 Hi, I am Zor. Welcome to Unizor Education. Today's topic is quantity among different fundamental concepts of mathematics. Well, everybody, I'm sure everybody knows what actually the common term quantity is, but in mathematics it's it's special as everything else. Quantity is a rather complicated object in math and I would like to start from a very philosophical standpoint. If you, let's say, consider a set of six buildings and six cows, well, they don't have anything in common, right, except the number six. So if you have two different sets, probably one of the very probably the only property which they have in common or they might have actually in common is their quantity. Okay, so fine, number six in case of six buildings is a property of a set and number six is a property of another set. Here are houses and the buildings and here are the cows and this is the common property. So what we can actually say that at least finite sets might have certain common characteristic. Now, how do we establish that the number of elements in a set of six buildings is exactly the same as the number of elements in a set of six cows or six anything? Well, for this purpose there is a concept which is called in mathematics one-to-one correspondence. So let's just spend a little time and explain what one-to-one correspondence actually is. If you have certain set of elements and I'm not actually talking about finite set sets anymore, I'm talking about any set which contains certain elements and you have another set of elements and you have the rule which from each element of one set you can always find a corresponding element of another set and what's very important is that different elements of this set which we can call the source if you wish, if they're different they are corresponding to different elements of another set. So if these are different then these must be different as well. So this is half of the definition of one-to-one correspondence. Another half is just an opposite. If you have all the elements of this set then for each one you should be able to find the corresponding elements of this one. So this is A, this is B. So we have a correspondence from A to B which is unique. Different elements should correspond to different elements from the image and it should be inversible which means for every element of the set B you can find corresponding elements of the set A and again different elements from B should correspond to different elements of A. So this is a mutual unique correspondence, unique because if sources are different the images must be different and it's mutual because the inverse is supposed to be true as well. Now if these rules, these correspondence between the elements does exist then we can say that there is a one-to-one correspondence between sets A and B. Now here is a very interesting detail. It's just one of the particular case of correspondence if you wish because if you take all the elements from A and map them to elements of B and if we assume that elements of B are completely filled up which means there is no element in B which is not filled by this particular correspondence then we can always say that inverse correspondence also exists because for each element of B since it's an image of something from A we can always say that the corresponding A element is the source which actually corresponds to this element of B. So in this case instead of drawing these two different errors, well actually I can use this, I can use the same but bidirectional error. So every correspondence can be bidirectional. Now this is a particular case of one-to-one correspondence. It's not actually necessary that it will be the case always and here is an example. Let's go back to the buildings and cows. So let's say you have a square building, a triangular building and a round building and cows are called let's say A, B and C. The correspondence between the buildings and the cows can be this and inverse correspondence can be this. This is our case I was just talking about but at the same time I don't have to really establish the inverse correspondence exactly in the same way as I established the direct correspondence. I can always say that in reverse they are this way. So there is a rule which are errors directed down which establish the correspondence between the buildings and the cows and two different buildings are corresponding to different cows obviously. Now the inverse correspondence also exists from A to triangle from B to a circle and from C to a square and it's also a unique correspondence because different cows correspond to different buildings. So depending on how I establish the correspondence it can be of this particular kind when it's just mutual or it can be of a different kind but in any case it's still one-to-one correspondence. So I would like actually to be very very clear that one-to-one correspondence is number one bidirectional and number two it does not have to be bidirectional in this particular way which means if element A corresponds to element B it's not necessary that B also corresponds to A. It can be something else like in this particular case but it's important is that both directions should be unique which means different sources should correspond to different images. Okay now let's talk about finite sets. Now in case of finite sets there is an obvious statement that only the number of elements is the same then we can establish one-to-one correspondence. Now why? Well obviously you can consider let's say you have two different finite sets. One is with three elements and another is with four elements. Now this direction can be unique without any problems because you have only three elements and you have four choices so you can always find a unique image for every source element but when you go backwards you have to have a unique correspondence between four elements and three images of these elements and obviously I think there is a name actually in this principle it's maybe Archimedes some kind of a mathematician. The principle is that if you have more elements in this set than in this then there is always something left it's like musical chairs you have a certain number of people and the number of chairs is less than the number of people somebody will be out of chair so there is some element in the bigger set which will not be which will not be corresponding to something which was not really touched before by this correspondence because you have three elements here three elements here but the fourth element will not have any unique image. So if the sets are finite then the one-to-one correspondence is related to the number of elements. Now another very important property of one-to-one correspondence is it's transitive now what it means is the following if a one-to-one corresponds to b and b one-to-one corresponds to c then a is in one-to-one correspondence with c now how can that be proven well actually it's quite easy let's say you take an element a now using the direct correspondence you get an element b now from b you get c now what you can say is that a to c establishes the correspondence from a to c now is it unique well let's just check if a and b are if a one and a two are different then b one and b two are different because this is a one-to-one correspondence between a and b now since b one and b two are different then c one and c two are also different so this correspondence for different for different elements from a produces different elements from c one and reverse is obviously true in exactly the same way so one-to-one correspondence is a trend that's very important now let's go back to quantity now we all kind of used to the fact that the word quantity is related to counting and obviously with finite sets we have established that the quantity is something which sets can have in common completely different sets can have in common and that's the number of elements that's the quantity actually by definition so if you want to say what's the quantity of elements of certain finite set well you have to establish the one-to-one correspondence between this finite set of elements and another set which is one two three up to m natural numbers so if there is a one-to-one correspondence between your set and set of the first n natural numbers then we can say that the quantity of the elements of this set is equal to m this is the definition basically that's it so the quantity is defined as a correspondence of the elements of our set with a subset of the first n natural numbers that's what quantity equals n means and from this actually we can say that the quantity of two different sets which have the same element which means they are mapped to the same there one-to-one corresponding to the same subset of natural numbers so if this is quantity n and this is quantity n and then we can say that the quantity of these two sets is the same they are equal in their quantities if you wish so that's the quantity of the finite set but this is just the beginning as always there are much more complex things and mathematicians always find something to to exercise their brains a little bit further so where are the complications complications are in the infinity okay now you know what the quantity is and you probably feel quite comfortable with finite quantities let's talk about infinity well the first infinity which we might deal with well first of all infinity is something which is a completely pure creation of our mind there is no such thing as infinity in the real life i mean yes there are some physical theories about infinite universe etc but this is all kind of theories you cannot in a normal way in a regular practical way establish the fact that something is equal to infinity so we are talking about creation of our minds infinity is in our minds so what's the first set with infinite number of elements which we can deal with well obviously this is a set of all natural numbers etc what's important is this in fact this is etc that's what make it infinite change the market all right um how big is it well it's infinite right but we would like to compare infinite sets in exactly the same fashion we compared the finite set which is one-to-one correspondence well let's talk about another infinite set let's talk about natural numbers but not all of them but only let's say even ones so it's two four six etc this is also infinite number and it's obviously smaller because this is a subset of this so we took only every second number so it should be smaller right well wrong it's not smaller first of all the words smaller or larger bigger or whatever they are not applicable to infinity there's no such thing as a smaller infinity however there is a way to compare different infinities we're just not using the word smaller bigger infinity we're using some other words which i will introduce in a little while so first of all let me talk about these two infinity infinities my statement is that there is a one-to-one correspondence between them and if there is a one-to-one correspondence between them we cannot say that one is more infinite than another they are both equivalent to each other so there is a new word called cardinality there's nothing to do with church cardinals so the cardinality is basically an equivalent of the word quantity but applied to infinite sets so my statement is that the cardinality of these is the same and because they are in one-to-one they can be put into one-to-one correspondence well how very easily now this direction is y is equal to two x where x is element of this and y is element of this now this is basically a correspondence a unique correspondence from this to this why is it unique well because it's defined on every natural number which is an element of this set and two different numbers using this function will correspond to two different even numbers so this is the rule basically that this correspondence should be number one complete it's defined everywhere and unique so two different elements would correspond to two different images now how about the way back well actually in this particular case we can have the way back using this function so for an element two of this my corresponding element would be one for four it will be two for six it will be three etc etc in this case by the way this particular correspondence is double double error because if this element corresponds to this using this transformation corresponding rule of correspondence then the the way back is exactly what this particular thing is so using these two correspondence rules we have established one-to-one correspondence between these two seemingly different sets because this is definitely a subset of this however both are infinite and both infinities are in some way equivalent which in in in more precise terms sounds like the cardinality of this set is exactly the same as the cardinality of this set which cardinality means it's a it's a replacement of the number of elements or quantity but four for infinities so the same cardinality means that there is a one-to-one correspondence between two sets but maybe all infinities are of the same cardinality maybe I can always find the correspondence between all the different infinities is it right well this is definitely wrong there are more infinite infinities and less infinite infinities if you wish and I'll just give a couple of examples for instance the number of points in a segment cannot be put into the correspondence into one-to-one correspondence to a number of natural to a set of natural numbers now this cardinality of this is definitely greater now why well I can definitely put this correspondence how well there are many different ways but for instance I have divided this segment in two halves and call this point corresponding to this then I divide this into two halves and I made the correspondence with a number of two etc so every time I divide by half and I can find from and for each number I can find a point so from this set to this I can find the correspondence which is unique and defined everywhere but not the vice versa why it's a different question and it's not easy but that's actually a true statement there is no correspondence unique correspondence from each point on the segment and and natural numbers so the next lecture I will actually talk about different examples well primarily infinite examples of different sets and I will compare their cardinality but for now I think this is enough to basically introduce you to a concept of quantity and cardinality applied to infinite sets just as an introduction to the whole concept so that's it for this particular lecture and don't forget that there are problems in the next lecture associated with this one which are very very important to understand this whole business of quantity and cardinality better okay so that's it for today unizord.com is where you can find this and many other interesting educational materials and parents are definitely encouraged to use this side to supervise and control the educational process of their student okay thank you very much and good luck