 Hi, I'm Zor. Welcome to UNISOR education. Today I will continue talking about well, quantity is a philosophical concept and more precisely, we will go through certain mini theorems related to the concept of quantity, which is usually addressed as cardinality in set theory, especially if we are talking about infinite sets. We will be talking about infinite sets. We have two different mini theorems lectures, this one and the next one, and in both lectures we will try to cover certain number of interesting facts about infinite sets, especially about their cardinality. Well, number of elements if you wish, just a plain talk, but you can talk about number of elements if this number is infinite. Anyway, so let me go straight to this example, to these theorems. I will use certain symbolics here. I will use n as a set of all natural numbers, which means 1, 2, 3, etc., to infinity. It's infinite number and set is usually written as the contents between the curly brackets. Now, I will use something which resembles absolute value as the cardinality of a set. Now, you know that cardinality is basically a generalization into an infinity, the concept of number of elements, quantity. Okay, so let me start with some very simple statement. There are different infinities. There are smaller infinities if you wish and bigger infinities. Now, what does it actually mean? What does it mean that two different infinities are equivalent to each other, or their cardinality is the same? Well, basically it's all related to one-to-one correspondence. So if you have two sets, a and b, and if you have one-to-one correspondence between them, which means there is one rule which puts into a correspondence for each element of a, certain elements of b, in such a way that different a's will map into different b's, so they can be distinguished. And there is another rule which goes in reverse. So for any elements from b, we can find corresponding elements from a, and also different b's will correspond to different a's, then there is a one-to-one correspondence. And in this case, we are talking about the same cardinality of these two sets. Now, if, however, situation is slightly different, namely, what if set a has a one-to-one correspondence to set b, but set b is a subset of a larger set c, and there is no one-to-one correspondence between a and c. In this case, it looks like the cardinality of a, which should be the same as cardinality of b, is smaller than the cardinality of c. In more plain language, the number of elements in c is greater than the number of elements in a or b, although both can be infinite. So these are infinities, and this is also the infinite number of elements, but this infinity is greater than this infinity of a higher rank, if you wish. Just as an example, I will just go slightly ahead of myself. These are natural numbers. C is real numbers, and these are natural numbers as part of the real numbers. So that's basically the situation I'm talking about. There are less natural numbers than real numbers. So cardinalities can be less than, equal, or greater than from each other. So we will use these symbols to reflect this particular property between two different sets. If the cardinality of one is less than the cardinality of another, it means that maybe the riser, well, most likely the riser, correspondence one way, but not the other way. Alright, with this defined or explained or commented about, let's go through the theorems. The set of all natural numbers is the smallest in terms of cardinality of any infinite set. So if you have any S infinite set and you have N, all natural numbers, then the cardinality of S is always greater or equal than the cardinality of N, which means that sometimes we can have exactly the same number, but in other cases it's greater, but definitely not less, which means that using this picture which I had before, so these are one-to-one correspondence. So this is N, this is S, and this is a subset. Let's put it as with the index zero, a subset of S which is in one-to-one correspondence with N. So what I'm saying is that if this is the natural numbers and this is any infinite set, then the situation is this, which means the set itself will contain a subset which will be in one-to-one correspondence, and that what it means that the cardinality of N is not greater, it's less than or sometimes equal. If S zero is the same as S, then the cardinality is equal. In all other cases it's smaller, but definitely cardinality of N is not bigger than the cardinality of S. And for this it's sufficient to assume that the S is an infinite set. Now, how can I prove this? We're talking about the theorem, so I'm going to prove it. Okay, now what does it mean, infinite set? Well, one of the properties of infinity is that it's infinite. It's not finite, right? Which means that if I will take one element out from the infinite set of elements, I will still have infinite set of elements left. So, if I will subtract one element S, the result would be still infinite number of elements in S. Why? Well, obviously why. There are only two cases. Either it will be infinite or it will be finite. If after subtracting one element from S I will have a finite set, well, then let me add this one back to the finite set. Finite set plus one element will be finite set. So, I take element and then I return it back. I cannot have an infinity. All right? So, that's why after I took it from the S, I still have to have an infinite number of elements. Otherwise, I cannot consider S an infinite set. Okay, that's very important because what we can do right now, from the set S, we will subtract element, I call it E1, and I will assign it number one. Then, since I still have left infinite number of elements, I will subtract another element from whatever is left. And I will assign it number two, et cetera. I will continue this process. And as a result, I have a subset S within the index zero and I have natural numbers in one-to-one correspondence. So, certain subset of my bigger set S is in one-to-one correspondence with the natural numbers, which means that the cardinality of S cannot be smaller than the cardinality of M, which means that out of all the infinite sets, and that's very important statement, out of all the infinite sets, natural numbers, set of natural numbers is the smallest, so to speak, which means its cardinality is less than or equal to cardinality of any other infinite set. And this is just the property of infinity. So, that's the first theorem. I'll put it here. Cardinality of M less than or equal to cardinality of S, where S is infinite. Okay. That's my first theorem. Next. The cardinality of a set that can be put into one-to-one correspondence with a set of all natural numbers will not change if one element is added to it. Okay. So, let's assume that I have two sets. One set M and another set A, and they are in one-to-one correspondence, which means that the cardinality of both is the same. This is cardinality of natural numbers. This is any set of elements, and since it's in one-to-one correspondence with natural numbers, I can actually represent it as a one, eight, two, eight, three, et cetera. That's my initial set A, which has the same cardinality as the set of all natural numbers. Now, what this theorem says is if I add one element to this, it will not change its cardinality, which means it will still be in one-to-one correspondence with natural numbers. Correct. So, let's add element B to A. What do I have? Now, I can put in this sequence B, A1, A2, et cetera. These are my elements. So, I used to have only A1, A2, and A3. Now, I have B, A1, A2, and A3. Now, is this set still in one-to-one correspondence with natural numbers? The answer is yes. I just have to make a different one-to-one correspondence. I will make this correspondence B as one, A1, A2, A2, A3, et cetera. So, this is just a different correspondence, but it's enumeration. It's still countable. That's why the Internet set, which has the same cardinality as set of natural numbers, will not change the cardinality if I add one element. How can I symbolically write it down? This one. So, N is set of natural numbers or equivalent to it any other set which has the same cardinality. I added one element and the cardinality of the set, which is the result of this, is still the same as it was before. Okay. Cardinality of countable union of a set of all natural numbers with itself is the same as cardinality of an original set of natural numbers. Now, we are talking about countable union. Now, what is cardinality of countable union with itself? Well, let's consider a finite example. For instance, you have, let's use numbers, let's use letters, A, B and C. This is your original finite set. Now, what does it mean union with itself? Well, it means this. That's what it means. It's a union of two sets, basically. A and A. So, you have all elements from the first and all elements from the second. The fact that they are the same doesn't really change anything. It's just the same element is repeated twice. But it's still a set and there is no problem with this. It's just element count twice. Now, what does it mean if I add it n times? Well, obviously, it's basically the same thing, n times A. I even used the multiplication here. But basically, what it means is A, A, n times, then B, B, n times, and then C, C. That's what it means. I regrouped. I collected A's together and B's together. Order is not really important here because we're talking about sets. Alright, fine. Now, what does it mean if I add the same set to itself? Countable. Infinite but countable number of times. Well, I cannot really put it in one string of characters because it will be an infinite here and then infinite here. But what I can do is the following. I can stretch it vertically instead of horizontally. I'll put it this way. A, B, C and then A, B, C and then etc. A, B, C etc. So my set is countable, infinite but countable number of times down and whatever number of elements it has across. Now, that's for a finite set. What if instead of a finite set, I have natural numbers? Well, we'll do it this way. I have 1, 2, 3, etc. to infinity. 1, 2, 3, etc. to infinity. And down also to infinity. So I have infinity in both directions. That's how I can put it on the whiteboard. The result of N times N. It's N times N. That's what it is. That's how multiplication actually works. So it's pretty much in agreement with general definition of multiplication as a multiple addition. Now, my question is, this particular set of infinite set of elements is infinite in each line and it's infinite in the number of lines. What's the cardinality of this particular set? Well, my statement is that it's still a countable set, which means I can put into one-to-one correspondence all elements of this set with 1, 2, 3, 4, 5, etc. I can enumerate them all. Okay. How can I do it? Simply. Let me just draw these lines. Now, each line contains finite number of elements. So instead of going into this direction or this direction, I will go into this direction and I will cover each diagonal separately. So I will start from this. So this will be my number 1. This will be my number 2, number 3. It will be number 4, number 5, number 6, etc. So sooner or later every element in this infinite matrix, if you wish, will be reached with certain number which it will correspond to. So that's how this both directionally infinite matrix can be still put into the correspondence with one directionally infinite set of natural numbers. And that's why the cardinality is exactly the same. Since I can put into one-to-one correspondence, this element corresponds to number 1, this element corresponds to number 2, this number 3, etc., etc. So every number which occurs here can be placed into a correspondence with some natural number. And by the way, it's not really important whether it's set of numbers or set of any elements which is equivalent from the cardinality standpoint to the set of natural numbers. So let me write it here. This is our another interesting formula. I mean, look at this arithmetic. It doesn't look familiar, right? But that's what it is. If we are talking about cardinality of infinities, infinite sets, that's what you should expect. Alright. Let's move on. Cardinality of a set of all subsets of an arbitrary set is greater than its own cardinality. Okay, this is a theorem which was proven first by the great mathematician Kantor, who was actually spent a lot of time researching all these concepts of set theory and cardinality, infinity, different ranks between infinities, etc. So what he has come up with is a very interesting statement. Not all infinities are equal, apparently. There are like more infinite infinities or less infinite infinities. Now, in this particular case, for instance, we have proven that the infinity of natural numbers is not greater than infinity of any other infinite set. Now, the question is, maybe all infinite sets have exactly the same cardinality as natural numbers, which means maybe all infinite sets can be put into one-to-one correspondence with natural numbers. Well, no. And that's exactly what this Kantor's theorem is about. It actually states that there is always a set which has significantly more number of elements, infinite but still more than the set of, let's say, natural numbers. So he offers a construction from each set you can build another set which is bigger than the original set, the bigger in terms of cardinality. So what is this set? He just basically offered a concrete set which is bigger. Now, here's how. Let's consider a set, any set. Well, it can be infinite, but it can be finite as well. And we will consider the set of its subsets. Now, what I exactly mean is the following. Let's say you have a finite set which contains two elements, A and B. What are its subsets? Well, it's empty set. It's a set, well, I put it in Karlin brackets as well. It's a subset which contains only element A, only element B, and elements A and B. So, this is very important. We started from a set which has two elements. Now, we are building a new set, elements of which are subsets of the elements of set A. So, in case we have like this two elements, then the set of all subsets has four elements. This is subset number one, this is subset number two, number three, and number four. Now, if I have, let's say, three elements, how many subsets and what kind of subsets we have here. Well, let's start from a subset which contains no element, then we will go to subsets with one element each, which is A, B, and C. Now, we will continue with two elements in each subset, which is A, B, B, C, and A, C. And finally, with three, A, B, and C. So, are these all subsets? One, two, three, four, five, six, seven, eight. Right, eight. Now, I will actually prove a theory that if A is a finite set with N elements, well, let me put it differently. If number of elements is equal to N, then the number of all subsets of A is equal to two to the N's degree. In case N equals to two, we have four. Two to the two's power of two is four. In case of three, we have eight. Two to the power of three is eight. Now, how can I prove this particular theory? Well, actually, I think it's one of the problems which I discussed before, but it's actually very easy to do. The easiest thing is to use mathematical induction by N. So, for N equals to, well, we checked it for N equals two, but for N equals one, it will be exactly the same. So, we have checked the initial condition for the beginning. Then, let's assume that this formula is correct for some N, and let's just make this step forward and check what it is in case we have one more element. So, if we have one more element, so let's say we have A1, A2, etc., AN. That's what we had before, and we assume that there are two to the N subsets. Now, let's have a new set. So, we had A1, etc., AN. Let's use a different letter. Let's use letter B. So, now our set is this. Now, how many subsets does this set have? Well, there are two categories of subsets. Either the subsets contain D or they don't contain D. Now, let's count them separately. How many subsets which do not contain D? Well, these are basically exactly the number of subsets which we had before for the N elements. So, it's two to the Ns. Now, how many subsets are with the B? Well, exactly the same, because with the B can be any other subset which we have already counted, right? So, it's, again, to the power of N. So, two to the power N, which is two times two to the N, which is two to the N plus one. So, this is the proof of the formula. So, we are adding one element, and by adding one element we always multiply the number of subsets by two, either the old subset without this element or old subset with this element, and that basically encompasses everything. Okay, so this is the formula for finite sets. The number of subsets is two to the power of N. Now, we are using this to introduce basically a purely symbolic, purely syntactical concept. If N is natural numbers, then two to the power of N is a set, sorry, set of all subsets of N. So, this is just a symbolic representation. It doesn't mean that we are really raising two to the power of N, because N is a set. So, N is a set, and two to the power of N is just a symbolic representation of a set of all subsets of this set N. And why we do it? Because for the finite sets, we have this correspondence between the number of elements in the original set and the number of all the subsets of this set. Alright, so this is just a representation of all the subsets. And now, the theorem which Counter has proved is the following. The number of subsets, or if you wish, the cardinality of a set of all subsets of any set S, finite or infinite, is always greater than the cardinality of original set S. Now, if I'm talking about finite numbers, basically, we have already proven this, because for finite numbers, if there is an N element, two to the power of N is greater than N. That's obvious. This is one, this is two, this is two, this is four, this is three, this is eight, etc. Obviously, this is greater. I mean, I can prove it by induction or something like this. But that's obvious. Now, for finite, it's obvious for infinite sets. We cannot actually use the term greater than in the old-fashioned way, because these are not numbers. Cardinalities are not numbers. They're both infinities. But what does it mean? This sign means that you can establish a one-to-one correspondence between S and subset of two to the power of S. That is possible. And two to the power of S will not be in one-to-one correspondence to S, which means that there is always something from this particular set which will not find the correspondence in this. Okay. The way how I'm going to prove it, it might be a little tricky or, well, there is a famous joke about this, actually. Consider a statement. I'm saying a false statement. If I'm saying this, is it true? Well, if what I'm saying is true, how can it be false? But if it's false, then the opposite is true, right? So it's kind of a statement which cannot be expressed as either true or false. The logic which I'm going to follow right now will be slightly similar to this. So try to follow it. It might be a little tricky. It's simple, but there is a little trick in it. Okay, here's how I will do it. Let's assume that there is a correspondence, one-to-one correspondence, between elements of the original set and elements of the new set, which means subsets of the original set. Okay. So element s1 is in correspondence to some subset of s. Let's say it's si1, sim, whatever it is. I'm running out of markers. Now element s2, we are assuming that there is a one-to-one correspondence between this and this. So for s2 there is another subset, which means certain, let's say, i k s i k plus 1, etc. Some other sequence of elements from s which constitute a subset. Okay. So let's assume that the research is one-to-one correspondence. Now what I'm going to do is I would like to divide my elements into two categories. Let's say black and white. Now if s1 corresponds to a subset where s1 is present, then I say it's, let's say, black. But if an element corresponds to a subset which itself is not present, then it would be a white element. So there are obviously some elements which correspond to subsets where they are actually represented and some other elements which correspond to subsets where they are not represented. Fine. No problem with this so far. Now next what I'm going to do is I consider a subset which contains only white elements. So only those elements which do not, which are not contained in the subset they correspond to. Now here is my question. Well this is a subset like everything else and it should have the corresponding element of the original set which it corresponds to. What is this element? Is it black or white? Is it in this subset or is it not? Well let's just think about it. This subset contains only white elements. Now if this element is among them, then it must be black like this one. But now how can it be black and be among all white elements? It's just impossible. Maybe it's white. But white is an element which does not belong to the subset it's in correspondence with. But this is a subset of all white elements which means it's supposed to be among them. It's also impossible. So you see we are coming into a contradiction. It cannot be white and it cannot be black. It's like in that particular statement which I had in the very beginning. I'm saying the false statement. It cannot be true and it cannot be false. So this is a contradiction basically. The contradiction is that if we assume that there is such a correspondence between elements and all the subsets. Then what I'm saying is that there is some subset which is this one. Which cannot have anything which it corresponds to. So that's why this set, set of all subsets is bigger in a way. So for each element of this set, set of all subsets. Not necessarily you can find the corresponding element from the original set. So this is not true. We cannot build a function from all the subsets to the elements of the set. So they are not equal. Cardinalities are not equal. But maybe cardinality of 2 to the s is smaller. Maybe it's less. Well, no, that's not possible because this direction always exists. Why? How can we find the correspondence between elements of a set and some elements of the set of subsets? Well, very easily. Because for each element s1 we can always have a subset which contains only one element s1. Obviously all the elements of s will have images. Because there is a subset which contains only s1 or only s2 or anything else. And obviously different elements from the s correspond to different subsets. So basically there is this relationship but not this relationship. And that actually proves that the cardinality of s is smaller than the cardinality of 2 to the power of s. So the cardinality of every set is smaller than the cardinality of the set of all its subsets. Well, that basically ends the set of mini theorems I would like to go through today. There is another set which I will do in the next lecture. Well, do not forget that unisorn.com contains lots of very interesting information including this and others. And what's very important, it's an excellent tool for supervisors and parents to basically be in control of the educational process of their students. Because they can enroll the student, they can check the exams score which this particular student got on each exam. And they can actually consider the course completed or not completed by this particular student to ask to repeat the particular exam, take it again, etc. So, welcome to Unisorn. And until the next lecture, I will continue with mini theorems about different cardinologies of different properties of infinities. Thank you. Good luck.