E, that means there's a positive integer A such that E equals A times D. Likewise, because E divides F, that means there exists some positive integer B such that F is equal to E times B. So I'm just unraveling the definitions of divisibility here. For which case then, how do you end with D divides F there? If you want to D divides F, you're going to have to find some integer when times by D gives you F. And so by combining these equations together, we don't get kept in planet, we actually are gonna get the equation that we're looking for. Because notice that both of these equations involve an E. If you substitute this into the other, you get the following, F equals EB. Now E is equal to AD. So notice we have F is equal to D times AB. Since AB is a positive integer, we get that D divides F and that gives us the conclusion. So therefore, divisibility is a transitive relation. And so now we've shown all three properties, reflexive, anti-symmetric, transitive. We've got the whole ugly rat looking at us right now. And this is the general template for proving something's a poset. You check the relation as reflexive, anti-symmetric, and transitive. Now, one more thing I wanna enter into this video about partially ordered sets is how do we visualize partially ordered sets? Cause we've talked about before how a relation can be visualized as a digraph, a directional graph here. It turns out because of the properties of a partial order, we can actually simplify the relational digraph significantly. And this gives us what's referred to as a haze diagram. The haze diagram for a poset X with relation less than or equal to there. What you do here is whenever, well, the vertices of your graph are gonna be elements of X, okay? And then you're gonna draw an X. Whenever there's a relation between X and Y, but there's no one who sits in between them. So if Y is the immediate successor of X, we then draw a line and no others. Now, unlike the relational digraphs we drew earlier, we don't draw arrowheads on these things. Let me give you an example of such a thing. So take this set X equals ABC and consider the power set, which because of the set containment symbol, the relation there, this makes it a poset. We talked about this in depth earlier. Notice that this set contains, this poset contains eight elements. The power set because X has cardinality of three, the power set will have a cardinality of two cubed, which is equal to eight. So there are eight vertices on this graph, which you see illustrated on the right right there. I'm gonna organize it in an upward direction. I'm gonna put the smallest element on the bottom. I'm gonna put the biggest element on the top and this thing always flows upward. This is why on a hazard diagram, we never draw arrowheads because the direction is always flowing upward. From small to big, you go up. I don't need arrows because I know that's how haze diagrams are drawn. Now the very smallest element in this poset is the empty set. The empty set is a subset of everything. Now, when we draw a line on the haze diagram, we only draw the line for immediate successors. So for example, the empty set is a subset of the singleton A and there are no other sets in between them because since this only contains one element and this contains zero elements, the only way you'd have to fit in between is you'd have to have less than one but greater than zero elements, which is not possible. So the singleton is an immediate successor of the empty set and in fact, all three singletons are. So we're gonna draw arrows like this. Then with regard to the singleton A, let's just consider it for a moment. It is a subset of A, B, A, C and A, B, C. It's also of course a subset of itself. Now in a haze diagram, we don't draw loops like with a relational diagram, we would draw a loop like this because the singleton A is related to itself. We don't draw loops on a haze diagram because since it's a partially ordered set, we know it's reflexive. I don't need to tell you with an illustration that's reflexive, I know that as part of the definition. So with a haze diagram, we draw no loops because those loops offer a bunch of, they just come, they make it more complicated and add so much encumbrance to the diagram that we don't need them because we know, oh, it's a subset. I know that it's reflexive, okay? Also, we don't draw arrowheads because for partially ordered sets, the relation is anti-symmetric. I know that the direction goes upward, it never goes downward. There's no two directions, it's always flowing up. So I don't need the arrowhead because I know the direction it goes, okay? That's again, simplification we get to the diagram there. Now again, the subset A is a subset of A, B, A, C and A, B, C. But the immediate successors of A are A, B and A, C. I'm not gonna draw a direct arrow to A, B, C. Instead, because the relation is transitive, right? Because the relation is transitive, I don't need a direct arrow to A, B, C. I just need a path. Remember with a relational digraph, if there was a path from one vertex to another, there must have been a direct edge if the relation was transitive. POSETs are fact transitive. And so because there's a path from A to B, from A to A, B, C always traveling upwards, I don't need a direct edge because the path, because there's a path, I know there's an edge. And so haze diagrams are designed exactly to take all of the pertinent information of a partially ordered set, but have it abbreviated and simplified. I don't need arrowheads because it's anti-symmetric. I don't need loops because it's reflexive. I don't need all of the possible edges because it's transitive, any path I also can infer is an edge. So we get this nice little cube, like you can actually label the vertices of a cube to give you the haze diagram for this POSET. Let's look at some examples using divisor graphs. These are often referred to as divisor graphs. So for example, if we look at the positive divisors of four, we're gonna get one, two and four. And by divisibility, one divides two, one divides four, but it's not the immediate divisor. Two divides four and there's nothing in between them. So this one, this divisibility graph is an example of a linear order, a total order, because the graph forms a line. One divides two, two divides four. And then the other relation is one divides four, but I get that because there's a path here. This is an example of a linear or sometimes called total order. The haze diagram will look like a line in that situation, which is why we call it a linear order. Total order because everything is comparable to each other. Now in general, divisibility graphs, haze diagrams, are not going to be a total order. Like for example, if you take the divisors of six, the divisors of six are gonna be one, two, three and six. And the haze diagram would look like the following. One divides two, one divides three. Two divides six and two divides three. All the other relations can be inferred from what's there. One divides six because there's a path there. There's nothing between one and two or one and three with respect to divisibility. There's nothing between two and six or six and three with respect to divisibility. This would give us that haze diagram there. Let's do one that's a little bit more complicated. Let's do the divisor graph for 24. The divisors of 24 are one, two, three, four, six, eight, 12 and 24 itself. Of course, every divisor graph will include the number and itself, it also include one. And as you're building the divisor graph, like a little hint I can give you is one always goes on the bottom and always goes on the top, okay? Then the next ones that are gonna be immediately above one are gonna be its prime divisors. The prime divisors of 24 are two and three. These are gonna be the things that are immediately above one because after all, the only divisors of a prime number are one and itself. So there's no other divisors that sit between them, all right? So then the numbers that sit above the prime numbers are gonna be those which have two prime divisors, like six, which is two times three and four, which is two times two. A repeated prime is perfectly acceptable there, all right? Then the next numbers are gonna be those with three prime divisors, like eight, which is two times two times two and 12, which is two times two times three. And then we keep on going. Those with four prime divisors, five prime divisors, six prime divisors until we reach the top. And so this gives us then the haza diagram for the divisors of 24.