 Well, we continue our discussion about the relations that we started yesterday and let me remind you in particular we discussed two special types of relations namely equivalence relations that is one type and second was a partial order and let us discuss a few more things about the partial order. Let us say that x is a set and say less than or equal to is a partial order on x. So, this is a partially ordered set by the way let me emphasize once again that though here I am using this notation less than or equal to it does not mean that it is the usual less than or equal to order among the numbers. It is any partial order this is just a notation and this is something which is fairly common in this study of this order relations. Usual partial order is denoted by this simple and the partial order among the numbers the usual less than or equal to relation is one special case of this partial order. We should also use the usual notations which we use in case of the usual partial order that you know for example, suppose we have two elements a b in x. Let us say with a less than or equal to b we will also this is same we can also denote this as by this b bigger than or equal to a. So, this will be just nothing but reverse partial order it is it basically means the same thing just a different notation. Similarly, a strictly less than b by this will be a less not equal to b and a not equal to b and similarly and this will we can denote this also by this symbol b bigger than a if there is some m in x such that a is less than or equal to m for every a in a and then this element m that is called upper bound upper bound will use the standard short form u b for a upper bound of or for a upper bound of a. Remember that of course in general set may or may not be bounded above in general set may or may not be bounded above it is bounded above it should have an upper bound also notice here that it is m it is not said that m belongs to a that upper bound did not be an element of a it is any element of x upper bound did not be in a. Second thing is that this upper bound did not be unique for example, suppose you take some element in x which is bigger than or equal to this m then that will also satisfy this because of transitive relation suppose you have let us say n and m is less than or equal to n then since a is less than or equal to m m is less than or equal to a will also be less than or equal to n. So, any other n which is bigger than or equal to m will also be an upper bound. So, upper bound is in general not unique. So, we will also discuss another similar concept what is called least upper bound least upper bound is called l u b again the standard short form least upper bound means what of course first of all it should be an upper bound because and it should be smaller than any other upper bound that is that is the thing. So, least upper bound is. So, let us let me say a number suppose let us say I will call that number let us say p not number any element p in x is called this least upper bound I will simply write l u b of a if two things first thing is that p is an upper bound of p what does this mean it means a less than or equal to p for every small a in that is the first thing it satisfies this property second thing is that if you take any upper bound for example m if m is an upper bound of a then m then p is less than or equal to m that is the meaning of second property is this if m is an upper bound of a then p is less than or equal to m this least upper bound is also sometimes called supremum just the same concept, but different supremum that is same as least upper bound and we shall be using both these terms quite frequently. Now, is it clear to you that in general an upper bound may or may not be unique, but least upper bound will be unique always that is clear for example suppose p and q are two least upper bounds then by the second property since p is a least upper bound and q is an upper bound you must have p less than or equal to q similarly reverse the rules so q less than or equal to p and by that by the anti-symmetry property p must be equal to q. So, even though upper bound may or may not be unique least upper bound will always be unique of course remember none of these things may exist all that we are saying is that if least upper bound exist it must be unique. So, let us just mention that least upper bound of a if exist is unique in a similar way we can define what is meant by saying that a set is bounded below what is meant by a lower bound and what is meant by greatest lower bound and in a similar way we can show that in general lower bound may not be unique, but the greatest lower bound will always be unique. Let us since all those things are very similar to whatever we have discussed in this case we shall not go into the detailed discussion on that. So, that means what is it that I am talking about first of all what is meant by bounded below lower bound greatest lower bound greatest lower bound and again just as this l u be the standard short form of least upper bound g l b is the standard short form of greatest lower bound and corresponding to this supremum greatest lower bound is also called infimum in a similar way one can show that if infimum exist it must be unique. Now, let us also discuss one more concept which is very similar to this upper bound etcetera it is called a maximal element it is slightly different from an upper bound let me talk of maximal element in x. Let us say that m belong to x is called a maximal element it means there is nothing bigger than this m it means there is nothing bigger than this m in x so one of the ways of writing that is that x if a belongs to x and m less than or equal to a that is if at all there is anything bigger than or equal to m then it must coincide with m then a equal to m. Now, try to understand that there is a difference between maximal element and an upper bound in upper bound for example, suppose m is an upper bound of x what will that mean it will mean that a is less than or equal to m for every every now this is not what we are saying for a maximal element we are not saying something like that we are not saying that a is less than or equal to m for every m all that we are saying is that if m is less than or equal to a then a must coincide with m that is there is nothing strictly bigger than m in x that is all that is called maximal element. I think if we take the examples it will be clear let me recall this example we had taken the example of this set of all natural numbers and in this we had defined this order m let me use the same notation again m less than or equal to n this means the not in the usual sense of less than or equal to this means m divides n this means we had seen that this is a partial order this is a partial order. Now, this time what I will do is that instead of taking the whole set n instead of taking the whole set n I will take only let us say this set suppose I take this set as x 1 2 let us say up to 10 and with this order with this partial order take first 10 natural numbers and consider the order m less than or equal to n means m divides n. Does this set has have any any maximal elements 10 is a maximal element fine is 10 an upper bound is an upper bound of x think carefully what is the definition of upper bound for an upper bound what we require that every element here must be less than or equal to 10 for example suppose you take the element 3 is this true is this true no that means 10 is not an upper bound 10 is not an upper bound that is clear. But 10 is a maximal element so there is a difference between a maximal element and upper bound and this is this happens when it is a partial order because certain elements may not be comparable at all for an upper bound what we require is that it should be bigger than or equal to every element in that set that means it must be comparable with every other element and bigger than or equal to but for a maximal element we are not saying that. But is this 10 only maximal element are there other maximal elements what about 9 what 9 is maximum are you saying that is this true are you saying this that is not right so 10 is not greater than 9 in this in this order so is there any element which is greater than 9 in this set x no so does that mean that 9 is a maximal element 9 is a maximal element is that clear 9 is a maximal element what about 8 it is also maximal element. So, now do you understand difference between a maximal element and an upper bound now this should be very clear in your mind because these concepts are important this is because this is not a total order in this x you will have two elements which may not be comparable to each other at all and once if that is the case if the elements are not comparable then there may not be any upper bound yes yes we have an example for example what are all the in fact you can list all the maximal elements we already see 8 9 10 all of them are maximal elements you can verify yourself that 6 7 these are also maximal elements what about 5 5 is not because there is 10 and 5 is less than ok let us take one more example of this type let us start with a set which contains let us say three elements and we will consider we will consider first this is a power set of this how many elements this power set will have 8 so what are those 8 elements since there are 8 we can list there are 8 elements we can list so empty sets then these three singleton sets A B C and then these three set A B B C A C and then the last set A B C which have we accounted for all now what I will do is that instead of taking this whole set we already seen that this is a partial order set with set inclusion that is this is a partial order on this set now instead of this I will take this set x as I will remove this last set from here now is that clear that ok sir let me let me first before going to that let me just again go back to this does this set have an upper bound with respect to this order and what is that this one this is an upper bound because this set is given any element here it is less than or equal to this so that is an upper bound and what I plan to do now is that precisely remove that element that is I will just remove this element from here and it is obviously now not 2 to the power it is x with respect to the same order now now does this set have an upper bound yeah it is which one which is the upper bound A B C no that is not here in the same but upper bound need not be in the set ok but now I am considering only this set now I am considering partial order only on this set I am not considering anything outside this ok fine this whole A B C will be an upper bound for this but I am not considering let us say elements outside this set so inside the set there is no upper bound ok all right does it have a maximal element think think carefully yeah for example ok let us look at this take this set A B right is there is there any set in this which is which contains this A B other than this right no other set is bigger than this right so this is a maximal element right and the same thing can be said about B C A C etc all these three are maximal elements what about this they are not they are not maximal elements right so is this is this clear to you now the difference between an upper bound at the maximal element see usually if an upper bound belongs to the set A itself then it is called a maximum element right if an upper bound lies inside the set A itself then it is called a maximum element of A but there is a difference between a maximum element and maximal element ok there is a difference between a maximum element and maximal element it should be in fact that is the first thing that should be very clear to you in your mind all right then we will go in a since the discussion will be very parallel to what we are discussed about the maximal element I shall not go into the details of that in a similar way you can define what is meant by minimal element what will be a minimal element we will say that ok I will say instead of this M let us say L is will be called a minimal element if instead of this M less than or equal to A we will have A less than or equal to L then that should imply L is equal to A that is if there is anything less than a given element then that should otherwise should coincide that is that is the difference of minimal element and minimal element is not same as lower bound it is not same as least element and similarly you can construct examples of that I shall not go into the detail discussion of that because that is that is very much similar to whatever we are discussed about the maximal element ok now we discuss one very important question when does the partially ordered set have a maximal element right and the answer to that is given by a very well-known principle of set theory it is called Jorn's lemma and this is lemma which will be used in many proofs of very well-known theorems but anyway before going to that let us first see what exactly this lemma says ok this says when does a partially ordered set have a maximal element so let us say that let x be a partially ordered set so if every chain in x if every chain in x has an upper bound remember chain means totally ordered set suppose you take a totally ordered subset of x so suppose every such totally ordered subset has an upper bound suppose every then then x has a maximal element we shall not go into the proof of this Jorn's lemma that is because it is actually one of the axioms of set theory we do not prove it in my first lecture I told you about axiom of choice and axiom of choice of one version of that is that if the you take a family of non-empty sets then its product is also non-empty another of course we can also restate it by making use of the concept of choice functions etcetera it can be shown that the axiom of choice and Jorn's lemma are equivalent that means if you assume axiom of choice you can prove Jorn's lemma and conversely if you assume Jorn's lemma you can prove axiom of choice but we shall not discuss any of these proofs that is because this is not a course on set theory because those kind of things will take lot of time we are basically just reviewing certain things in set theory which we shall require in the real analysis course and let me again say that this Jorn's lemma is used in many proofs of the standard theorems not only in analysis but in many other courses for example in linear algebra in order to prove that every vector space has a basis you have to make use of this Jorn's lemma whenever you will learn that proof you will see that you require Jorn's lemma. Now let us go to the next topic we will also spend a discussion about the partial order for the time being we now want to say something about the number of elements in a set and how that is how one goes about that way so again suppose let us say a and b are two sets empty or done empty I do not bother about it right now then we define what is meant by say that a is numerically equivalent to b and we shall use this notation for that notation we shall use this notation a is numerically equivalent to b if there exists a bisection between a and b that is there should exist a map from a to b which is one one and onto so let us say that a it is said to be numerically equivalent to b this is the rotation if there exists f from a to b which is okay either you can say one one and onto or a bisection okay basically this is a notation which says that a and b have the same number of elements that is the meaning of saying that a is numerically equivalent to b now let us take this set of all natural numbers let me also remind you we have we have defined what is meant by a function right in general a function is a rule which takes suppose we are taking to set function rule which takes from a to b a is called domain and b is called codomain if this domain is the set of all natural numbers then that function is called a sequence right sequence is nothing but a function whose domain is the set of all natural numbers then this set b can be anything it can be real numbers rational numbers complex numbers any other objects matrices if it is a matrices it will be sequence of matrices if it is functions it will be sequence of functions it sequence of real numbers etc this is a concept of sequence which we shall discuss in more detail little later for example when you take a sequence of real numbers what is meant by a convergent sequence etc but here all that I require you to know right now is that if if you have a sequence then you can arrange the elements of the sequence like this x 1 x 2 etc that is you can label the elements of the sequences x 1 that is x 1 is the thing but f of 1 x 2 is the thing but f of 2 etc all right now coming back to this numerical equivalence suppose this is the set n that is the set of all natural numbers suppose we take only the first n natural numbers we take the first n natural numbers we shall denote that say we shall use some name for this suppose I will call it j suffix n that is 1 2 etc up to n this is called segment of n now before proceeding further let us make one very obvious observation obvious in certain sense suppose I take two such segments j m and j n j m and j n when will this happen j m is numerically going to j n if and only if m is equal to n okay so that is the first observation so if and only if m is equal to n okay let me just write this as a theorem again I will not go into the discussion the proof of this proof is fairly easy okay in fact if m is equal to n there is nothing to prove right it is an identity map okay you can take j n to j n as identity map that will be 1 1 and on 2 but only thing is that if j m is numerically equal to j n then proving that m is equal to n that will require some work but it is also not very difficult or another way of showing the same thing is that if m is different from n then there can be no bijection between j m and j that is little easy to show but that will also take some time but I suggest that you try to do it on your own okay if there is any difficulty then we shall discuss that in the class okay alright now then let us go to next concept suppose we take a as any set okay a is any set then a is set to be finite set to be finite if a is empty if a is empty or a is numerically equivalent to one of these segments okay a is numerically equivalent to one of these segments if a is equivalent to j n for now for such finite sets suppose a is a finite set we will define what is meant by cardinality of a and instead of writing this full cardinality we shall use this short notation cardinality of card a this is this we shall define as this number n okay this number n if it is non-empty if it is empty we will define it to be 0 so cardinality of a to be defined as 0 if a is an empty set is equal to n if a is numerically equivalent to j is equal to empty set that is right thank you if a is equal to empty set and it is n if a is numerically equivalent to j n okay and can a be numerically equal to two different segments is it possible that a is numerically equivalent to j 4 and j 7 7 no right so that means this number n will be unique if once a is a finite in fact that is what is this observation okay that is what is this observation okay j m is numerically equivalent to j n that will happen if and only if m is equal to n so given a finite set it can be numerically equivalent to only one of the j n's exactly one of the j n's or else it should be empty or and what we have called is cardinality of a roughly not roughly it basically means the number of elements in a set a cardinality of a is nothing but the number of elements in a set and what is what is the definition it is that number n such that a is such that there is a one-man correspondence between a and j n okay between a and j one thing follows from this okay and that is the following if a is a finite set and suppose you take any proper subset of a okay then there can be no bijection between those two sets right so why this happen because if a is a if you take any set any subset a of b okay then b also will be numerically equivalent to one of these j m's okay one of the j m's and then or one of the j m's and then again we use this there will be no bijection if m is different from n okay if m is different from n so we can say that a finite set cannot be numerically equivalent to any proper subset of itself right of course one can first show it here okay in fact it follows from here if you take any proper subset of this okay it will be numerically equivalent to some j m where m is different from n and hence there can be no bijection between those two sets okay so let us just make this observation a finite set cannot be numerically equivalent to a finite set cannot be numerically equivalent to its proper subset and once we define what is meant by a finite set the definition of infinite set is clear okay whatever is not finite is infinite okay so we will set a is of called an infinite set if it is not a finite set okay so infinite this means not finite what does it mean first of all it means that it is not empty okay so it means first of all it is not empty and it is not there is no 1 1 2 map between a and any of these segments for no n a is numerically equivalent to j m that is the meaning of saying it is not a finite set okay it is an infinite set and okay one obvious example of an infinite set is this okay set of all natural numbers it is easy to show that this is not numerically equivalent to any of this because if you take any such segments you will obviously find one element which is outside this okay we will see that this property is not true for infinite sets okay in case of infinite sets you can easily find subsets which are numerically equivalent to the even sets and one can very easily find such subsets but just for the sake of completeness we will just take one example like that let us say take the set n 1 2 3 etc okay and we will take some yeah fine we will take another set suppose x is 2 4 6 etc etc so this is a proper subset of x right sorry proper subset of n and what is the what is the 1 1 2 map between n and x yeah so we just define f from let us say n to x by f of n is equal to 2 n okay for then f is 1 1 and on now the next obvious question okay should such a thing happen for every infinite set here we have given an example okay n is an infinite set okay n is an infinite set and n has a proper subset which is numerically equivalent to f okay now the obvious question is suppose you are given not necessarily n but any infinite set okay will it always contain a proper subset which is numerically equivalent to itself okay the answer is yes it is not obvious it will require some work okay and let us now proceed to do that work and to do that we will need one more definition and let one result among the infinite sets we make some one further subdivision okay we say that a set is countably infinite okay if it is numerically equivalent to n okay it is if it is numerically equivalent to n so we will say that this is the definition let us say a is a set a is set to be countably infinite countably infinite if a is numerically equivalent to n okay that means there is a 1 1 and on to map between a and the set of all natural numbers for example this set x 2 4 6 etc this is a countably infinite set okay and okay we will also use another term what is called countable countable means finite or countably infinite okay countable means finite or countably infinite in fact there is a reason for this word countable as the word suggests it means that elements in the sets can be counted okay because what is the meaning of countable finite means it is numerically equivalent if it is empty set we can forget about it if it is not empty it is either numerically equivalent to one of these zanes or to the whole of n okay if it is equal to one of these zanes I can write element it has n elements and I can write those elements as a 1 a 2 etc if it is numerically equivalent to n we can still write the elements as a 1 a 2 etc but only thing is there will be no last element okay so when the set is countable its elements can be arranged in certain order okay first we can talk of what is first element what is second element etc and they can be counted in that order and that is why it is called countable so if a using the language of a sequence if a set is countable you can view the elements of the set a as a sequence okay as a range of a sequence okay if it is countably infinite if it is finite then you can just write those elements as a 1 a 2 a n etc okay then before proceeding further let me also say something about this terminology unfortunately this terminology is not followed in all books certain authors use some other terms what we have called countably infinite is also called denumerable denumerable okay that is Rudin uses the word denumerable and what is worse is some books use countable for this denumerable what is called what what we had countably infinite some books use countable for this okay and then what we have called countable they call at most countable okay so that that is something you should remember when you are looking at a book okay right in the beginning you see how those terms are defined and then then see subsequent things okay but we shall follow this notation okay for us countable denumerable is same as countably infinite and countable means finite or countably infinite okay now let us go to what we were having in mind namely about showing that every infinite set has a proper subset which is numerically equivalent to itself but even before going to that we need one more fact let me write it as a theorem it is the following every infinite set has a countably infinite subset see by the way before going to this okay suppose I make a following statement that every countably infinite set has a proper subset which is numerically equivalent to it is that clear to you right because if it is a countably infinite subset I can write the elements as a1 a2 etc then take the subset like this a2 a4 etc that will be and you can easily establish a one-one correspondence between this okay alright now is this clear to everybody okay now suppose I prove this theorem will it imply that every every infinite set has a countably infinite subset okay and that countably infinite subset has a set which is numerically equivalent to itself okay so will it immediately mean that every infinite set has a proper subset for example suppose let us say you you started with a set a okay suppose these are infinite set that you started with okay and suppose b is a countably infinite subset okay right now this b will have a proper subset suppose I call that set as c so let us say c as a proper subset of b such that b is numerically equivalent to c okay right then from that can we say immediately that can we say immediately that a has a proper subset which is numerically equivalent to a yes yes yes tell me how that's right right you what you do is see that you already know that there is a map one one and onto map between b and c okay then for those elements which are outside b you just consider identity map that is take a map see there is a map let us say f from b to c this is one more and onto okay right now what I will do is that I will consider a map from let us say g from a to okay I'll write this set later okay the map is defined like this g of x is equal to f of x for x in b and equal to just x for x in a minus b okay that is those elements which are outside b I will take g of x as simply x and those element which are inside b I will take g of x is equal to f x okay right is it obvious that this will be one one and onto okay but now what is the what is the range of this map okay see because the elements which are in b those go to c okay right and the elements which are in a minus b those will go to a minus b right so c union a minus b right right and is that a proper subset of a because we are removed the elements so this is not same as b right not yeah a minus right okay we are removed the elements which are in b but not in c okay which are in b but not in c so this is a proper subset of a and that is numerically equivalent to a okay so once we prove this theorem okay let me write it as a corollary okay every infinite set every infinite set has a proper subset proper subset that is numerically equivalent to itself of course we have not yet proved a theorem but assuming that we proved a theorem then this corollary will immediately follow from this theorem this is the proof of that corollary which I have discussed just now okay and what it means is that this is the fact which differentiates between finite and infinite set okay a finite set cannot have a proper subset which is numerically equivalent to it and infinite set will always have a proper subset which is numerically equivalent to itself and that is why in some books this is taken as a definition of infinite set okay this is taken as a definition of an infinite set and then you define a finite set as the one which is not infinite etcetera okay that is another approach of dealing going for the whole things okay let us now come to the proof of this okay let us say that the set is let let a be the given set okay let a be an infinite set we want to show that a contains a countably infinite subset okay first of all can a be empty it cannot be empty right so since let a be infinite then a is not empty and what is meaning of saying that a is not empty there exit at least one element so a that a is not empty let suppose I call that element even let even belong to now consider a minus a1 okay that is remove that element a1 from the set a okay can this set be finite why why if it is finite it will be that there will exist some n such that a minus a1 is numerically equivalent to jn okay right that means there is a 11 and on to map between a minus a1 and this set 1 to n then what you can do is that you can map this a1 to n plus 1 okay then a will be numerically equivalent to the segment jn plus 1 and that will make a to be finite right which is not the case okay so this is an infinite set if it is an infinite set it has to be non-empty okay so a minus a1 this must be non-empty this must be non-empty okay then what follows from this again if it is a non-empty there must be some element in that I will call that element a2 okay I will call that element a2 okay so then then a2 belongs to a minus a1 okay then what is to be done after this is clear okay now consider a minus a1 a2 a1 and a2 are different now okay because a2 is not same as a1 right a2 is taken from a minus a1 okay so it is different from a1 okay a minus a1 a2 is also not finite by the same argument again okay and then proceeding in this way at the end stage suppose you follow this procedure for n steps you will get the elements a1 a2 an then consider a minus a1 a2 an that is also not empty that is also not empty okay so what it means is that you can find one more element there that element is a suppose you call that element a n plus 1 okay remember a n plus 1 is different from all of this a1 a2 an okay and this is the unending process you can continue in this way continuing in this way what you will get you will get a sequence okay you will get a sequence so you can fill up the details will as simply say proceeding in this way this way we get a1 a2 etc an okay and all these elements are distinct okay all these elements such that a i not equal to a z for all i not equal to z now what is the next thing to do you just collect all these elements a1 a2 a that is a countably infinite subset of a right okay so let b be equal to a1 a2 etc etc then b is a countably infinite subset of a and that completes the proof of the theory okay we will stop with that we will consider some more properties of this countable set etc in the next class.