 In this lecture, which will be answering these questions, what is the cardinality of infinite sets? Are all infinite sets equivalent? Which has more elements than natural numbers or real numbers and can a set be equivalent to its subset? When we were small kids, the first thing we learned maybe were alphabets, but after that we were taught how to count. One, two, three, four, five, and we can go on and on counting. Have you ever wondered how many natural numbers there are? These numbers are called natural numbers or counting numbers because they develop naturally and they are used to count. But first let us define the cardinality of a set. We say that the cardinality of a set is just the number n, such that A, the set A, for example, has five elements. So as an example, the set of vowels A, E, I, O, U has cardinality five since it has five elements. Now for a finite set, finite sets are easier to handle. Take note that if A is a finite set, n of A is equal to k, if A can be placed in a one-to-one correspondence with a set one, two, three up to k. So a one-to-one correspondence is simply a function. It's a correspondence between two sets A and B, such that for every x in A, there exists exactly one element in B mapped to it and for every B in B, there corresponds exactly one element in A. So for example, for the set A, E, I, O, U, its cardinality is five and we can place the set A, E, I, O, U in a one-to-one correspondence with the numbers one, two, three, four, five. It can also be with five, four, three, two, one. This correspondence need not be unique. So it's just a pairing off. Like in real life, we want everyone to have a partner bawat isang tao, isang partner lang. Hindi pwede ang two is to one or one is to two, isang tao, dalaw ang kanyang partner, hindi yun fair, di ba? So we want it to be a one-to-one pairing. Now, we define two sets to be equivalent if they have the same cardinality. So equivalently, A and B are equivalent if they can be placed in a one-to-one correspondence. So for example, the set A equal to A, E, I, O, U and the set B with elements one, two, three, four, five are equivalent. So take note also that if A is finite and B is a proper subset meaning there are elements of A which are not in B then the number of elements in B is strictly less than the number of elements in A. So for example, if A is one, two, three, four, five and B is the set two, four then take note that the cardinality of B is two and the cardinality of A is five. So the cardinality of B is strictly less than the cardinality of A. Now the question now is, is this also true for infinite sets? In particular, can infinite set possibly be equivalent to a proper subset? So maybe this difficulty imagine but we can see later that this is true. So let us first look at the familiar number set, the set of natural or counting numbers. So n is the set one, two, three, four, five the kth element is k and then the next element is k plus one. This is an infinite set since it goes on and on. What is the cardinality of this set? When we have a finite set we can easily count but if we have an infinite set can we define the number of elements? Of course we can. We just have to assign a certain symbol. What is the cardinality of the set of natural numbers? We call it the aleph null which is represented by the Hebrew alphabet aleph. It's also called aleph not or aleph zero. Let us now answer the following question. So you've always heard that infinity plus one is still infinity. What is infinity times infinity? Now this is something not so well defined but if we have aleph null what is aleph null plus one? What is aleph null plus two? What is aleph null plus aleph null? And what is aleph null times aleph null? So maybe this is a bit too complex but I can easily show that all of this will be equal to aleph null. Now David Hilbert is a German mathematician and one of the most influential mathematicians he gave the paradox of the Grand Hotel or the Hilbert Hotel problem which presented counter-intuitive properties of infinite sets. So I'm warning you this is counter-intuitive so it's not the usual thing that's the usual way of counting. So this is the Hilbert Hotel problem. Imagine a hotel with infinitely many rooms which can only accommodate one person per room. So and for a certain night each room is occupied. So if a new guest arrives can he be accommodated? Kasha pa ba siya? What if two guests arrive? Kasha pa ba sila? What if infinitely many guests arrive? Kasha pa ba sila? So take note that let's number the rooms as room one, room two, room three and remember there are aleph null rooms so this will be infinitely many. Each room is occupied let's call them guest one, guest two, guest three and so on. So suppose that one person arrives and he wants a room, he needs a room so can he still be accommodated? So anong gagawin ng hotel manager? Remember that all the rooms are already occupied by one person. Two persons cannot share a room so dapat one is to one lang and there are aleph null guests there are aleph null rooms. So can he still be accommodated? So what do we do? Assuming that all the guests are nice you can request them to transfer to the next room that's what you will do. Ask guest one to go to room two as you guest two to go to room three and so on. In other words guest n who is in room n will now go to room n plus one. Take note that you have infinitely many rooms so every room has a room beside it so everyone can move one room let's say to the left for example and that will vacate room number one meaning the guest who arrived can still be accommodated. So what will that mean? The number of rooms is aleph null the number of guests is aleph null plus one with the arrival of one guest but there is still a one to one correspondence between guests and rooms so what will that mean? Aleph null plus one is still equal to aleph null so adding one does not change the value of aleph null. Now what if two guests arrive? Can they still be accommodated? So similar to our solution for one guest arriving we simply move guest one to can can you guess? Guest one will move to room three guest two will move to room four guest three to room five and in general guest n will move to room n plus two. So if you're occupying room ten you're just going to move to room twelve. If you're occupying room one million just go to room one million two and you have infinitely many rooms aleph null number of rooms but with this movement the first two rooms will now be vacated so therefore the two new guest will be accommodated. So what does this mean? Aleph null plus two is still going to be equal to aleph null. So in general suppose k number of guests arrive k can be any number they can still be accommodated by transferring guest one to room k plus one. So example if there'll be ten guests arriving you ask guest one to move to room eleven you ask guest two to move to room twelve and so on. In general if k guests arrive you simply ask the guest in room number k room number n to move to room k plus n. So what does this mean? This will mean that aleph null plus k is equal to aleph null. So it does not really change the value of this of aleph null. Now imagine we have a hillburt hotel with infinitely many rooms and suppose hillburt has a bus so let's call it the hillburt bus with infinitely many seats and suppose the hillburt bus arrives and all the seats are occupied. So what does this mean? It means that the hillburt bus has aleph null passengers. So suppose they come from somewhere and they want to be accommodated for the night k kasyapa ba ang mga nasa hillburt bus? If yes this will actually mean aleph null plus aleph null is still going to be aleph null. Kaya pa ba? Kaya pa yan. So how do we do it? So what we do now it's not that simple. What we do is you ask the person, guest one in room one to move to room two. You ask guest two who is in room two to move to room four. Guest three where do you think should he go? The guest in room three will now go to room six. In general the guest in room n will go to room two times n. And if n is any natural number 2n is still going to be a natural number therefore all the guest can move to room two times n. So what does that mean? We have occupied only the event number rooms and if you think of all the event numbers this will still be infinite. So what will be vacated will be the odd number rooms room number one room number three room number five and so on. So therefore you still have infinitely many rooms vacated. And these are now ready for occupancy. So the passengers in the Hilbert bus can now be accommodated. So what does that mean? It means that alefnal plus alefnal is alefnal. So far is that clear? So let us review. We have said that alefnal plus one that's the one guest arriving is still alefnal. Alefnal plus two is still alefnal. Alefnal plus k is still alefnal and now alefnal plus alefnal is still alefnal. Meaning two times alefnal is alefnal. And you can also show somehow that alefnal plus alefnal plus alefnal is still alefnal. So what you can do is you vacate all the rooms so ba'yagil lang siyang only the you just leave all the multiples of three rooms vacant in which case infinitely many can still fit in. Now before we deal with the question what is alefnal times alefnal? Pwede pa ba yon? Let us now consider the number sets defined by n, the set of natural numbers, w, the set of whole numbers which will start from zero and then one, two, three, four. E is the set of even numbers that will be zero, two, four, six, eight. The kth element will be in two times k so these are the natural numbers or the whole numbers which when divided by two will leave no remainder and zero is an even number and o is the set of odd numbers one, three, five, seven. The kth element is 2k plus one. In other words, odd numbers are just even numbers and you have z, the set of all integers. These are the whole numbers together with the negative of the natural numbers. And q is the set of rational numbers so they are called rational because we can imagine them. These are just numbers of the form a over b where a and b are integers, b is not zero and in terms of decimals these are just terminating or non-terminating but repeating decimals. Example three-fourths is a rational number one-fourth, one-third. One-third is non-terminating but repeating one-fourth is a terminating decimal that's 0.25. Now, but we also have the so-called irrational numbers and these are numbers which are not rational meaning we don't use them in our everyday lives. Example will be pi squared of two and these are in terms of decimals the decimals which are non-terminating and non-repeating. So the set of real numbers is just the set of rational union the set of irrational numbers. So if we represent this using event diagram you see that these are two disjoint sets rational, irrational. Inside rational will be the natural numbers inside the set of whole numbers inside the set of integers. And take note that the set N is a proper subset of W the set W is a proper subset of Z and Z is a proper subset of the rational numbers because there are rational numbers which are not integers. There are integers which are not whole numbers and zero is a whole number but it's not a natural number. So, an tanong what is the cardinality of these sets? So what is the cardinality of W of Z of even numbers of rational and irrational numbers and the real numbers? So for example the set of natural numbers is contained inside the set of whole numbers but you can establish a one-to-one correspondence to show that a certain set has cardinality alef null you simply have to show that you can pair off one set with a set of natural numbers which is one-to-one meaning bahwat isang part isang bawat natural number I make a partner do sa isang set and vice versa. So if you look at the set of natural numbers one, two, three, N and you look at the whole number zero, one, two you just pair off one with zero two with one three with two in general N with N minus one. So in other words there is a one-to-one correspondence so kahit mas maliit tignan yung natural numbers than whole numbers actually they have the same number of elements. This is counter-intuitive unlike infinite sets if it is a proper subset it always has less elements but for infinite sets it's a different thing. Now let's look at the set of even numbers two, four, six, eight and so on we can include zero and then the odd numbers be one, three, five two N minus one you can easily have a pairing let's say one the natural will go to two even two natural will go to four three will go to six N will go to two N so if I give you let's say 10 as a natural number it will be paired off with 20 and let's say I'll give you an even number 100 what is the natural number which goes with it it will be 50 so take the this well defined because if N is natural two N is even and if M is even you divide it by two it's still going to be a natural number so what does it mean? even numbers this will be a proper subset of the natural numbers but they have the same number of elements so parang nakakasyak but it's true because we have shown that there is a one-to-one correspondence between natural and even numbers for the odd numbers it can also be similar so even if parang kalahati lang yung even ng natural they have the same number of elements meaning one-half of alefnal will still be alefnal the same manner that twice alefnal is still going to be alefnal how about for natural numbers and integers? take note natural numbers parang kalahati lang siya ng integers and we can use the same reasoning pag kalahati lang or times two siya ng isa and one is alefnal the resulting set is still going to have cardinality alefnal so the one-to-one correspondence will be if you have let's say the number zero so integers we can pair it off with zero then what you do now you cannot pair off one with one two with two what will happen? if you have this pairing of positive integers with natural numbers there'll be no more partners for the negative integers so you have to make sure lahat make a partner para lahat masaya so what you do is you divide z into two parts or n into two parts which will be even and odd and now you have to map the natural numbers which are even with the positive integers and the natural numbers which are odd with the negative integers and this will ensure that all of them will be paired off with one element so therefore this will be a one-to-one correspondence and this will mean that the set of natural numbers and the set of integers are again equivalent again the set of integers is seemingly bigger than the set of natural numbers because you have negative of the natural numbers but eventually they have the same the same cardinality now it's a little bit more tricky for rational numbers what you do is you can just represent the rational numbers as an array so you have the numbers one and then two and then three then form all possible combinations so you can have the first row will simply be one over one one over two one over three and so on the first column is one over one two over one three over one and so on so you will have alefnal rows and alefnal columns so you will have how many elements all in all it will be alefnal times alefnal so what is alefnal times alefnal if we can show that there is a one to one correspondence between the rational numbers which we have formed this way and these are only the positive rational numbers then it will mean that rationals and natural numbers are actually equivalent now you cannot make the pairing one goes to one over one two goes to one over two because for the first row your natural numbers will all be used up the same manner you cannot just use the pairing with the column so what you do is the trick is you imagine an infinite string and then number that with one two three for the natural numbers and you string the rational numbers diagonally so if you do it this way take note that one over one will be map to one one over two will be map to two two over one is map to three and in general n over m is map to it's something not simple n plus the quantity m plus n minus one times m plus n minus two over two a bit complicated but it works and these are one to one correspondence so what does it mean? aleph null times aleph null is still aleph null so let us recall again review the even numbers and the odd numbers are proper subset of n n is a proper subset of w z q but all of these sets have the same cardinality aleph null hence they are all equivalent so these only a property of infinite sets an infinite set may be equivalent to a proper subset and this is not true for finite sets in fact you can define infinite sets to be infinite a set is infinite if it can be placed in a one to one correspondence with a proper subset of itself so we say that a is countably infinite if it is equivalent to the set of natural numbers meaning aleph null so theoretically it's still countable because the numbers are discrete so you can count one, two, three and so on until you die but you can continue counting and counting it will never end but you can count okay so but is the set of real numbers also countably infinite meaning does it have a cardinality equal to aleph null so anong mas marame real numbers or natural numbers so this time maybe you can say that real numbers but it took some time to prove that okay so it's intuitive mas marame ang real but remember earlier you thought that natural numbers have more elements than even numbers so there is a German there was a German mathematician George Kantor who developed set theory he studied infinite sets and their cardinality and showed that there are more real numbers than natural numbers specifically he showed that the cardinality of n the set of natural numbers is less than the cardinality of the set s which is the open interval from 0 to 1 meaning the set of all real numbers between 0 and 1 take note if a number is between 0 and 1 it can be expressed as a decimal of the form 0 point then you have you have numbers following 0 point something and this can be terminating it can be non terminating for example 0.25 is between 0 and 1 0.333 dot dot dot is also between 0 and 1 so Kantor argues what we call natural argument to show that we cannot establish a 1 to 1 correspondence between n and the set s and the argument is by contradiction so we call it now the diagonal argument so what do you mean by argument by contradiction so he first assumed that n and s will have the same number of elements what do you mean by a set has a left now elements it will mean that you can theoretically enumerate all so suppose you can enumerate all elements between 0 and 1 so take note that in this argument if you list down a number let's say r1 is 0.00012345 let's just keep it finite first r2 is 0.04256437 r3 and r4 are given by 0.29578356 and 0.52679013 we take note that we can actually create a new number from this 4 which will be distinct from the 4 numbers enumerated so we take note in r1 the first decimal digit is 0 in r2 the second decimal digit is 4 in r3 it's 5 and in r4 the fourth is 7 how do we form a new number which is not the same as the previous 4 numbers what we do is we simply choose the decimals the first decimals should not be equal to the first decimal of r1 the second digit should not be the same as the second digit of r2 the third digit is not equal to the third digit of r3 so for example the first decimal is 0 just choose any number not equal to 0 for example 3 the second decimal of r2 is 4 for the second decimal choose anything not equal to 4 for example 1 so if we proceed this way take note we form the new decimal 0.3148 and it's not the same as r1 r2 r3 and r4 so if we extend this argument for all the decimals between 0 and 1 we actually can show that you cannot enumerate all the numbers between 0 and 1 so this is how Cantoy did it you have the set the number 1 is mapped to r1 for example you can enumerate all real numbers between 0 and 1 so 1 is mapped to let's say r1 of the form 0. we express the decimals in the form a11 a12 a13 and so on 2 is mapped to r3 r2 3 is mapped to r3 in general n is mapped to rn and rn is expressed as 0.a and 1 that's just one number a and 2 a and 3 and so on and then we now show that you cannot enumerate all because you can actually create another number which is not in the list so we are assuming that all the numbers between 0 and 1 are already in this list so how do we create that number so let r be the number let's say 0.b1 b2 b3 and so on that can go forever where b1 is not the same as the first digit of r1 b2 is not the same as a22 which is the second digit in r2 and bj in general will not be the same as the jth digit of rj and take note because this number is not equal to r1 r2 r3 because they differ by at least one numeral it cannot be equal to any of the given numbers r1 rn and so on so what does that mean? the set of natural numbers is not equivalent to the set of points between 0 and 1 the set 0 and 1 is just a subset it's a small part of the set of real numbers but it has a lot more elements than the set of natural numbers so we cannot place the two sets in a 1 to 1 so which means that the set of natural numbers is strictly less than the cardinality of the set of points between 0 and 1 question is which has more elements? the set of numbers between 0 and 1 or the whole real numbers the whole set of real numbers take note the set 0, 1 can be expressed as a finite line segment and the set of real numbers can be expressed or can be represented by an infinite number line you are familiar with the number line so 0, 1 is finite r is infinite can we place them in a 1 to 1 correspondence? so let's do it geometrically so for example if you have two finite line segments you can place them in a 1 to 1 correspondence by just get any point outside the two lines and then connect the point to one point on one given line and then just connect this point and extend so it intersects the second line and this will be the partner points so if you continue this way all the points in the first line will be partnered with another point in the second line so there's no problem if you have finite lines but you must also do the partnering in the correct manner if you just draw perpendicular lines it will not be a 1 to 1 correspondence because some points in the longer line will not be mapped two points in the shorter line now how about if you have an infinite line and a finite line so what we do is you simply bend the finite line and you can now map it this way get the midpoint of that arc and you simply map with a fixed point and then pair it up with the points on the curved line and then try to see where it intersects the line on the number line now this can be easily seen if you remember the tangent graph you have the line pi over 2 or negative pi over 2 to pi over 2 is mapped actually to an infinite line meaning if you have tangent x and you simply let x be between negative pi over 2 and pi over 2 then it will generate the whole set of real numbers this is just a 1 to 1 correspondence one point between negative pi over 2 pi over 2 to one point between negative infinity to positive infinity so there is a 1 to 1 correspondence between a line segment which is finite and an infinite line again counter-intuitive kasi mukang mas maraming points ang infinite line but actually they have the same number of points and it's not alef null this number is greater than alef null now cantor went into depression several times in his life and they noticed that each time he was depressed he would be thinking about one question is there a number between the cardinality of the set of natural numbers and the cardinality of the real numbers which we call now alef 1 so up to the day that he died and he died in a hospital I think thinking about this problem it was that he never was able to prove it that there is a number between alef null and alef 1 but he was able to establish that there are different degrees of infinity alef null is the smallest infinity it's the cardinality and the set of real numbers will have a bigger cardinality which we now call alef 1 now cantor's big question is now called the continuum hypothesis which states that there is no set whose cardinality is strictly between that of the natural numbers and the real numbers it seems to be a very simple question but it was included in the 23 problems posted by Hilbert and this was the first problem these were important math problems during that time which were unsolved some have been solved but for the continuum hypothesis I cannot say that it has really been solved well it has been shown that at this point you cannot prove nor disprove it so it seems that cantor's thinking over it was not really in way it is a difficult problem to prove it with existing Zermella Frankl's set theory or without what we call the action of choice perhaps if we change this set theory or the action of choice maybe we can prove or disprove so again the things that we learned now are counter-intuitive but we learned something that in mathematics we actually have to be open-minded because what seems to be obvious is not maybe that obvious and what seems to be real may actually be not real but we can always show it and we can prove or disprove our hypothesis but if on a certain night you need a hotel remember that there's always a room in Hilbert's hotel