01 Naive Set Theory

um we I have some uh organizational uh things to say um basically ER there will be recordings available on YouTube um and there will be a a Discord Channel um er for questions and discussions I I will I will check it regularly um and in addition there will be notes available in in my web page um the the recording of the ER of last time is a missed something like uh five minutes uh so I'll just repeat the um the introduction I I said back then amatan oh I see somebody cannot hear [Music] and do you know for the recording so it should is your mic so I I so I should uh is it okay now yeah it's good okay um so amatan most of my life I I did mathematics in in Academia I I did a PhD in in the Hebrew University of Jerusalem and then a few posts in in Europe and um two years ago I I started the working in in cryptography and um there there seemed to be a need for a a mathematical background for people working in the field and that's how this uh this course emerged I now work for the ethereum foundation this is a this is a course that is hybrid partly I mean h participants are in the the ethereum office in Berlin and some online so we will meet on a weekly basis for two hours um and uh this is expected to take something like a year um last time I gave uh a motivation and overview of what we do in the course ER and today ER I I will I will start ER talking about sets um what happened historically is that in the early uh 20th century mathematicians discovered a growing number of paradoxes and as a result they they got into a a project of a a new foundations for mathematics and they converg to what is called ER Now set theory ER as the the solid foundation for all modern mathematics but also incorporating all the previous Corpus of of math apart from a very small small things that were discovered to be wrong um and the the the formal set theory is called a Camelo celo Frankl um choice and this is abbreviated as zfc and this is a set of ax a set of proxs that is that is now the standard foundation of math and in which basically um all modern mathematics can be uh can be written and also theoretically let's say h can be verified by a computer so basically it's it's a set of axioms and deduction rules and er every mathematical statement can be written as a I mean every proof can be written as a a sequence of um deductions from from the axioms so it could in theory be checked by a computer so this gives a very a very solid foundation of mathematics ER and um what most mathematicians think about logic is it's good to know that we we have solid foundations and that in theory we we can prove all the stuff that we do in in a computer and and so on but other than this we don't care and set theory is a a research domain for logicians but this is a very small fraction of mathematical Community maybe an interesting fact is Frankel was one of the founders of the math department in Jerusalem so Jerusalem typically have a strong mathematical logic group what we will do is a what I call what is typically called naive set theory and um what I mean by that is H we're not going to write down all the axioms and be very formal this is this is not very useful but we will still talk talk about sets because they are um fundamental in every mathematical treatment ER and the slogan here is that sets are um the machine code of a modern mathematics and now what is a set so from the axiomatic point of [Music] view we we have a a language a logic um in which we can write statements um the way to write statements involves um a bunch of symbols um so we will write H this symbol this is for all and then we have exist um H negation ER implied or um uh if then ER implied and being implied this is called if and only if and did I miss something not element of uh we will soon get to the element of um yeah so this this is basically the the The Logical symbols in which H one can write statements but we will often replace some of these symbols by words so this is a for all this exist and this is [Music] not um this you could say um if then and this is if and only if and I will abbreviate this as as if sometimes so I a set um s is a collection of elements um in the syntax to to write a set um is with a early brackets and then the specification of um of all the elements let's say typical ways and to write a set one is that you can you can specify all the elements I mean this is of course Possible only if s is finite let's say one two and three h and you see I I put curly brackets in the beginning and the end and I separate the elements by a comma and another way to write a set is by by some rule so I can write s is the set of all n such that n is bigger than bigger or equal than two so you see here I have I have a formula H this you can regard this this part as like the the entry condition to uh to the set and uh entry conditions are written in in logical logical formulas ER of course I I don't want to be too formal and and and sometimes instead of writing the the entry condition in this in this way um I can write s is H the the set of all numbers two three four and then three dots three dots means a it's an abbreviation instead of writing a form formal entry condition where the entry condition should be obvious for the for the reader now [Music] um one of the the syntax the syntax we have for sets is that for every object X um either X is an element of s in which in which case we write um we write X belongs to s or or X is not is not element of s and then we write uh X does not belong to S okay so uh so this is this is H this is the basic uh a syntax of of sets and um one thing that that is is important to notice I mean um sets um or maybe um when considering elements um in a set we ignore repetitions that is for example the set S as specified by the following H following formula is equal to the set 1 2 3 right uh and the the order of the order of elements the order or the collection of elements uh in s has no particular order so what I mean by that is that um the set one to 3 is equal to the set 3 2 1 Okay so um if um a set s is finite um we write hash um s for the number of elements in s and again I I I ignore repetition right so uh so in this in this case for example the number of elements in s is uh is three and [Music] um we say that um h two sets so s is included in T and this is denoted as s in t if for any element a small s in big S the the element small s is in fact in in the set t er and and note that um we have four for sets s and t s is equal to T if and only if s is included in T and this is end um T is included in s okay so this H this kind of observation although trivial is is is a very useful tool H H to to break the equality into into two pieces it's a bit like you can say that two numbers n and M are equal if and only if n is smaller or equal to M and M is is smaller equal to n okay so this this is another a way to to break the equality between numbers [Music] um and with regard to sets we have um a basic operations one of them is is a the union I take H let's say a sets A and B and I I Define the union this will be my notation for definition I Define the union to be the set of all X such that X is an element in a or and this this is an inclusive or h x is an element in B right so potentially X could when I say inclusive I mean potentially X could be an element in both A and B and this this would still be considered an element in the union and in mathematics we always use inclusive or and the notation for this is this V and I can also talk about the intersection um if I have two sets A and B then the intersection is the set of all X such that X is an element in a and X is an element in B and I can Al also talk about uh the difference or complement and this is um a minus B is the set of elements X that belong to a but do not belong to B any any questions about this part yes I have a question about the first uh precondition for a set that you'll uh the super set is the the set of natural numbers right is that implied by the small n or oh H you mean this in this yeah yeah and yeah I I I didn't mention I will I will soon talk about famous sets but yes I if I wanted to be precise here I should have said n n is a natural number okay and N is bigger bigger or equal to two yes okay thanks so this is a good reminder maybe ER we talk about some famous famous sets so uh the first as uh as was mentioned this is the the natural numbers 1 two 3 and so on um then we have the integers this is um 0 1 - one uh 2 - 2 and so on and we have the the rationals this is the set of all qu I A over B such that A and B are integers and B is non zero H and we have the real numbers um I guess one way uh to quickly Define it is is the of all decimal expansions so it's all numbers of the form h a do uh a A1 A2 and so on such that a is an integer and H for every I bigger or equal to one a i is a digit yes apparently there's some confusion whether zero is a natural number or not sometimes apparently it is and sometimes it's not how what so in my convention it is not this this is T this is typical typical convention but it's true some some uh in some places it is included [Music] um apart from the real numbers we can also talk about the complex numbers I don't know if H any of you I I I assume that you heard about this but I will I will Define it this is all the the numbers of the form a + uh bi I such that um A and B are real numbers so this ER this I is like a formal symbol and we will talk about later we can talk about the the operations um of complex numbers it's not hard to Define but for now we only care about sets so this is a um this is the set of complex numbers and of course you can see that I have inclusions um coming back to Union and intersection I don't have to uh do a union or intersection only of two sets I can take any collection of sets so of course if I Define Union and intersection for two sets then inductively I can Define it for every finite number of sets but even if I have an infinite collection of sets I can talk about the union of this collection and the intersection of this collection how do I do it I simply generalize the formula so yes speak up could you please highlight the difference between the rational number and the real number because it seems like so so you see in in the real numbers the decimal representation first of all need not be f okay maybe we start with the rationals the rationals are all quotients of integers okay and it's true that every quotient of integers you can have a decimal representation and this is the reason that the rationals are included in the reals but it is not true that every decimal representation and especially I mean basically only if it is infinite and nonrepetitive so like like Pi Pi cannot be Pi has a decimal representation that is that does not have any Cycles I mean does not have any fixed cycle that you can so in this case I mean this needs to be proven and so on but it is not a rational number it cannot be now we will talk about I mean the the Greeks thought that all all possible numbers are rational and er then there is a famous a famous proof that the square root of two is not a rational number okay so yeah so for us ER and this this is a quick a quick definition but in fact we will not deal a lot with the the ER real numbers and the complex numbers we will talk about them for intuition but what we do is finite fields and then reals and complex numbers are not not used coming back to the the union intersection if I is a an index set is let's say a set that you can think of as a set of of indices this is and for every I in I um we are given a set AI then we can Define the union of all the AI this is this is denoted like this the union over all the IND the union of all the AI for every I in I this is the set of all X such that there exist an index I in I for which X belongs to AI you see what I did here I used the the the syntax that allows me to um H to write a condition an entry condition to the set with the uh The Logical symbols and uh and this is this is a valid a valid definition and dually I have the intersection of all all the AIS this is the set of all X such that for all I in i x belongs to a I Okay so so if I am allowed to write a formula I can also do this uh the following trick I can Define the empty set it is denoted by Fe and it is the set of all X such that X is not equal to X okay it's it's a legitimate definition and and of course the number of elements the number of elements of the empty set is zero but in fact um and this is a maybe a nice uh nice think from the the foundational point of view I can Define all the natural numbers using the empty set how do I do it um I Define one so so suppose we are like in a word in which we only have the sets and the and the basic operations that we talked about uh so in particular I have the empty set I can Define before I Define one I Define zero to be the empty set and I Define one to be the set whose unique element is the empty set and I can go on I Define two as the set with two elements one of them is the empty set and the other is the set containing the empty set as a unique element in other words a a one is the set containing a unique element that we previously defined as zero and two is the set containing two elements that we previous previously defined as zero and one and a generally I Define assuming that I defined a the number n n minus one I can Define as the number n minus one Union the set containing as a unique element the number n minus one which in other words is the is the set of all previously def find numbers or natural numbers okay so this H this is just it's not we are not going to um to use this definition this is just to show you how mathematics can be constructed completely from sets um I mean in this way we defined this is inductively and so I I get the definition um of the natural numbers as sets and we will see later how you can Define the integers and the rational numbers as sets as well we will not we will not do the construction of the real numbers but there is a way to to also construct the real numbers sets so in this fashion you construct all the basic mathematical objects as as sets from the from the empty set right this is like an an Anor ER and then this this this gives you a way to to view all the all the mathematical Corpus in this axiomatic way any questions this is so n is is a is of course a set of sets yes yeah I had actually a question about the definition of the empty set because uh it's defined as elements that are not equal to themselves but if you take for example in JavaScript I think it's JavaScript there are some languages where you have objects uh like n n not a number so clearly that's not a number but it's still something that is conceptually understood um not a number is not equal to itself uh so yeah I I mean maybe I'm miss something or I'm trying to apply something that doesn't really belong uh but it seems to me that it doesn't quite work well I mean we are not mathematics separates itself from any real world domain ER precisely for the the purpose of being being decisive and so I mean on the formal level there is an axom in set theory in in zfc that says that every X satisfies X was X right right so I guess for for my my counterargument to be receivable we would need a formal definition of JavaScript objects yeah I mean uh and I mean eventually once you put this basically in every in every logic uh in every mathematical logic set of axioms you have this axum that X is equal to X for any X otherwise it doesn't make any sense got thanks okay so now what can we do with with this syntax of sets I mean this is this is a nice uh nice thing to be be able to define the natural numbers um I I now want to be able to talk about a ordered elements I mean remember this the the elements in in a set are not ordered but many many of the mathematical objects we we deal with require some order of the elements so er in ordered so let's put it like this given two objects A and B the ordered a tle of A and B is the set I will call it o ab and this is the set that contains two elements one of them is the set containing the unique element a and the other is the set containing two elements A and B and this I mean of course we we will not eventually H we will not eventually use the ER this formal definition of ordered pair ordered tapel or ordered pair but H but this is another demonstration for what we can do in sets and and um the the the power of the the syntax we have so the observation here is that um for objects a b a prime B Prime o AB is equal to O A Prime B Prime if and only if a is equal to a prime and B is equal to B Prime so how I mean how do I prove that H um we we have two directions to prove right so let's say if o a is equal to O A Prime B Prime then I take an element in the set in the the left hand side um the set a is is an element in in OAB so this must mean that the set the singl ton a is an element in O A Prime B Prime which I remind you is the set containing H the single tone of a prime and the the two set element a prime and B Prime H but the singl ton a is a set with one element so it cannot possibly be equal so the singl ton a must be equal to either the single tone a prime or to the other element which is the set with two elements but this cannot happen right the Singleton a I mean a set with one element can never be equal to a set with two elements so um we must have that the single tone of a is equal to the single tone of a prime and since this these are sets with only one element I get that a must be equal to a prime if a is equal to a prime then I also know that the set a b is another element in a in o a um in the ordered pair of A and B and and therefore I know by by the assumption that the set a comma B must be an element in the h in the set I mean in O in O A Prime B Prime which again I remind you is the set with these uh two elements and this means that um by the the similar argument that AB must be equal to one of these two elements but it it cannot be equal to H the first element to the the set the single tone containing only a prime because a comma B is is a is a set with two elements so it must be equal to the other set with two elements a prime B Prime uh but since we already know that a is equal to a prime we must have h b is equal to B Prime okay so this is the implication um what I mean if if if I know that OAB is equal to O A Prime B Prime then I necessarily know that h a is equal to a prime and B is equal to B Prime I'll just write it again OAB equals o a prime B Prime implies a is equal to a prime and B is equal to B Prime um conversely if I know that a is equal to a prime and B is equal to B Prime then clearly o a a is going to be equal to o a prime B Prime right this is look at the the definition of of O uh OAB um if a if I a is equal to a prime here if a is is equal to a prime and B is equal to B Prime then immediately I get that OAB is equal to O A Prime B so so this this concludes the proof yes can you hear me yeah yeah uh but this only works if you have an enumeration between those two sets right like if you can count the the elements yeah if I mean but okay so I I mean I don't want to be bogged down too much in the axiomatic setup but uh kind of the the underlying assumption is that I can recognize I mean a set with one elements and a set with two elements yeah but if the set is infinity no I mean a a itself could be a itself is just an object but which could a could be an infinite set but the set containing a as a unique element is is always a finite set you see what I mean I mean like like the set containing the natural numbers as a unique element is the number of elements in in this set is one okay so this this is kind of the the syntax that you need to get used to um that um right we we we can talk about a sets as as elements in other sets Okay and in fact from the the axiomatic point of view that's all all you can do because all your objects in the universe are sets and you can define a set of sets and a set of sets of sets and so on okay I have another question yes having the set a is equal set a prime I mean from the number number of element perspective doesn't imply directly that a is equal a prime we said that um a set a is equal to a set B yeah if and only if for every a in a a must be in B yeah so is so I take I take this this a it is in a right and it must be it must be an element here yes but there is only one element in in the right hand side the one element is a prime oh yeah okay does answer so uh so if a a a belongs to the set with one element a prime then it it must mean that a is equal to a okay ER so we will denotation [Music] um I will write h a comma B for this OAB and this is the the ordered pair of A and B it is a mathematical object that has the property that if if you take a um a two sorry four objects a b a prime and B Prime and then you have equality of the two ordered pairs if and only if a is equal to a prime and B is equal to B Prime and notice that we needed a new object for for this property because the the set a comma B does not satisfy this property I mean because it could be that a comma B is equal to a prime B Prime but [Music] um for example a is equal to B sorry a is equal to B Prime and and and B is equal to a prime right so so it's because this this um sets do not do not encode the order of elements right so so we need a new mathematical object that that does take care of the the orders any questions before we take a break okay so uh let's take 10 minutes break you can uh you can look at the notes and and so on and come back with questions about what we did what we did with ordered pairs all right so maybe maybe one simple question if it has the order pair has three elements um is it then a the the super set of set with a set with a and set with a b c no the the ordered pair is a set with two elements but regardless of the number of elements no regardless of the number of elements in I mean a could be a set with infinite elements B could be a set with finite elements but o AB is a set with two elements you you can you can know that because there is one comma okay thanks okay but indeed I mean if you look at the well okay that's that's the issue all right so given given this notion of ordered pairs and we can talk so um if a and b are sets in the cartisian product of A and B is the set a * B the set of all top ordered pairs a comma B such that a is an element in a and b is an element in B okay so this is a h this this was a um in invented by the cart the philosopher and it is one of the the cornerstones of um of the the modern set theoretical treatment in in mathematics um and for example I mean maybe a simple example I take a to be the set with the 01 and B the set with 2 three then a * B it has four elements and they will be 0a 2 0a 3 1A 2 and 1 comma 3 um right because I allow h i allow every element in a to be in the first coordinate and every element in B to be in the second coordinate now it's useful to see see what happens geometrically so let's say a is um this is called the unit interval it's the set of the set of real numbers that are bigger or equal to zero and smaller or equal to one and let's say B is the interval 2 3 right this is the set of real numbers that are bigger equal to two and smaller equal to 3 then I take the interval a I put it on the x-axis the interval B I put on the Y AIS so this is zero this is one this is two and this is three and a * B is this a a purple perple area maybe I should have said um if I take um I mean I take a is equal to the real numbers B also equal to the real numbers then a * B is what we call the the plane this is denoted as R * r or r r squ and this is a I mean formally it's the set of all a comma B such that a A and B are real numbers the set of all ordered pairs of H real number in the first coordinate and real number in the second coordinate and um we have a picture for this we uh denote by this X and Y AIS and uh a point here typically is a comma B right and so so uh here if if a and b are a a only subsets of the the the real numbers then a * B is going to be a subset of of R squ and uh and the picture is is uh written here [Music] um any questions about the cartisian product okay so using this notion of cartisian product um I can Define the following um a given two sets a and b a function um f from A to B denoted F colon a arrow B is um a subset of the cartisian product a * B such that two conditions are satisfied one is um for every a in a there exist B in b such that the pair a comma B is an element in F and two if a comma B is an element in F and a comma B Prime is another element in F then B is equal to B Prime this is a formal definition of functions um and let me give the the the informal definition um a function f from A to B is a rule that Associates to every uh let's call it [Music] input element a in a a unique output elements output element B in b h and um B is the noted as F of a so F of a is is simply a a a a notation for this this output element so you see informally speaking I I think of a as the set of inputs and B is the set of outputs and a function is um is a rule that gives for every input a unique output so it has to be the defined on every every element of the declared set of inputs and it has to has to give a unique output for for that input H so so here in the formal definition if if you look at it this is just a a saying the informal definition in the in the language and the syntax we have in set theory what do I mean by that I [Music] mean instead of saying a rule because I don't know what is a rule formally speaking so a rule would be simply a collection of TPP a collection of ordered ordered pairs uh the first coordinate in the ordered pair needs to be in the uh in the input set and the second in the output set and the two conditions are are exactly saying well the first condition says that for every element in the input set there exist an element in the output set such that such that the ordered pair is a a um an element in the in the function f and two that um I cannot have the the second condition basically says that um for every input so the first condition says that for every input you have an output and the second condition says that for every input the the output that exist is unique is this clear is this uh now um question case b is empty yes and then there will be um so okay if it could also be that a is empty right if um if a is let's say let's start with a if a is empty then um a * B is necessarily empty right and um um if a a b is non empty we have a unique function um from A to B so from the empty set into into B and this is a this is completely legitimate because um what I need is like is a a collection of of pairs I mean a subset of a * B I take um f to to equal to the empty set this is a subset of a * B and it satisf and it satisfy the two conditions okay for every a it it we say in mathematics that it satisfy it vacantly because the first condition is for every a in a something needs to happen but if if there is no a in a then the condition is very vacantly satisfied right and the second condition says if I have a pair a comma B in F then something needs to happen this condition also is satisfied vacantly um on the other hand if a let's say let's say B okay let's say a is not empty and a a b is empty [Music] um then of course as before we we still have a * B is equal to the empty set but now um there is no function f from A to B there there can be no function from a to be and why because you can you can see it intuitively if if a is is a nonempty set of inputs I need to give you a a for every input I need to give you some output but I have no output to give you so so then there there would be no function question maybe yeah in Parabola also functions H not exactly so okay so this is this is a good opportunity let's let's go back to the the what is called real valued functions and see what is a function and what is not function so um I take F from the real numbers to the real numbers um let's say f of x is equal to x² this is a function this this def the function right for every x i for every input X I give you an output X squ the output is is a valid element in the I forgot to say um this is called the domain and this is called the range of the function um but if I look at um let me just draw it like this a a circle or ellipse I mean this the circle of of radius one is is the set of all tles X Y such that x² + y² = to 1 this is not the graph of a function right because it's not a graph of a function I cannot write it there is no F such that um let's call the C the graph of f is the graph of f is simply X is simply F in the the formal definition and and there is no function f let's say f of x whose graph is the the circle and the reason is I mean you take a a one x and you need to give to give it two outputs right for for this x you will need to give two outputs in order to create a graph so you can but you can separate it into two functions this this would be the graph of one function and this will be the graph of another function right or you can separate it in in other ways and the same goes for an elliptic curve elliptic curve is a is a solution set it's like the like the circle it's a solution set of a poly nomial in two variables and uh you can separate it into two h two functions and we will do it but this again it will take time um if you have let's say something like this then you can separate it into H this and this and the equation I mean you have y s equals um maybe put it here have Y2 = X Cub + a x + B with some condition on the A and B and you can write a a y y = s < TK of X Cub + a x + b and and the the one below would be y = minus X Cub + a x + B and then these are functions so why in this case I could write um y as a function of X and right and and the these would be H functions from the real we can save from the real numbers to the real numbers or we could save from the ER the positive real numbers to the real [Music] a maybe remark how do we know that two functions are equal so two functions F from A to B and um G from um let's say b or not B from C to D are equal if um first I need that the the domain and the range of these two functions would be equal okay so a must be equal to C and B must be equal to D and two um I I need that for any a in a f of a is equal to G of a okay that's it this of course this is equivalent to say that um as subsets of of the cartisian product they're they're equal so um f is this is same same saying f is equal to G as subsets of the cartisian product A and B Because fa is remember fa is is this h b this would say this would tell us that a comma B is a legitimate uh legitimate pair in F and this would tell us that um a comma B is a legitimate pair of of G right so it's it's the same H but but note something something important is that I could potentially Define the either the domain or the the range of a function to be different than a so example and maybe a question with the first point a equal C the start domain being equal just I just want to know if if there is there any function where the start domain can be different but produce like the same I mean I could okay so this is the example I mean um I mean if I take maybe yeah I I I I can look I can look at F from the real number to um the real numbers um X goes to x² right this we said maybe the one goes from the comple number to but I don't even have to go that far I can do from the positive numbers let's say R bigger equal to zero to either are let's let's do in this case and and the formula is the same right X goes to X squ these are two different functions okay even though the their formula is the same and even a third function I go from the real numbers to the non non- negative numbers real numbers and the formula is the same X goes to x² right I can do this because I know that x² is always non negative and F is different than F Prime and F Prime is also different they are a FR Prime is different than F Prime and F is also different than F dou Prime these are three different functions so you can think of it like in a in a type language like in C a function I mean if the output is declared to be an integer or declared to be a float it's a different function right so this is a this is this is this is the the convention about functions in mathematics as well yes so to that same extent you know when you said uh thetic curve it cannot be a single function because you've got uh a negative side and the positive side uh we could actually just Define a function that goes from R to R squ and you know give like output topole that would plus minus but in that case what you mean is that uh it's not the Topo is not the same branch and therefore uh actually could it not be actually no so it's a good it's a good point um I could Define and this is called the parameterization I could Define a a a function from R to r² um that who whose whose image whose graph kind of is um is the the elliptic curve H for example if you want to do it for the ccal um you can define t goes to I think it's cinus t c t right and so this is a function from R to r s um you can even say do it from from the interval 0 up to 2 pi but you can def you can simply repeat it right because cinus and Senus are um periodic and the image of this function I mean will be exactly the circle because cinus square plus sin square is always equal to 1 okay so you are right I mean there is a function for which um the the elliptic curve or the circle um are the the the image of the function but it's not it's not a function from R to R so maybe the the to be precise on on what we what I meant earlier elliptic curve and circle cannot be described as a graph of function from R to R yeah that makes sense okay um right so given two functions um say F from A to B and G from B to C their composition is the function um this is it's simply a notation this is G composed F it's a function from a to c defined by a yes A a to c defined by a a g Circle F evaluated on an element a is G evaluated on F of a okay so this is a this is exactly what um uh what you would think that a a composition of two uh function routines in a programming language does right you you need that um the input of of the second function of G would be a the exactly the output of the uh of the first fun function and then you can uh you can plug it in and this note that we say F composed with G but we write a g g Circle F okay so this it's a bit confusing but that's that's the convention and and and the reason is is H is because of this uh um formula uh it's convenient to write the G Circle F proposition if m f is a function from from A to B G is a function from B to C and H is a function from C to D then a prior I can do two things right um I I can do a age composed G and all that composed F and I can do H composed G composed F and the proposition claims that these two functions are equal here this is called associativity associativity of composition um how do we prove associativity this this is very easy I I mean remember we need to show the two functions we have two functions um right the the left hand side and the right hand side and we need to show they are equal how do we show they are equal well um we need to say we we need to show that for every input element um the two functions agree so H composed with G composed with f evaluated at a is equal to H composed G composed F evaluated at a so it's a lot of brackets and we need to start peeling them off um the left hand side is equal to H composed G evaluated at F of a right and and this is equal in turn to age composed sorry evaluated at um G evaluated at F or F of a and the right hand side is is going to be similar and um I first um do so this is age of G composed F evaluated at A and G composed F evaluated at a by definition of composition is H I mean this is G of f of a okay so we get we get the same expression um what else any questions about this in this notion of composition in functions all right so maybe one more thing before before we end um for every set um a uh there is a function which I will denote as the identity identity a this is a function from a to a and defined by a identity a evaluated an element a is equal to a right this is the function that doesn't do anything and exercise it's a very easy exercise for every function F from A to B now I can do two things I can take F and compose it with the the identity function on a this is like diagrammatically it means I I do the identity function on a and then I do F I claim that F composed with the identity function on a is equal to F and the second thing is I can I can take F and then I can do the identity function on B this is diagrammatically it means I do F from A to B and then I do the identity function of B from B to B and this is also equal to F okay so I guess this this is a this is what I wanted to H to cover for today um are there any questions or requests for next time okay and uh see you next week