 Welcome back to our lecture series math 4220 abstract algebra one for students at Southern Utah University As usual, I'll be your professor today. Dr. Angie Misseldine Lecture for we are still in section 1.2. It turns out there's a lot of stuff to say about sets and such Although in this lecture, we're not gonna on these will be sets, but we're really gonna focus on the equivalence relationship Side of the section header there. And so what is an equivalence relationship? Well, one answer is this is one of my very favorite topics As as one who studies algebraic convictors partitions is something that is dear and true to my heart They they come up with sure rings and a lot of other stuff. I do my own personal research So we're gonna talk about equivalence relationships in this lecture and there's a lot that can be said here This one mostly will present the definition and some examples of equivalence relationships So remember before we had to find what a relationship is right? What is a relationship? say that we have Two sets like say a and b and so a relationship are was some subset of the direct product of two sets And so then whenever the element a comma B is inside the inside the relationship are we would say that a Rb things like that. Well, it turns out using our as your relationship symbols kind of corny Typically, we'll use a different symbol like in this case a little twiddle will say like a twiddle B or something like that So that's the general definition of a relationship We then proceeded after we define this to define what a function relationship is In this lecture, we're gonna talk about a different relationship Which is often referred to as an equivalence relation so an equivalence relation is You often use something that in some regard looks like an equal sign. So you don't typically use equal itself because that's typically reserve for equality of sets But we could use things like a congruence symbol like when you do modular arithmetic the triple equal sign there or congruence like maybe if you do geometry or Approximately or just a single twiddle or twiddle line There's a couple of different options one could do here But you often use a symbol that in some regard resembles an equal sign here now an equivalence relationship is going to be a relationship on a single set x so that means is it's a subset of x cross x Equivalence relationship always relates things from the same set to another element of that set and so it's a relationship on a set with itself That satisfies three properties, which we could call the axioms of an equivalence relation That if something's an equivalence relation the following three properties must hold and without one of them It's not an equivalence relationship. The first one is referred to as the Reflective property the reflective property says that every element is related to itself So if you take any element x inside the set x will be Related to x or as this is an equivalence relationship. We'll say that x is equivalent to itself and when one thinks of an Equivalence relationship you often want to think of With the following idea that we're trying to generalize the notion of equality So when it comes to equality of like numbers, right x is equal to x. That's a property we want for general equivalence relationships The second property is referred to as the symmetric property Which says that if x is related to y then y is related to x the the order doesn't matter Who's on the left who's on the right? It doesn't matter whatsoever And this is again a property we see with equality that if x is equal to y then y is equal to x That that's a property of equality equivalence relationships are generalizing this principle and lastly For an equivalence relationship. We require the transitivity property the transitive property This tells us that if x is related to y and y is related to z then x is related to z as well And we see a similar property for equality if x equals y and y equals z then It must be that x equals z So as equivalence relationships are trying to generalize the notion of equality You see that these three axioms of an equivalence relationship are exactly these three properties of quality that we demand when We Equations and such so when one has an equivalence relationship some related terms that we're going to find here Is if you have a typical element x inside of the set that has an equivalence relationship Then you often draw something like x Inside the box there so bracket x bracket sometimes people write things like x bar Or something like that what this is described as will be called the equivalence class This is going to be the set of all things to kind of wrap around to the other side there all things We want all elements of x which are related to x there now x of course is going to be inside of this set Because of the reflexive property Equivalence classes are never empty. This is the main reason why we have property one here x always belongs to the equivalence The the x always belongs to its own equivalence class But then you'll also contain any other member of this set that's equivalent to x and Because of the symmetric property and the transitive property It doesn't matter which element you use inside the set here an element that belongs to an equivalence class We call a representative and if you interchange x with anyone it's equivalent to that as you pick a different representative It'll describe the same equivalence class. It doesn't matter which representative you choose Oftentimes there's a natural selection on who you choose here and let's see some quick examples of this So let's take as our set x the set of ordered pairs of integers So a and b are both integers such that b is not zero So we take the set of ordered pairs of integers so that the second Coordinate is never zero and we define an equivalence relationship on those elements. They're ordered pairs So we have two integers here, right? We say that the pair a b is related to cd if the product ad equals bc All right, so we're gonna take the first term times this So we take in the first in the first pair we take the first term and in the second pair We take the second term we multiply those together. It's ad that should equal The product of bc Which rides you take the second term from the first one in the first term of the second one So if you always take the product of the two numbers in different positions We want that to be the same and if that happens we say the two Pairs are equivalent now. We want to show that this forms an equivalence relationship. How does one do that? Well, we have to check the three axioms. We would begin by showing that The operation sorry that this relationship is reflexive. So to show the reflexive property We have to take a generic element. So we would say something like let a comma b be inside of our set x So it's just a generic element the only thing we can suppose about this element a comma b is that it's inside of x Which means that a and b are both integers and b is not zero. All right, so then notice the following then The product a b is equal to a b That is a fact of equality like we mentioned earlier a b equals a b and therefore We get that a comma b is Related to a comma b because that's what's expected right if you take a times b. That's equal to b times a so we get Equality there and so therefore we get that a sorry a b is Related to a b that proves the reflexive property So that's how one always shows the reflexive property you're going to just take a generic element of the set and argue by the definition of the Relationship why that generic element is related to itself. That's generally a fairly easy thing to do Symmetry, how does one do symmetry? So to prove the symmetry The symmetry axiom what you do is the form you're gonna select two elements So we're gonna say a and b and then c and d. These are elements in x such that a Comma b is related to c comma d So to prove symmetry you always have to assume a Relationship that two elements are related to each other and then from there you want to argue that I can reverse the relationship So by assumption we did we assume there's a relation there to get started because symmetry is it is a conditional statement if Then type statement, so we assume the if part and we're gonna prove the then part of based upon that Hypothesis we're assuming so what do we do next? Well, we now assume that these things are related so unravel the definition so This tells us that a a d Whoops a d equals b c And so then we might have to you know twist some things around a little bit thus We see that See b Equals d a so you kind of have to turn that thing around a little bit But these are just properties of equality and so therefore we get that Cd is related to a b Like so and so notice looking at the definition of this object right here if a b is related to cd That means ad equals bc. So using properties of equality. I was able to take this equation and switch it around I had to use the commutative property multiplication and I use the symmetric property of equality But these two these two equalities are equal to each other this equality right here is the one associated to this relationship and This equality right here is the one associated to this relationship It's a subtle thing, but that's what one would do to show the symmetric property here And so for the symmetry property you assume One pair is related and then you have to show that the reverse relationship is also present All right the transitivity so transitivity requires we assume we got three elements in the set that are related So we'll say something like the following let a comma b C comma d and e comma f be inside of x such that a b is related to cd and We want that cd is related to e f So we assume that we have two Relationships here and so then we want to infer from this why a b is is related to e f here now The next thing to do is to unravel the definition. What does it mean for a b to be related to cd? So then this tells us that a ad excuse me is equal to bc And that's what the first relationship tells you this one right here. The second relationship right here tells us that cf is going to equal De Assuming I did all those correctly right ad equals bc and cf equals de Hmm now this one's a little bit trickier than the previous ones like how do we how do we relate these things together? Interesting well with this one. We're I'm gonna try to combine all of these things together. So we might say something like the following. Okay, hence We see that adf is Equal to the following. So adf Adf let's see. Well ad is equal to bc. That's great So adf is equal to bc f That's the first part and then cf is related to de so that will equal bde and So we have that these things are equal to each other via the transitive property of equality You're probably gonna use that since equality is coming up here And then the next thing to notice is that well, we're looking for and this is a strategy I can't overemphasize when one does something like this whenever you're working on a proof if you ever get stuck Sometimes it's a good idea to go to the end of the proof and work backwards. So we say something like therefore a comma b is related to e comma f That's what we're trying to prove and it's important that we pay attention to what is that we're trying to prove here But what does a b relate to e f even mean that would mean something like thus A f is equal to b e so this is the statement. We're trying to show a f equals b e We're kind of there. We have a df equals b df. That's exactly what we need except for we have this extra D How do we get rid of the D? Well, we can say something like the following. Okay, well, couldn't I just like? divide by D Well, one has to be careful. You can't just divide by any integer. You could only divide by integers that are not Zero oh Yeah, well D's not zero right because D was the second coordinate and by definition that second component can't be zero Aha, we see why that's part of the definition now This is something you should also pay attention to that when we have a definition There's usually a reason why those things are put as part of the definition. It's not just some arbitrary thing It's like we only will accept purple numbers. Why purple, right? There's a reason and same thing with a proof Right if you're trying to prove if this then that there's probably a reason we listed all of the assumptions We did if there's an assumption We didn't use in the proof then it means we didn't need it and we can make a stronger statement by removing it So the fact that we had to have the second component be non-zero means we probably needed eventually so we could say something like by Canceling oh Boy, are there two L's in canceling you're now going to see why I am not having a PhD in spelling here by canceling D which does not equal zero we then get I'm going to erase this thus We get that AF equals BE and we finish the proof showing the transitivity property And so this is not the most polished proof you ever going to see here If you want to look in the lecture notes, you can see a better version of these proofs I'm trying to write this out in real time so you can kind of see as your as yourself Need to become a mathematical proof writer I want you kind of see the thought process that goes into writing a proof believe it or not writing a proof is a Creative process some people often think that oh math is so so objective There's always a right there's always a wrong and it's very clear-cut what that is And then people like oh art is so subjective, right? There's these subtleties nuances, you know, it takes a real talent to become a great artist It turns out that mathematics is both Objective and subjective there are some things that are very very objective right clear right clear wrong But it turns out that mathematics is also a creative process writing a proof is like creating an art form of some kind It's very difficult to teach proof writing because it's a creative process How do you teach someone to be creative students be creative? You know, I can say things like that, but this creative endeavor takes a lot of practice And I can try to guide you along the way, but it's very difficult. So give yourself some patience here And so that's that's this is an example of how one writes a proof. It's a creative process this now shows that this set of Pares of integers is an equivalence relationship and we actually call the set of equivalence relationships the rational numbers a rational number is just a pair of Fractions, right? I should say a pair of integers, which we call a fraction We often denote a column B instead by the symbol a divided by B That's what we usually do But we say that two fractions are equivalent to each other exactly in that context one half. Oh boy one half equals three 36 yeah, I can do that because One times six is equal to three times two This is the typical relationship on the rational numbers and so by placing on a relationship on Pares of integers we can then create the set of rational numbers that we know and love