# 03 Equivalence Relations

- Channel: [Berlin Ethereum Meetup](https://streameth.org/berlin-ethereum-meetup)
- Date: 2024-10-07
- Duration: 1:46:18
- Watch: https://streameth.org/watch/yt-AgTRQI1zUoY
- YouTube: https://www.youtube.com/watch?v=AgTRQI1zUoY

## Description

Equivalence Relations 

Lecture notes to be found here: 
https://drive.google.com/file/d/13HvwSMr0YsJrUfZr3xPe5jXiOqTLP5k5/view?usp=sharing

## Transcript

okay so um so I I would like to cover today ER this H this topic of equivalence relations um I I wasn't sure it's the right time to introduce it because it is a it is a bit tricky to understand I mean not necessarily because it's hard but because it's a it's a bit different than uh than What er what you are used to and it takes some time to digest um but I think we can try to do it today and and come back to it later on if needed and the basic motivation is H we defined I guess the basic motivation that I can I can explain at this point comes from a foundational question that is we defined H the natural numbers as sets and we would like to Define um the the integers and the rational numbers also a sets um so let me just postpone the the definition of of integers and even of um negative Russ numbers but let's suppose we we only want to Define positive IR rational numbers and so this I I will denote as as Q bigger equal than zero um and intuitively I would like to say that this is all the quotients A over B such that a a and b are natural numbers and B is non zero however we have a problem because um if if I do it from a foundational point of view then 1/ two is a different element than two over four which is right it's it's like I mean if if from like just using this A over B as an official symbol um then then before doing anything A over B I mean one over half one over two is different than two over four which is different than 48 and so on so I I would like to to have an operation that identifies a um for let's say formal quotients that I want to consider to be the same um so this you can think intuitively as kind of a gluing procedure we we we have some set and we want to ENT to to kind of pinch some some subsets in it into into actually one element ER but what does it mean to be equivalent so intuitively so we need a a notion of equivalence let me denote it as a TIA so we would we would want to say for example that 1/ two is equivalent to 2 over 4 um but what is the what what is the the the basic prop what are the basic properties that we expect from um an equivalence notion um this is given in the following definition um okay maybe I'll postpone the definition what we want as properties basically we want it um I mean for notion of equivalence on a set X we want first that for any x in x x is going to be equivalent to X right otherwise it wouldn't make any sense second we want that um for any X and Y in x if x is equivalent to Y then Y is equivalent to X this is um this is called Symmetry by the way and the first prop property is called the reflexivity and the third property is um is that for any X Y and Z in x if x is equivalent to Y and Y is equivalent to zed then necessarily X is equivalent to zed right we we would like to be able to H to move around with uh with this equivalent so if I have I know that X is equivalent to y and Y is equivalent to Z and this you can imagine is like I have some kind of path from X to Y this is the the equivalence between X and Y and I have some path between Y and Zed then I can I can concatenate the path H from X to Y and Y to Z and get A A A New Path from from X to Z so this property this is called the transitivity this is a uh I mean the whole the whole H three properties I guess you could say are are what um what intuitively we we want from um a an identification relation on a set and and so to to be more formal slightly more formal H let X be a set a relation on x is a subset let's call it R of the cartisian product of X with itself okay so R is is a set of pairs a a and we we if a comma B is a pair that is indeed an element of the relation R we denote a r b okay and a relation r on X is an equivalence relation if um it has the the three properties so um I'll just rewrite it for for any a in x H we have a relation between a and itself this is the same again in in our notation it's the same as saying that the the the pair a comma a belongs to R to um for any A and B in X um a comma B belongs to R if and only if B comma a belongs to R so this is again it's the same as saying that we have a r b if if only if we have b r a and third this is the transitivity if or or any elements a b and c in X if a is in relation to B and B is in relation to C then um a is in relation to see and when relation is an equivalence relation um we denote um a till the B whenever h a is in relation to B which is again is the same as saying a comma B belongs to the relation uh I have a question as the like the the need for I like the the first the Symmetry was if you already have uh the sorry if you have the Symmetry and the transitivity do you need uh no sorry the yeah do you need the reflexivity yeah okay you cannot you cannot deduce formally reflexivity from Symmetry and trans if you think of it I mean because to to get reflexivity you need to know that a is in relation to a yeah how would you do that I mean suppose a is in relation to B yeah then by symmetry you know that b is in relation to to a exactly but how how do you how do you I mean how do you construct I mean how do you deduce that a is in relation to a well because of the transitivity a is a is in relation to B which also means that b is in relation to a so by transitivity a is Rel to no so so transitivity you you should think of it as a triangle right yeah I have a b and c a is in relation to B yeah then you get but what is your C A so the triangle is a segment because a segment is a singular triangle oh I I see what you mean a is in relation to B B is in relation to a and then you're right that a should be uh in relation to but we don't okay this is true if you know that that a that there exist some pair a comma B for which a is in relation to B then you can indeed deduce that a is in relation to a but you up you don't know that there is even one pair you take an element a and you don't know that there exist even one element B for which a comma B is is so you you really need this is kind of an enor right yeah because indeed like if your equivalent is only about one point yeah you don't have yes yeah that's it's a good remark any other questions on the definition okay so as examples the first example is I claim that if f is a function from X to X then it defines a relation on X I mean I I I I will say that um a is in relation to B if and only if this is like so it's like a definition right um if and only if F of a is equal to B right so in other words I mean we we actually have defined a a a function as a subset of the cartisian product of X with itself so in this definition I mean you can view I mean if we View f is simply a subset of x * X then it is a relation I mean relation is simply any subset of pairs of EX with with itself however it is not necessarily uh an equivalence relation ER can you can you think of over what can go wrong what what can uh what can be a problem for a function to to Define an equivalent relation um I guess B projectivity has to be because you need to be able to have reflexivity yeah but even even if you have a bctiv of the function you don't NE get an equivalence relation right because let's say I I oh yeah I mean just take a this is it's always good to do like examples with the finite sets they have a b and c I can Define um a bjective map that shuffles a b and c so let's say a goes to a goes to b b goes to C and and C goes to a right and then even even if it's bjective I mean you see that I don't have the for example I don't have ref reflexivity so so um unless you have I meanless you have the identity function then you don't have an equivalence F the function has to be from the same like from the same set to it onto itself yes of course that's that's that's kind of a prerequisite otherwise we don't have a notion I mean one can give a more General notion of relation between the two sets X and Y but we don't we don't do it now so this is almost never an equivalent trans relation however um there is one main example um of equivalence relation that we will consider so um so take X to be the set of integers and and we said a a equivalent to B um if and only if a is equal to B mod n this means that um when I write a as um when I do long division of a by n i mean I get let's say h q1 n plus R1 and I do long division of B with with n i get some Q2 n with R2 um then R1 is equal to R2 the remainder of long division of a by n and and long division of B by n is same and I claim that this is an equivalence relation so let's see it I claim that Tia is an equivalence relation [Music] um so first we need reflexivity so reflexivity is is immediate it means I mean of course a a is always equal to a module n um symmetry is also immediate right a I mean if a is equal to B module n then immediately A B is equal to a module n and transitivity is also is also easy I mean uh if a is equal to B module 1 and B is equal to C module n then I mean the remainder of a by long division with n is the same as the remainder of B with long division by n which which is the same as the remainder of C with long division so um so a is equal to a to c okay so so now that we have this H um notion of equivalence relation we would like to to do the gluing procedure and H for this we do a construction as follows so so let X or let Tia be an equivalence relation X for an element x in x Den not bracket X as the set of all elements Y in X such that X is equivalent to Y this is called H the equivalence class of X and um let's let's inspect this uh this notion um and note that um if so so of course the equivalence class of of an element is simply a subset of X right and if I have two elements a B in the equivalence class of X then a is equivalent to X by definition or actually I wrote it in the other direction so X is equivalent to a and X is equivalent to B but by symmetry I get that um a is equivalent to x and by transitivity I would get that a is equivalent to B so here I I have by symmetry A is equivalent to X and I also know that X is equivalent to B hence a is equivalent to B by by transitivity okay so so the the equivalence class of X when all the elements in the equivalence class of X are equivalent to each other and um there is another property of this this equivalent cluster X um and that is what if I look at at different equivalence classes so um let x and y v two elements in X um and consider the equivalence class of X and the equivalence class of Y if so these these are two two subsets of B right so if their intersection is non empty then there exist some element Z in X such that Z belongs to the equivalence class of x and z belongs to the equivalence class Y and this is to say that Z is equivalent to X and Zed is equivalent to Y but then I get as before by by Symmetry and transitivity that X is equivalent to Y right I mean if Z is equivalent to X then X is equivalent to Z this is symmetry and by transitivity X would be equivalent to Y but if x is equivalent to Y I claim that the equivalence class of X is the same as the is as set is simply the same as the equivalent class of Y um so this this implies that the equivalence class of X is simply equal to the equivalence class of Y so if I found one element that is common to two equivalent classes then they they they completely identic and of course the complement of this uh of this property is that if X and Y are not equivalent then their equivalent equivalence class is is completely disjoint you cannot find even one element [Music] so otherwise or maybe let say it like this thus if the equivalence class of X is different than the equivalence class of Y then necessarily the equivalence class of X intersection with the equivalence class of Y is the empty set so you can think of it this way let's do like a a simple diagram this is X and I have some equivalence relation on X and this equivalence relation divides um divides X into a this joint subsets so this would be maybe the equivalence class of X1 this would be the equivalence class of X2 this would be the equivalence class of x3 for example um so this if you want to to get to get like a exactly this this drawing in in a concrete example ER take the integers um and the equivalence relation um is modu three so so two two integers are are equivalent if they are equal module 3 and then what are the possible equivalence classes and all possible equivalence classes are um first I have the equivalence class of zero the equivalence class of zero is H well zero of course belongs to the equivalence class of zero and then any any integer that is um equal to zero module 3 so I have 3 minus 3 um 6 - 6 and so on by the way of course since a three is equivalent to zero in this in this relation then the equivalence class of zero is is the same as the equivalence class of three the second equivalence class is the equivalence class of one this I have one I have M uh I have four I have minus two um and so on um sorry did I want minus two yeah I I yeah I substract three yeah okay and of course um this is equivalent to the for example the equivalent class of for this of course you have infinitely many ways to to write the same equivalence class and the third equivalence class is the one of two I have two five um minus one and so on okay so if I look at this this is kind of a canonical ER way to to write equivalence classes and um I get that the the collection of all equivalence classes is um is what is called the partition of the set X note um for any for any equivalence relation um TAA on E um the collection of all equivalence classes I will call it Q um this is the collection of all equivalence classes X such that X is some element in the in the main set the ambient set X um this you should you should note that when I write this definition I mean I take the the set of all equivalent classes of X of the relation this set this set has a lot of repetitions right I mean whenever two elements are equivalent their equivalence class would be the same but nevertheless I can write set because we in set we remove repetitions um and the collection of all equivalent classes of um of the relation Tia is satisfy or is what is called a partition um and a partition me simply means that um first um any two elements in q I mean elements in here are simply um to some equivalence classes are in either are either equal or disjoint and second and if I take the union of all elements in in Q when I take the union of all equivalence classes uh that appear in Q then I get back my original ambient set X okay and the the picture again is like equ so equivalence classes or equivalence relation always give you some some partition of the the the original set into a a disjoint disjoint part that cover the entire settings any questions so far okay so I I I will simply be slightly slightly formal with this H equivalence classes and I would say um the set of all equivalence classes is called the quotient of X by TAA and noted X so x divided by TA or X modu of TI this this is our notation for the set ke so in in in other words when I I mean the the way to glue um to glue elements in a set According to some equivalence relation is simply taking the the the collection of all equivalence classes right so and and again intuitively think of of the integers and the equivalent relation module 3 we want to to glue integers according to this equivalence relation so we simply take the collection of all equivalence classes with respect to this relation and um what we get is actually three it's a set of three elements right let let's just uh scroll back to this module 3 example um if I take I mean the quotient integers modulo the equivalence relation given by module 3 is simply the set of all equivalence classes but the set of all equivalence classes I can write it as the set of equivalence class of zero equivalence class of one and equivalence class of two right this is It's a set with three elements three distinct elements of course I I could also write it as the set of equivalence class of a say four set of equivalence class of um two and the set of the and the equivalence class of um what do I have here so this is one I need something for three so let's make it a minus minus three right this is an equivalent way this this is a a the same way I mean just a different way to write the same set okay but in any case this is a set of three elements so this is what this is the the the reason that the intuition is is that I glue I take a lot of I mean a lot of numbers that are the same model 3 and I I I think of them as one element so that's that's the the the reason behind the term quotient now I claim actually that the process I did with a a starting with an equivalence relation and considering then the equivalence classes which are a partition of the set ex and then considering the qu is simply the the all the elements in the partition this process can be reversed namely if you give me a let me write it down so conly suppose X is a set together with a partition let's let's call this Partition p p what is a partition it's a collection of subsets of X so it's a set UI I is some element in an index set I where each UI so for each index I in in in big I UI is a subset of x and such that I mean this is the two properties of of a partition first if I take the union of all UI for I in the index set big I I get the set X and two if um well for any i j i and J in big I either UI is equal to UJ or UI is disjoint with UJ okay this is the the defining property of a partion and if I have such a I mean such a a structure on X I can define an equivalence relation Define a relation F on x by a a setting a equivalent to be if and only if there exist some index I such that A and B belong to the same UI and this is easily seen to be an equivalence relation um the Tia is an equivalence relation let's let's check it I need reflexivity reflexivity I I take um so let a be some element in X since the union of all UI is X there exist some index I in in big I such that a belongs to UI right what does it mean to take the union of a collection of sets it means I look at all elements that belong to at least one of the sets in the union so if the union is X then it means that for any any element in in in the ambient set X there exist some um some UI for which it it came from sorry so once I know that a belongs to UI then I know that um by by the definition of the relation that a a comma I mean a equivalent to a sorry because the the relation said A A and B are equivalent if they belong to the same set to some set UI so a a and a is I mean you take the pair a a comma a both of them are I mean assuming I mean we have that a is an element of UI so obviously a the other a is also an element by definition it means that a is equivalent to to okay kind of it's a it's trivial once you just unravel the definition it's not anything sophisticated um to see simmetry symmetry is obvious because I mean it um clearly if a is equivalent to B then there is some UI such that both A and B are in UI but then but then obviously um I mean uh B is equivalent to a I mean the the the the relation we defined is is immediately symmetri and and transitivity is also the same it's also very easy transitivity if a is equivalent to B then um a are elements in UI and if B is equivalent to C then b c are elements in some UJ but then B is an element in UI and an element in UJ so I has to be equal to J so let's say UI is equal to U UJ and if UI is equal to UJ it means that a B and C are are simply in the same in the same set let's say UI um and this means that in particular that a is equivalent to C so the picture is that I give you a set any partition of the set gives you an equivalence relation and as a result you can take the quotient the quotient is simply just the different the different equivalence classes or the different ER sets in the partition and if I give you an equivalent relation you you get a partition it's it's an equivalent way of describing the data right so every every equivalence relation gives gives rise to a partition every partition gives rise to an equivalent relation and once I have one of these types of data I can take the quo which is simply either the set of all equivalence classes or simply the the collection of all the the set the ER it's time for a break but uh before we we do that do you have any questions I do have a notation question so people have questions connected to the the content um okay the notation I have the question I have if you could scroll back back to when you define the quotient quotient uh yeah right here yeah you define x divided by till equals like you have you have column equal Cube yeah and in other places you you've used equals Death yeah it's it's the same it's the same it's the same I mean I I tend to like this colon equal a bit a bit more because it gives you Direction I mean it means this is annotation that is not Define I mean we we now Define a new notation and we use something that we previously defined um but ER but you you can yeah you can write either other questions so let's meet in uh in 10 minutes okay uh let's get back um we we talked about the formal definitions of equivalence class and and quotient and er I'd like to to illustrate it with a few examples we start with um very easy examples I mean I take X to be the set of a the numbers one up to seven and let's draw it like this one 2 3 4 five 6 and seven H then I can I can illustrate an equivalence relation by by this uh by diagram and what I get is that [Music] um the quotient of x by this equivalence relation is um the equ equence the equivalence class of one the equivalence class of two and the equivalence class of five right here I I simply mean whenever two circles are connected by a line then I consider I consider them equivalent ER and and of course this qu set can be written in in other ways for example I could do the equivalence class of four and the equivalence class of three the equivalence class of uh seven and that's that's the same the same thing now going back to to this foundational ER question of of H how to define the the integers and the rational numbers given the definition of of the um of the natural numbers so this is the main example um I take X to be the cartisian product of natural numbers with itself and Define a relation done on X by setting a pair a comma B is considered equivalent to a pair a a prime comma B Prime if and only um A+ B Prime is equal to a prime + what I'm doing here is um I I think of of the pair a comma B is representing or I mean I wanted to repres presentent the the quantity a minus [Music] B so I should say um maybe if we want to be more more formal on the the foundational level I mean on on n um defined as sets we we have we have the operation plus I mean to do addition of numbers when the numbers are defined as as sets like the the zero is the the empty set one is the set containing the unique element being the empty set um remember that um the number n is simply defined to be all the numbers up to uh up to n minus one right this was the the definition we had of n is H is exactly this I mean you you can see it like zero is a is the empty set one is the set containing the empty set but the set containing the empty set is simply the set containing zero and and two is one Union the set containing one but what what is that one is the zero Union One this is the set of 01 right so you see it's it's it's it's a bit funny but I mean in this definition of a a natural numbers as sets the the element n is simply the collection of all um of all all numbers that previously defined beforehand and then of course um we have a a a a an operation of addition I mean n plus n+ one is defined right n+ one is simply n Union the set containing the unique element n and by induction I can Define n+ k n + K would be n + 1 + 1 + 1 K * yeah so we do have we do have an operation of addition on the set of natural numbers defined in this way and now we want to to to define the the integers so how do we Define the integers I I kind of want to represent the the quantity a minus B and if I can represent the quantity a minus B then of course I take for example I take a equal z and b equals some some integer I get all the negative numbers but of course A minus B A can I mean if I simply take all pairs a comma B then I get a lot of redundancy so I need to I need to Define an equivalence relation to identify [Music] pairs a comma B that would represent the same quantity a minus D and to do that I simply say Okay I I Define an equivalence relation by saying that a a two pairs like this ab and a a prime B Prime are equivalent if um a a plus b Prime is equal to a a a prime plus b and this is well defined even from a foundational point of view because I I do have this notion of plus between natural numbers as defined as as sets so let's let me just show you that this is an equivalence relation um Gilda is an equivalence clearly we have reflexivity I mean a comma B is always equivalent to to a comma B for any any element a comma B in in the set X I guess I remind you the set X is simply n i mean n cartisian product with itself um a symmetry if a comma B is equivalent to C comma D then by definition a plus d again just keep keep this in mind I mean a a comma B represent a minus B and C comma D represents C minus D and we want to think them as equal if if when I I mean I don't have a a substruction of natural numbers as defined on the foundational level of sets but I do have addition so I can say that a minus B would be the same as C minus d if a + D is equal to C+ B and assuming that I have this relation then H then obviously C comma D would be equivalent to a comma B because I can simply reverse the the equation I mean h I mean c a plus b is equal to A+ D okay so I have symmetry and transitivity um if a comma B is equivalent to C comma D and C comma D is equivalent to e comma F then I'll just write it in one line a + D is equal to C+ B and C+ um C plus f is equal to e+ D and what I want I want to show for for transitivity that um AB is equivalent to EF but to do that it's the same as saying that a plus f is equal to e+ B but from these two equations I I I can add them two together and I get that a plus um plus f plus d + C is equal to e + B plus C+ D and er C+ D and C+ D can cancel out so I get that a + f is equal to e+ B as as I want is it okay Andre um okay so I get an equivalence relation right and now Define the integers to be the Set n * n modul this equivalence relation and you see that um the equivalence class I mean the class um er the equivalence class of a pair a comma B is the the set of all pairs C comma D such that um I mean okay I'll just repeat the definition but this this is this means that a plus d is equal to um C plus d C+ B right which is again it's it's the same as saying I mean that a minus B is equal to um to C D okay so I simply in this way I have defined um a representation of the of the integers using only what I had previously defined as as as sets now okay ER the next example is the the rationals but ER you have any any questions about ER about this example yeah I have a bit of a problem because okay I understand that's an equivalence class that contains all the number there are basically uh all the results of substraction that end up with into same numbers for example let's say two being 3 minus one and uh yeah five 7 minus 5 yeah but I'm still having a bit of a conceptual Le to do to understand that this is indeed a definition of the substruction itself ah so I okay so this I didn't claim I claim that I can Define the the set of integers I didn't claim that now defining the operation of substruction okay so you're defining the set of integers but did you had you not done that already when you defined every single Set uh so every single number via set well okay I mean that's you're you're in principle right that I could have defined the integers by just declaring a um I mean a for formal symbol for every every natural number I declare formal symbol of I don't know for every natural number a I declared a bar to be some some element some some set that is distinct from a right and then I take the set of integers to be all the all the natural numbers a and all their distinct distinct sets new sets a bar but um but in this definition I I mean I I mean this definition of of the integers that I gave here ER would would it would be much more easy to to work with when you want to Define now the operation of additions addition and subtraction let's say addition is very easy because if you have a pair a comma B you have a pair um okay so let's say I guess well I I I don't want to get into too too many details of these foundations but this would be let's say one option to define the integers I think in the example of the rationals that we see now you you will be maybe more convinced that this is kind of a canonical way to H to do it so let let me let me sketch it but just to generally answer your question this definition of the integers is much more easy to work with when you actually want to Define the Operation of of addition and and substraction um sorry I'm going to no no no it's fine uh but for this definition of the substruction you are using the addition already because you have a plus I'm using addition of natural numbers not addition [Music] of okay so let each be the set of integers cartisian product with the set of nonzero integers so I write it like this integers minus the set the the set with the single element zero Define an equivalent what a relation K on X by setting um a pair a comma B is equivalent to a pair C comma D if and only if um a * D is equal to um c times B so here I'm secretly assuming that we have a um H we have defined a multiplication of integers so this is just to be complete assuming we have defined a multiplication of integers um uh multiplication of integers again I I don't want to do all the details of this foundational constructions but you can imagine it's not that hard to do to Define multiplication of integers because I know how to define let's say I can Define addition of integers in a very similar way that I I Define addition of natural numbers and then I can define a a multiplication is just addition M yes multiple addition so given given the a definition of multiplication of integers I can define an equivalence relation um like like written here and now I think of ab the pair a I think of it as a divided by B and of CD is C ided D and I say that they are equal if again like in a in elementary school I mean if a a * D is equal to C * B and this is an equivalence relation so sta is an equivalence relation I can do the details but I think you can do it as an exercise if you want um well obviously the pair a comma B is equivalent to itself then um for reflexivity I mean if the pair a comma B is equivalent to the pair C comma D then just reversing this equality give you I mean then we know that a d is equal to CB CB which is the same as as saying that CB CB is equal to a d h but then by definition this means that CD is equivalent to um to AB okay so the the the relation is clearly is clearly symmetric as well um and and transitivity let me just write it intuitively I mean if you know that a this AB is equal to CD and you know that CD is equal to EF then you manipulate the equations and you get um that AB is equal to e f right I I I leave it I leave the details to do as an exercise it's it's very similar to to what we did before and I Define the rational numbers as the quotient you take integers I mean a pair of integers such that the second coordinate is non zero and you mod out by this equivalence relation okay so um and in a similar way one can Define addition and multiplication on the rational numbers I mean you have poi right way to so because sorry the density of Q in R it's a but of course when we Define um our from Q we we don't we don't know in advance Ian we don't have in advance the density of of q&r okay right but it it is a so okay but um this is this is a it's a nice reading of how to define kind of axiomatically or from foundational level um let me just even erase this you you can read like the the the the full definition of deding cuts in in many places in the internet um I want to do a I think two more examples yeah so so now I'm getting out of the the realm of this foundational issues and we we assume that we have the real numbers ER we have the real numbers and I can consider the set X to be the uh the real dimensional plane so this is R2 this is R cartisian product with r which I mean as you you so in high school it's like the the set of all pairs X Y such that X and Y are real numbers Define an equivalence relation dilda on X by a a setting X comma y equivalent to Z comma W if only if x² + y² is equal to z s + w² and I I I leave it for you as an exercise to to check that this is indeed an equivalence relation now the equivalence class of a pair x x comma Y in R2 is the set of all pairs Z comma W such that um so I guess just to maybe stress that the what we fix and what we uh so so suppose I I I take some particular point x x0 y z and I want to consider I I want to identify the equivalence class take a particular point x0 y0 and I want to identify the equivalence class of x0 y0 so now I let Z and W vary and I'm looking for all pairs Z and W such that Z ² + w² is equal to x0 s+ y0 s so let me denote x0 2 + y0 square I denote it as as r or maybe r z um this is a positive number so in fact I could um I could even denoted I mean it would be a it would be some some square of of a nonzero number and the set of all ZW such that z s + w² is equal to r0 squ are simply the elements on the circle with radius r z all and z a w whose distance from the origin is r0 and in fact somewhere on the circle lies also x010 y z because x0 y0 also satisfy that um x0 s + y0 square is equal to r0 squ so you see the equivalence class of an element is all the all the elements that sit on a circle with centered around the origin passing through this uh through this element and um we have many different equivalence classes right we have this this white this white circle is representing one equivalence class the perel representing another and the the the blue will represent even another equivalence class so now I want to consider the the quent what is the quotient the I mean of course we can say on a formal level the quotient is just the the the the set of all circles in the origin that are centered around the origin and every circle is a different equivalence class but instead of a a set of circles I mean it this is just a a a set of points and I can I can pick in any Circle a representative um I mean I can I can choose a a a line passing through the origin let's say this line and I choose one element in every Circle that intersects this line so in this way I can identify I I I can identify the set of equivalence classes of of this example with a a simply one given line that passes through the origin okay and this you can let's say you can see intuitively that this kind of preserves the geometry because over the if I take the set of equivalent class of circles I have a notion of distance between two equivalence classes that is the the difference between the radius and instead of um looking it like this I can I can also look at the line and on the line I choose the points and the distance between any two points would be the same as the distance between their representing circles um so so so in this way I I do pres I mean the the question set by the equivalence relation kind of inherits a geometry or at least inherits a notion of distance and I can represent this this set of equivalence relations with a distance in a in several different ways in fact of course I I didn't have to choose this line I could have chosen another line like this and I take I take all the points here um any questions on on this what miss the beginning of this section this was what does Define what so I take I take the set the set a of all um I mean the set of um um T I mean ordered pairs of real numbers the plane and I Define an equivalence relation like the one in in blue okay not clear where where it is going ah no this this is just to give you a um to give you an example in which the the set of equivalence relations inherits the additional structure that that came from the original original set on which the relation was defined right on on on the set of of pairs of real numbers I have a notion of a distance this is the idian distance and then if I do a nice equivalence relation this is not always true but for a nice equivalence relation I get some induced ER induced notion of distance could we use this to you know that we're effectively the creating a partition of the plane right yeah it is a partition uh which is parameterized by the distance of any point on the circle to C could we actually use this as a definition of uh the real numbers because we take any Circle centered in it will be circular because because because in order to even Define the set x uh right so okay ER I let me let me just give you a a a a sketch of of something that we will see H we will see later on but this is a this is called the projective line a I stress that this is just a sketch because I don't want to get into H the whole details but I think you can you can already understand um some um I mean the the the direction of of of where we're going to use equivalence relation in this construction I mean um so so I take the set X to be the set of all points in the plane that are not the origin this is like the punctured plane and um Define an equivalence relation on X by um by setting X comma Y is said to be equivalent to X Prime comma y Prime if and only if there exist some nonzero scolar such that XY is equal sorry well such that x x Prime y Prime is equal to Lambda X Lambda Y in other words um I I simply say that X comma Y is always equivalent to Lambda X Lambda y for any for any nonzero Lambda and it's easy to check that this is an equivalence relation and so now I take um I Define the projective line to be the quotient of this set X or let me let me just write it explicitly the the the set of all points in the the two dimensional plane that do not include the origin modul this equivalence relation why the origin these are because if I include the origin then any point will be equivalent to any point if if Z 0 is included in the in the set then take any n any nonzero number Lambda Scala LDA and you get that I mean Lambda times yeah so wait uh do I yeah if if you take a z0 to be in the set and you declare the the equivalence relation like like I did here um yeah yeah well it yeah I guess it would not be true that everything is equivalent to everything um but I I will get a a I will get some some point ER for which is not equivalent to any any of the other points to anything else in fact um and and what are the I mean how do I describe the equivalence classes the equivalence classes I mean one particular equivalence class of let's say some XY is the set of all points that sit on a line pass passing through the origin um and and passing through the the point XY so an equivalence class in this H in this H relation can be described as simply a line passing through the origin and I have many lines passing through the origin right each of these lines would be a different equivalence class so the the the quo set is the set of all lines passing through the origin and I can I can use a a slightly different geometric description by saying I take the circle with radius one this is called the unit circle on the circle with un with radius one for every for every Vector I mean for every H element in the circle um I identify the the so-called antipodal element this is element when you take a a minus on both of the coordinates and the the projective line can be equivalently described as the unit circle when I identify two antipodal points okay and if you identify two unipodal points in fact you get an you get another Circle because you can you can say for example you you identify a these antipodal point this is like taking a a taking the circle dividing it into two halves and gluing gluing them H one on top of the other but of course this point is identified with the antipodal points here so I get again the circle um okay so this is just to give you a bit of a a a a glance to the future but we will H we will come back to it later and uh sorry I I went over time but are there any questions on uh on what we did here all right then uh see you next weekit I have a question actually that's not mass related next week is a holiday so is there going to be CL H which holiday by the way yeah Ascension ah okay they going up exactly okay there's going I I differentiate between going up and going down vacations right yes this is going on vacation I love the going on vacation okay so yeah let's let's let's take a vacation next week