 My name is Zor. Welcome to Unisor Education. Today's topic is abstraction. If you ever wondered what mathematicians do, and you would like to hear it like in a couple of words, I think the first thing which comes to mind is they are engaged in abstraction. What does it mean? Well, let's consider you have certain objects. You have one object, maybe a real object, and you have another object. And you would like to know certain properties, or you would like to think about the qualities of these objects, etc. You noticed that an object on the left has properties A and B, and an object on the right has properties, let's say, B, C, E, D. Alright, the property B is common in these two objects. Maybe there is something which can be derived from this fact. So what do mathematicians do? Well, they create an artificial new object which has only one property B, and research it. For instance, they have come up with the conclusion that if there is an object with a property B, it has a quality lowercase B. What does it mean? It means that this particular object also has this quality B, and this object also has quality B, because both of these objects have capital B property. There are two advantages to study this particular artificial object rather than individual prototypes, as we can call it. Number one, this is simpler. Less properties, and it's purely artificial, so it contains no extra things which we might not really like. And the second one, whatever you bound with this particular object as a property, this lowercase D, for instance, it's applicable to many prototypes which share the same common property B. Let me give you an example. Let's consider first integer numbers and operation of addition. So you have all integer numbers and operation of addition. Positive and negative. On another hand, you can consider all rational numbers and operation of multiplication. These are two completely independent objects, but there are certain commonalities between these two. For instance, this operation, the addition in integer numbers is completely self-contained, two integer numbers will give an integer number. We are within the category of integer numbers all the time, whatever the operation addition we are performing on whatever members. Same thing here. If you take two rational numbers and multiply them, you will also have a rational number. Another property. For every number, there is an inverse number which being applied to gives zero. Actually, the very similar story exists here. For every rational number, there is an inverse which being multiplied always gives the number one. What's so special about zero and what's so special about one? Zero in addition doesn't change anything being applied in both sets. One on any rational number doesn't change it. So these are very similar properties, zero for addition and one for multiplication. Okay, fine. We found that there are certain commonalities between these two objects. What do mathematicians do in a case like this? Very simple thing. They invent a brand new object. So they forget about integer numbers and operation of addition. Forget about rational numbers and operation of multiplication. They invent a brand new object called a group. What is a group? It's not a set of numbers. It's just a set of anything, whatever you want. Sort of certain elements. Together with a set of elements, they're saying, well, let's assume there is some operation. I will use the circumflex for an operation. So there are elements and there is an operation, binary operation, which means any two elements of this group can be applied against this operation and you will get another element of the same group. So let's just symbolically write it this way. Where x, y and z are all elements of the same set, same original set. Okay, so we have defined our integer numbers with operation of addition really looks like it. Elements are numbers, integer numbers, and operation is addition. Our rational numbers with operation of multiplication also looks like it. Rational numbers are elements and operation of multiplication is this operation in general, which we are saying. Well, there are add some properties. Let's assume that there is always some kind of a unit element which being applied to any other will give the same and if you will apply it on another side, it will be also the same thing. We can add a couple of other things, for instance, for every x there is an element which we call inverse. I'll put it with a bar on the top which will give this unit element. Remember, with an integer number and operation of addition, you have a negative number which gives zero. For a rational number, you have a reverse one over it which being multiplied will give to one. Basically, this is equivalent of zero and one for addition and multiplication. This is equivalent of negative and one over operations in our prototypes. We have two different prototypes but both of them really are just particular representations of these more general cavologies. Now, let's just prove a very simple theorem about groups. Well, it's not really a group yet in mathematical sense. Group also requires associative law, etc. But let's not talk about this. We're not studying the groups right now. We are actually demonstrating the concept of abstraction. So we have abstracted two different prototypes integers with operation of addition and rational with operation of multiplication into one concept. And let's just prove something within the framework of this concept. It's simpler. It doesn't have all the peculiarities of numbers or addition versus multiplication. I'm not talking about this. I'm talking about any abstract operation which just have these properties. And the theorem which I am going to prove is the following. There is one and only one element, unit element, which has these properties being operated upon, any other element, it will give itself. Well, here is the proof. Just using these two properties, since x can be any element, let's assume that we have two different unit elements, i1 and i2. Both have the same property, which means i1 being applied to any element on any side will give it the same element and i2 will have exactly the same property where x can be anything. Now it's very simple. Since all these equations are true for any element x from the group, this equation is true for element x equals to i2. And what happens in this case? We have i1 operation i2 being applied against x will give i2. We use this one. Now let's use this one for x equals to i1. What do we have? We have i1 operation i2 will give i1. Well, now look at these guys. The left parts are the same, which means the right parts are the same. So i1 is equal to i2. Well, it's such a trivial theorem it doesn't really deserve to be called a theorem. But what's important is that by doing this I have proved there is only one number zero. Which has exactly the same property of being added to any other number will not change the value. So if you have something like x plus a equals a among integer numbers there is only one x which is equal to zero. Similarly, what I have proved among the rational numbers if you have something like this in this case a is an irrational number. So if you have this equation among a rational number you have only one solution. So I have proved that these and these two equations have one and only one root which is in this case a unit element for multiplication which is number one. So in this case is a unit element for addition which is zero. So that's what mathematicians do they do abstraction they prove something in their abstract theory in the abstract set of elements or whatever they come up with and then they propagate it back to the prototypes from which they abstract these colleges. It's easier that way.