 Hi, I'm Zor. Welcome to Unizor Education. I would like to continue discussing certain topics related to quantity and cardinality of sets. And I would like to present these topics as the set of problems, which I would like actually you to try to solve yourself first. And right now I'll present my own solutions, opinions about these problems. They are actually quite important to understand what actually the quantity as a concept, as a philosophical concept, if you wish, really is. From the mathematical standpoint, it's a very good exercise actually, because it really lies in between the mathematics and philosophy, if you wish, because we're talking about infinity. Most of these problems are related to infinity, because the finite sets, they are kind of simple, and the quantity as a concept is very, very simple. It's just a number of elements, no big deal. But as far as the infinities are concerned, well, this is a little tricky. So, try to spend some time yourself just thinking about these problems, trying to offer your own solutions, and here is what I think about it. Okay, problem number one, prove that cardinality of a set of all integer numbers is equal to cardinality of a set of all natural numbers. Okay, so all integer numbers are, I'll do it like this. So it's from minus infinity to zero to plus infinity. These are all integer numbers, positive and negative. Now, the natural numbers are one, two, three, et cetera. My task is to find one-to-one correspondence between these two sets. Okay, I think something which really quite natural in this particular case is start from the center, and let's try to find the corresponding natural number for all the integer, but we will start from the center and then we'll go left and right, left and right in both directions. So, zero, so this is integers and this is natural numbers. So, zero will correspond to one, then one will correspond to two, minus one will correspond to three. I'll use plus, so plus two will correspond to four, minus two will correspond to five. Now, will we find for every integer number, positive or negative, using this technique, the corresponding natural number? Yes, the answer is yes, because for every integer number, sooner or later we will find its place in this particular sequence, plus three, minus three, plus four, minus four. So, eventually we will reach any number, integer number, positive or negative. And since we have plenty of numbers, actually infinite number of numbers, we will definitely find the corresponding natural number for the integer which we are looking for. So, what we can say right now is that the correspondence, the unique correspondence from the set of all integer numbers to a set of all natural numbers exists, since for each integer we can find the corresponding natural. Now, how about in reverse? Well, the simplest thing which I can do right now is say that since natural numbers are subset of integer numbers, then it's definitely for each natural number there is an integer which is basically the same number, for one, one, for two, two, for 25, 25, etc. It will not cover all the integer numbers, but that's not actually required. The one-to-one correspondence is not necessarily working both ways. It can work one way using one rule and another way using another rule. At the same time, now this is a simpler approach. Now, at the same time, a more complicated approach would be to find the one-to-one correspondence which is bidirectional, which means every image can be as a source and then the source can be an image of that element. Now, how to do this? Well, let's just think about some kind of formula which we can come up with. So, from zero, again, back to integers and natural, from zero we have one, plus one we have two, minus one we have three, plus two we have four, minus two we have five. Now, if you see for every positive number, the corresponding natural would be twice as big. So, this is basically the rule. So, for zero, it's one. For each positive number, it's double. For each negative number, it's double negated. If minus one double would be minus two, we negate it will be two and plus one. So, this is the formula. And it's actually reversible. So, in this particular case, it's y is equal to 2x for x greater than zero and y is equal to minus 2x plus one if x is less than zero. So, that's basically the rule. Now, and y is equal to one for x equals to zero. All right? So, is it reversible? Well, it is. These are even numbers. These are odd numbers. So, for every even number, I can say you have to divide it by two to get the corresponding integer. And for every odd natural number, you have to subtract one, you get the even, then reverse the sign so you will get minus and then divide by two. So, basically, this is the formulas which establish, excuse me, the one-to-one correspondence which is not only one-to-one correspondence but is a reversible or inversible one-to-one correspondence when the source of the image is the original source. All right. So, that's it for this first problem. It's a little bit, maybe, too lengthy an explanation. When you basically get a grip of it, you will find it's much easier. First of all, I did not have to go through the second part to establish the correspondence between natural numbers and all integers because I know that since natural numbers can be mapped into integers just in a natural way with an equation, basically. What's the corresponding for 5? 5. What's the corresponding for 25? 25. What was important is to find the correspondence between the, so to speak, bigger. I will use this word bigger but it's not really a valid word in this case. But anyway, seemingly bigger set which is integers to its subset which is natural numbers. If we establish the correspondence between a set and its subset, then you don't have to really go in reverse because the reverse obviously exists with this natural matrix of subset into set. Okay, number two. Prove that set of all integers, integer numbers divisible by 7 is countable. In other words, its cardinality is the same as cardinality of set of all natural numbers. So again, we have integer numbers but not all of them. We have 0, 7, minus 7, 14, minus 14, etc. Or if you wish I can put it like minus 7, 0, plus 7 to extend to both left and right alternatives. Well, actually the way how I have written it right now, this is already a map. It's a correspondence between these numbers, all integer numbers which I have written in this particular sequence, and natural numbers. This is the correspondence. So once I can enumerate them, I will use just yet another word enumerate. Once you enumerate the elements, it means actually you have put them into the correspondence with the old natural numbers. Now how can I get a formula if you wish? Well, let's think about it. For 7 you get 2. So you have to subtract. So what's the next one? For 28 you get 6. And for minus 28, for minus... Okay, let me just write it this way. So for 0 you get 1. For 7 you get 2. For minus 7 you get 3. For 14 you get 4. For minus 14 you get 5. For 28 you get 6. And for minus 28 you get 7. Alright, so what's the law? What's the rule here? Can I establish it using some kind of a formula? Well, you probably have to divide it by... So let's talk again positive and negative separately. Because we know that every negative is the next after the previous positive. So we can just ignore this one and consider this. 7, 2, 14, 4, 28, 6. So these are going with... Oh, I forgot 21, sorry. This is 21. Not 28. That's what actually something which I missed. So these are going up by 7 and these are going up by 2. So to get it you have to what? Divide it by 7 and multiply it by 2. That's what it is. So the formula is divide by 7 and multiply by 2. 4x greater than 0. From 7 we get 7 by 7 times 2. For 14 you get 14 by 7 2 times 2, 2. And for 21 we get 3 times 2, 6. Okay, so for positive we have this. For negative it will be... Well, you have to change it to a corresponding positive sign. Also divide it by 7 and multiply by 2 and add 1. That's how you get... Instead of 2 you get 3, instead of 4 you get 5, etc. So this is for x less than 0. So this is an attempt to go by formula. But again, I really don't have to do this. What's actually enough is just write it down one under another and obviously you see what's the principle of one-to-one correspondence. So in the future I'll try to avoid all these formulas because it's not really needed. What is needed is just to demonstrate the idea. And the formula can be derived... You can derive it yourself without any problems. Prove that cardinality of a set of all integer numbers is equal to the cardinality of all rational numbers. Now, this is much more interesting and quite frankly less obvious. You have all the integer numbers on one side and all the rational numbers are significantly... Well, bigger, I'll use again the word bigger, I'll look into the bigger set. And obviously integer numbers can be a subset of rational numbers because every integer number x can be represented as a rational number x over 1. So we obviously have a mapping from integers to rational numbers. So we have every integer number will have its image among rational and obviously different integer numbers will have different images. So from integer to rational is easy. But how about from rational to integers? Now this is not as easy. And the gain obviously there are many different ways to do it but let me just offer something which I will consider important. Here is how. First of all, let's consider only positive numbers separately from negative numbers. Since we know that rational numbers can be positive and negative and integer numbers can be positive and negative. So what I will do is for rational number zero I will map it to an integer number zero for everything which is greater than zero I will map them into integer... Now this is rational and this is integer. So I will map them into positive integers and negative rational numbers I will map to negative integer numbers in exactly the same way as positive. So I will talk about only positive numbers. Only positive rational numbers I have to map into integers. Now how can I do it? Okay, what is a rational number? Rational number is basically positive rational number. It's two positive integer numbers with something like a separator if you wish. I don't want to say it's division. But whatever it is, sign of division or separator because I can always put it this way as a rational number basically any rational number is a set of two integer numbers. Now we are talking about positive rational numbers and we are talking about two positive integer numbers. It doesn't matter how I write them down. But here is how I would like to map this set set of all these pairs of positive integer numbers into a set of integer numbers. Okay, here is how. I will use the term weight. It's the sum of these two. Now how many positive rational numbers exist with weight one? Well, actually none. Because zero over one is equal to zero. It's not positive and one over zero does not exist because you cannot have a rational number because zero is a denominator. Okay, how about with weight equals two? Well, we have only one rational number which is equal to two, one one. Now, for weight equal to three, I have actually two rational numbers. But finite, mind you. I always have a finite number of rational numbers with a specific weight. For weight number four, for weight equals to four, I have one-third, two-second, and three-firsts. So I have three. Well, obviously the number, as the numbers grow, I will eventually find any rational number among these. But in this particular case, I can enumerate them because I can put the corresponding number of these to one as an integer, positive integer number. These will be mapped to two and three. These will be mapped in four, five, and six, etc. So what I would like to point here is that for every rational number, eventually I will come to a line which will state that its weight is such and such. For instance, rational number m over n. My weight will be m plus n. So eventually I will come up to this. And eventually it will find its number because I have an infinite number of integer, positive integer numbers. So eventually it will come to some kind of a number of this particular rational number. Now this integer number is its image. So that's how I put into the correspondence all rational numbers to positive rational numbers to all positive integer numbers. Now same thing with negative, obviously, and that's how the whole relationship is established. So for every rational number, I can always find the corresponding integer number which means that there is a one-to-one correspondence between all rational numbers and its subset which is integers. Now by the way, I don't know about you, but first time when I thought about this and somebody actually showed me that the number of rational numbers is exactly the same, or I shouldn't say the number is the same. The cardinality of the set of all rational numbers is exactly the same as the cardinality of all natural numbers or all integer numbers, whatever, they're all the same anyway. I was kind of surprised. All right, now let's talk about something a little bit more interesting. Proved that cardinality of a set of all points of a segment does not depend on its lengths. In other words, cardinality of a set of all points of a short segment is exactly the same as cardinality of a set of all points of a longer segment. So we have two different segments. This is a long segment. This is a short segment. Now, how can I build the one-to-one correspondence between these points and these points? Well, it's actually very easy. Here it is. Since this is shorter than this, I will connect this to this, so they will cross somewhere, and this is a very important point. Every line which I draw signifies a correspondence between this and this. Now, every point in the longer segment, let's say this, can find its image in a shorter segment by connecting to this center. This is an image. And vice versa, every point on the short segment using the line will find a unique image in a short segment. Obviously, different points will correspond to different points in the image, both directions. So this actually proves that the cardinality of this set and the cardinality of this set are the same since there is a one-to-one correspondence, and this is the way how I build the correspondence. Again, it's maybe a little bit counter-intuitive, just a little bit, that there is the same number. I mean, using a common language, not the mathematical language, both longer and shorter segments have the same number of points. But again, they are the same because there is a one-to-one correspondence. That's an interesting property of infinity. Things which are not necessarily seem to be equal to each other, they still have the same number of elements. Okay, proof that cardinality of a set of all points on arc is equal to cardinality of a set of all points of a chord between the ends. So if you have an arc and you have a chord, no, this is obviously part of the circle, then the number of points here and the number of points here, these are two sets, and this particular problem says that the number of points have the same cardinality in those cases. Well, again, we used to think that the straight line is shorter than anything else. Yes, it is shorter from the length perspective, but from the perspective of one-to-one correspondence, these two sets have exactly the same cardinality because we can build a one-to-one correspondence. Well, there are many ways to build the correspondence. One way is to use the center, for instance, and draw these lines. These lines establish the correspondence. Different points on the arc would correspond to different images on the segment, and vice versa, different images on the segment would correspond to different points on the arc. Is this the only way to build this correspondence? Well, no. You can have, for instance, perpendicular lines, all the perpendicular lines to a chord. They also establish the one-to-one correspondence between points on the arc and points on the chord. So these are different ways, but whatever the way you choose, both establish a bi-directional one-to-one correspondence which is actually completely reversible, which means the image of this would be an image of that if you go to a different, into opposite direction. So these numbers, the cardinality of this set and cardinality of this set are the same. They're equal to each other. The cardinality of a set of all points of an interval is equal to cardinality of a set of all points of a straight line. Okay. Now, first of all, now, this is even a bigger contrast. I'm talking about interval, which is basically the segment without those ends. And a straight line. How to build the correspondence between these two sets. Well, our original idea, when I had a shorter segment and a longer segment, I just connected the ends and used that as a center. Here it's not really so easy. However, what I can do is a slightly different, a slightly modified, I have two steps, correspondence. Remember the transitivity. If A is in one-to-one correspondence with B and B is in one-to-one correspondence with C, then A and C also have one-to-one correspondence. So I'm not going from here to here directly. What I will do first, I will make a semicircle here. Also, no end points here. This is the center. Now, we know that the cardinology of this set and cardinology of this set are exactly the same. So, instead of proving for a segment or interval, rather, I will use the semicircle. Now, with semicircle is a little easier because I can use its center. Now, I presume that this diameter is parallel to this line. I can position it the way how I want, basically. So I position it in such a way that diameter is parallel. Now, there are no end points here, right? So I don't really need this diameter. I'm just staging that this will be parallel to the line. Now, I will do this. This is my mapping. This is my one-to-one correspondence. And as you see, the further my point is on the line, the closer my image would be here, or vice versa. Again, this is a one-to-one correspondence between all points on the line and points on an arc without these end points. I don't need end points because the line would be parallel but not cross anything. So these points will not actually correspond to anything. But all other points, so the whole arc without these two edges, for each point on the arc, I can find the corresponding point on the line and vice versa. The cardinality of a set of all points in the plane is equal to cardinality of all sets of points in a straight line. Now, this is also something which is kind of not really very obvious. You have all points on the plane and all points on one line. The first seems to be significantly bigger than another. But apparently, their cardinality is the same. And that's what I'm going to prove. Look at it this way. What is a point on the plane? A point on the plane can be characterized by actually two points via projections on co-ordinate axes. So instead of considering all points on the plane, I will consider all pair of points on two lines. So what I can say right now already that the number of points on the plane, the cardinality actually of the points on the plane, is equal to cardinality of points on two lines. Okay, that seems to be much easier, right? So instead of all the points on the plane, you have only points on two lines. And they have to prove that the number of points on two lines is in one-to-one correspondence with the number of points on one line. Well, this is easier. Now, how to do this? Well, consider it this way. For instance, I can reflect back to a couple of previous problems. Number of points on the plane, sorry, on the line is equivalent one-to-one correspondence with number of points in an interval, right? So two lines give me two intervals. And number of points in two intervals obviously is equal to number of points in some interval because I can always just put them one-to-another having a bigger actually interval. And bigger and smaller interval are always equivalent to each other as far as cardinality is concerned. So I'm referring back to these previous problems which I have solved and that's how I prove that the number of points on the plane is equal to number of points on two lines, which in turn equals to number of points in two intervals, which in turn equal to one particular interval, which in turn equal to one line. So I'm switching from two lines to one using intervals. Okay. So that seems to be easy, although might be unexpected. Prove that cardinality of a set of all infinite sequences of zeros and ones is greater than the cardinality of a set of natural numbers. Okay. We have a set of sequences. We have two characters, zero and one, and we have different sequences of zero and one. This is one sequence. This is another sequence, et cetera. Now, all these sequences, infinite sequences by the way, all these sequences make up a set, a set of all sequences. Now, the statement of this problem is that this set is more numerous than the set of natural numbers, which means that I will try to enumerate them. This is number one, this is number two, et cetera, this is number n, et cetera. If I will try to enumerate them, I will not be able to do this. Well, how? Very easy, actually. There is something which is... It's some kind of a trick, actually. One of the famous mathematicians, Kantor, actually came up with this trick. He says, okay, let's assume that we can actually write one sequence under another and enumerate them, which means put them into correspondence with natural numbers. Let's assume now, what I'm going to prove that there is one particular sequence which is not included into this enumeration. So no matter how we try to enumerate them, there is always one sequence which is not part of this enumeration, which means that we cannot actually place into one-to-one correspondence, set of all the sequences and natural numbers. Now, how to find which sequence is not among these? Very simply, take the first sequence and take the first number, which is zero, in this particular case, and reverse it. Now, from the element number two, take the second number in this sequence and reverse it. From the element number three, you take the third one and reverse it, etc., etc. So you go diagonally, and from each sequence, you take the element on the nth place, whatever, if sequence has a number n, corresponds to natural number n, take the nth element in the sequence and reverse it. And that's how you build a new sequence of zeros and ones. And what's interesting is, it's different from the first sequence because I could change the first element. It's different from the second because in the second place it's different. It's different from any nth element because on the nth place I have reversed from zero to one or from one to zero. So it's different from any one of those, which means it's not part of this sequence. So this new element of my set, a new sequence of zeros and ones, E is actually not part of this enumeration, no matter how tricky I try to enumerate all these sequences. So it's not countable. Countable means it can be put into a one-to-one correspondence with natural numbers. So the number of old sequences is not countable. And what's very interesting actually is, let's think about what each sequence might represent. It actually might represent a real number written in binary system. So every real number can be represented as finite or infinite, periodic or aperiodic either decimal or binary or in any other numerical system written sequence of characters. In this case I'm representing in the sequence of zeros and ones, which means it's binary actually. And what's interesting here is that the number of different sequences of zeros and ones, which basically represents the number of real numbers, is not countable. I cannot put it into one-to-one correspondence with natural numbers. So rational numbers, if you remember one of the previous problems which we discussed, is actually countable. Remember the weight, numerator plus denominator, and that's how we counted it. With infinite sequences of zeros and ones, we cannot count them, no matter how well we count, we can always find something which is not part of our count itself. So that's actually a proof, albeit not exactly like 100% rigorous, but it's kind of a proof that real numbers are not countable. There are much more, if you wish, again using the regular language, there are much more non-rational real numbers than rational real numbers. Well, this concludes this particular set of problems. I hope you have a better feel of what infinity actually is, and there are different infinities, like infinity of the real numbers is significantly greater than infinity of rational numbers, which is the same as infinity of natural numbers. All right, fine. Enough about infinities. Don't forget Unisor.com is a great source for all these educational materials and a very good, actually, tool for parents and supervisors to control the educational process of their students. Thank you very much, and good luck.