 Hi, I'm Zor. Welcome to Unisore Education. This lecture is about operations. Well, basically, let me start with a very simple example, and then we will all know what we're talking about. Okay. Example is addition, addition of let's say integer numbers. This is an operation, and there are many other operations which behave like addition in some way or another, and mathematicians as they always do, they abstract things. So the term operation actually is some kind of an abstraction from such operations as addition, multiplication, rotation, etc. So let me start from, well, it's kind of a definition, but not a very rigorous one, but at least you understand what I'm talking about. First of all, operation is a function. Now, we all know that functions can have domain where the values of arguments are taken from and co-domain where the values of the function actually are located. So what are domain and co-domain of operations we will be talking about? Well, we will be talking about two different kinds of operation, the unary and binary. So for unary operation, this is some kind of a generalization or abstraction of such operations as let's say negation when from a number 25, you get minus 25 from 13, you get minus 13. So these are functions of one argument and what's important is that the domain and co-domain are exactly the same. So in case of let's say negation, we can say that this is a function which has a domain, all real numbers and the co-domain, all real numbers or it can be integer to integer or something like this. So basically domain and co-domain are exactly the same set. Basically, that's it. That's all the most general definition of unary operation. Now, as far as binary operation is concerned, again, this is something which is an abstraction from addition or multiplication when we have two different arguments of the same type and the binary is also of the same type. Now mathematically speaking, you can say for instance like addition, this is a function which is defined on set of all pairs of real numbers and the result is another real number or it can be a set of pairs of integer numbers and the result would be let's say integer number or something else. Now, this set of pairs, by the way, it's ordered set of pairs because there is first argument and there is a second argument. In case of addition, doesn't really matter because the addition of five plus three and three plus five are exactly the same. But there might be some other operations which do not have this important property. In this case, we're always talking about the first argument and the second argument, and the set of all pairs of these arguments is called a Cartesian product of these two sets. So, we have unary operations and binary operations. In case of unary, the domain and co-domain are exactly the same. In case of binary, there is one particular set which serves as the component of Cartesian product for the argument and it's the same set for the results of the function. Okay, unary and binary and we have learned about their domain and co-domain. Basically, that's kind of a definition of at least the operations which we will be talking about. Then again, obviously since we have certain properties of such operations as negation, rotation, or addition, or whatever, we will try to generalize. Let me repeat again that mathematics is kind of form of art where mathematicians are experimenting with different creations of pure mind. That's why we're talking about these generalizations, et cetera. If you can think about some practical implementation of operation of addition, yes, we all kind of use the ending in our day-to-day life. But abstract operations which are unary, binary, whatever we will be learning, most of them don't really have any implementation in real life. Pure creation of the mind. That's why mathematics is a form of art. Okay. Let's talk about symbols. Okay. Symbolically, operations can be expressed in many different forms. For instance, if we are talking about unary operation, we can define it in some simple form, like y is equal to f of x where x and y belong to the same set, whatever the set is, domain and co-domain. Domain and co-domain are exactly the same, as we know, for unary operations. So that's one of the ways to describe it. Another way to describe it is something like this. It's an operation f, which for each element of some set s has the corresponding function for result of the operation on this element in the same set. Sometimes it can be symbolized using something like this. All these symbols basically mean exactly the same thing. There is some operation, which is a function, which is basically a rule, which for each element of some set, which is a domain, puts into a correspondence an element of the same set. Yeah, something like this also. And again, x and y belong to the same set of s. So these are all symbolical demonstration symbols of unary operation. Now, in case of binary operation, that's basically very simple. In exactly the same fashion, we can say something like this. In case of binary operation, when you have two arguments, so you can have something like z equals f of x, y, where x, y and z belong to the same set s. Or you can have something like f as Cartesian product with s is result in s. Or f of x, y results in z, or any combination thereof. Sometimes, actually, we might use something more intuitive symbol for abstract operation. Now, this intuitive symbol basically is borrowed from addition or multiplication. So you know that if a and b are two numbers, we can have something like this as a result of their addition. So in a more abstract sense, we can invent some kind of a symbol which symbolizes this operation. So plus for addition, dot or a small asterisk for multiplication, well, I can say that something like this, on sign, number sign, whatever you call it, can symbolize an operation. So x and y, that is an abstract symbol for exactly the same thing as everything else before that. So this function f, which has two arguments, can actually be represented symbolically as x. Some symbol which I have chosen, it's not like general acceptable. I just chosen to use this bound sign, number sign as a symbol of operation. And this is another argument. And here, it's very clear that there is a first argument or the second argument, there is an operation and there is a result of this operation. Okay, let's go on. Now, let's talk about unor operations first. Now, obviously we know that some unory operation converts an element from a domain to another element of the same domain. Now, I can actually have either the same operation or a different operation, which is defined on exactly the same set. Let's talk about different operation, which is also defined on the same set. I can define their consecutive application on any element. So if there is an element x, let's say, which belongs to the domain s, I can first apply f and x will go to y. And then I can apply g to the result of the first operation to get the next element. So symbolically, it can be written like this, y is equal to f of x, z is equal to g of y, or since y is, in turn, the function of x, so z is equal to g of f of x. So if I have two operations, f and g, I can always define their composition if they are defined other than the same domain. So there is a composition. And let me just define it this way. I'll use the symbol commercial f, whatever, for m% for the composition. Okay, so the composition of these two functions is a function which applies to any element x is consecutive application of one function, which is the closest to the x and then the second one. And obviously, I can define a different composition, f and g of x, which is first you apply g and to the result of the g you apply f. Are they the same? They might be the same, but might not. And let me just give you a couple of examples. Examples are interesting in this case. So we are talking about the unary operation, actually two different unary operations, on the same set. And we are composing a combined operation from these two, and we will try to find out whether we can reverse the sequence of operations. Now, from the first example, which I mentioned, the unary example, negation. Negation is actually, you know, very, very simple operation. Another operation can be, let's say, multiplication by five. So first, you multiply by minus one, which is negation, and then you multiply by five. If you start with any number and you first multiply by minus one and then multiply by minus five, it's exactly the same as if you will multiply by minus five first and then by minus one. That's the property of multiplication. Now, but not all different operations, unary operations on the same set, have this property of interchanging their places. And here's a very interesting example. Let's consider an operation of rotation, rotation of points on a plane. Let's say by 90 degree. Counterclockwise. Now, what does it mean? Let's put coordinates. That's easier. And let's have, let's say seven and three. Now, if I turn the whole plane, all the points on the plane, by 90 degree counterclockwise around the center, then this particular point, let's call it X. Now, the perpendicular to it would be this. And obviously this would be seven and this would be minus three. So I just turn by 90 degrees and obviously the whole rectangle will turn 90 degrees, which means this will go to this and this will go to this. That's why this is minus three and this is seven. So we have point Y. So from seven, three, we got to minus three, seven. So this point is seven on X axis and three on Y axis. This point is minus three on X axis and seven on Y axis. All right, this is one transformation. One operation, which is turning by 90 degree counterclockwise of all the points on the plane. Now, let's consider another reflection relative to let's say Y axis. So reflection relative to Y axis. Now, what happens with a point Y? If I reflect it to point Z, obviously this is three, seven. So after reflection, minus three, seven, will go to three, seven. Okay, great. So these are, this is F and this is G. We have two different transformations of the points and I have applied first the F transformation, F operation to the point seven, three, got minus three, seven and then this. So basically what I can say is that point seven, three will go to three point seven as a result of first was F and then it was G. So G composition with F, this is an operation which transforms point seven, three, X point, two points, three points, three, seven, which is Z point. Okay, great. Now, let's do the opposite. Instead of GF, I will use FG, which means first I will apply the G transformation, which is reflection. And then F transformation, which is rotation. Okay, so reflection relative to Y axis from X, I will get Y prime, which is seven, will go to minus seven and this will be three. So my seven, three will go to minus seven, three. This will be the result. And now I turn 90 degree counterclockwise, which is this way. And obviously it will be in this point. So minus seven, three will go to minus three, seven. Or minus three, minus seven, this point. Minus three and seven. So in this case, as you see point seven, three was converted by F composition with G into minus three, minus seven. So as you see, the result is not the same. In one case, when I'm composing first F and then G, result is this point. And if I'm doing in reverse G and then F, the result will be different. So if two different compositions produce the same results, if operations F and G produce the same result, regardless of what order we apply them, then we call them commuting with each other. So F and G might have this property of, regardless of the order, so F and G is exactly the same as G and F for any argument from the domain. Then we call them commutative. We are talking about these operations are commuting, or commute each other. Example is obviously two different multiplications, or let's say two different rotations by 90 degree and then by let's say 180 degree or something like this. These operations commute. Now how about let's say one operation would be, addition of X would go to X plus five. This is one operation, so any number I add five, that's my operation. Another operation would be I multiply that number by X. Now, do they commute or not? Well, let's just see. If I want to do F and G of X, what is it? So first G is applied, so it's two X and then I add five. So it's two X plus five. What about G and F of X? First I apply F, which is plus five, and then I multiply by two, so it's two X plus 10, which is different. So these two operations do not commute, but two multiplications do, or two additions, for instance, do, by different numbers. Okay, so we talked about commuting unary operations. Now let's go to, oh no, one more, one more. So there is another property, it's called associativity. So what is this? Basically it's this, if you have F and G and H, you have three different operations. All unary operations. Well, by definition, what is this? By definition it's F and G and H of X, which is F of G of H of X, that's where it is. Well, but you know what, I can define it differently. I can define it this way, in which case it would be, first this should be applied to X, which is, again, first H, then G, and then F to the resultant. Well, same thing, right? So it looks like it doesn't really matter how I define where I put the parenthesis here. So in this case, we're talking about associativity of unary operations. All right, so we talked about commutative property and associative property. Now let's switch to the binaries, binary operations. We also have a concept of commutative and associative properties. Binary operation, now, if order is not important for the result, for any pair of X and Y which belong to the domain, then we are talking about this particular operation F being commutative. Now, if instead of this notation, I'm using something like this, what it means is, is this. Now, are all binary operations commutative? Well, you know about addition or multiplication of two different numbers, right? Well, apparently, not exactly. Sometimes we have a completely different case. And here is the case when it's not true. Let's consider as our domain, and actually not our domain element which is supposed to be Cartesian product with itself. Set of all strings. So we have set of all strings and we define operation of concatenation, all right? So if you have a string ABC and you have, let's say, I'll use plus sign but it doesn't really matter what the sign actually is. Let's call it X, Y, Z. The result is ABC X, Y, Z. This is what concatenation actually mean. If you have one string of characters incatenated with another, you get basically all characters from the first, continued with all characters from the second. Now, is this operation commutative? Absolutely not because X, Y, Z plus ABC would result in a completely different string. So that's where the order is important. This is the first argument, this is the second argument of the Cartesian product. So in some cases, operation, binary operations can be commutative, in some cases not. Now, let's talk about associative property. Now, associative property is the property of the order of three different operations. So if you have, let's say, operation F, and then you do this. You first do operation on these two and then the result of this, the result of the first application on X and Y is used as the operand for the second application. Now, at the same time, we can do different, in different order, we can first apply operation on Y and Z and then use this as the second operand for the next application of operation F. It looks a little better if I will use a different symbolic, different symbolics here. Okay, this is the definition of our binary operation. Now, if I have three arguments, I can either use this, that's what this means. Or in this order, it's X. So right now, the difference is where exactly I put the parenthesis. Now, again, we all know that if this binary operation is, excuse me, addition or multiplication or something like this, then the order of parenthesis is really not important because operation is associative. How about our string operation, the concatenation? Well, apparently, if you have three different strings and first you apply to the first two and then the result to the third string, you will get what? First you get A, B, C, D, E, F plus G, H, I, which is equal to A, B, C, D, E, F, G, H, I. Now, if you do it in a different order, so you put parenthesis here, what happens? Well, it will do like this. It will be A, B, C plus the result of this operation, D, E, F, G, H, I, which is equal to A, B, C, D, E, F, G, H, I, which is exactly the same as this one. So concatenation, although it's not commutative operation between two strings, still has associativity as its main property. Now, together with the property of having this operation commutative and dissociative, I would also like to introduce a concept of a unit element in this case. Here is what unit element is defined with. Okay, I will use the following symbolics. I will use letter E to define a unit element if this is true for any element X. So the operation which is applied to the element X and one particular unit element, if it results in the same X, that's the definition of this unit element. If this element exists, which has this property, then it's called a unit element. Now, what's the unit element for addition? Obviously, zero. No matter what number you choose, in the beginning, you add zero and you will get exactly the same number. For operation of multiplication, the unit element is number one. You multiply number one by any number and that number remains the same. All right, so that's just the definition. Nothing's more than that. There are certain properties which we can go into, but may do a little later, of unit element. Another interesting property is inverse element. Inverse element is the following. Y is called an inverse to X if the result of the operation of one element on its inverse is the unit element. So first, it assumes that there is a unit element. That's number one. And secondly, if for an element X, there is an element Y which results in operation in the unit element, then this element Y is called inverse to X. Now, is inverse element always exist? Well, no, there are cases when there are no inverse element in some cases. Even for the same operation, for some elements, you might actually have an inverse element and for some other element, you might not. So it all depends. Now, in case of addition for any element, the reason inverse, which is the negative element for five, it's minus five. In case of multiplication, the reason inverse element for everything which is not zero, for multiplication to get one, you have to combine the number and one over this number, but not if r is equal to zero. So in some cases, in some operations, inverse elements do exist. In some cases, they're not. Okay, what's the inverse operation, for instance, for let's say, oh, by the way, inverse operations can be introduced in unary case as well. And here is what I meant. If you have one operation, then you have another operation which being applied to the result of the first gives exactly the same thing, then these two operations are called inverse to each other. Let's consider the following operation. It's called identity operation. For any x, the corresponding result of this operation is exactly the same x. For any x in the domain. Now, what does this mean? It means that f and g is an identity operation. And g, f also. So basically, it's the combination of these two properties, not just this, because this means that the f is the first one and the g is the second one, but it can be another way around. In this case, we definitely can say that the g and f are mutually inversed to each other. If the f is inversed to g, g is inversed to f. So in case of unit operations, this inverse, also the concept of inverse operation exists in the same fashion as in binary operation when we're talking about inverse elements. And now there is a very interesting connection between unary operations and binary operations. And here is the connection. Let's say you have a binary operation. I will use this sign, this symbol as a binary operation. Now, let's assume that you have an element E such that application of our binary operation on any x and this particular element E results in the same x. Let's define a unary operation. Now, why is it a unary operation? Well, by definition, unary operation has one argument and in this case it's x. So for every x I put into a correspondence the result of this operation, the result of a binary operation of x and this particular element E which happened to be unit element of a binary operation. Well, obviously this is since I have this I of x is equal to x, right? So I is identity operation. So from a binary operation which has a unit element we basically derived a unary operation which happened to be inverse. Okay, so the existence of the unit element for binary operation actually results in existence of identity operation on the same domain. Now, how about more general approach? Let's say I have a base fixed element of a domain called B. Now, this is a unary operation. So if I have a binary operation and I have chosen a specific base element it actually defines a unary operation as a result of application of a binary operation on any x which belongs to the set and this particular fixed element B. So if B happens to be my unit element of my binary operation then f-attacks would be identity function. But let's consider this. What if I have chosen two different elements, two different base elements? One is based on B and another is based on C. Now, what is a combination of these two unary operations? First, I have to apply g to the x and that would be x plus c. Then result would be B. So first I apply g which means I do the binary operation with c and then I apply f which means the result of the first operation would be applied to a binary operation with a fixed element B. I would love this to be this. Why? It just looks better because in this particular case I can say that the composition of two different unary operations derived from this binary would be a unary operation which is derived from the binary operation on the basis. I mean, that looks much better. However, I cannot usually say this unless I have associativity. So if my binary operation is associative then I can actually do this. And that's actually why associativity is a very important factor. Associative binary operations produce the corresponding unary operations with fixed binary which really very harmoniously, if you wish, can be described very harmoniously because the composition of the unit operations would be a unit operations derived from a composition of the base elements. Otherwise I wouldn't be able to do this. And now, applying to, for instance, case of inverse elements, what if B and C are inverse to each other? Which means B and C are equal to a unit element of the binary operation. Well, let's say if binary operation is addition, maybe it's five and minus five. If it's a multiplication, maybe it's two and one half or whatever else. So in this particular case, I can say that the composition of these two would be, so if B and C are inverse to each other then C operation B will give me a unit element and then X with a unit element would be just an X. So in this case, H of X would be an identity operation with the X plus unit which is exactly X. So in case my operation is associative, then my derived unary operations have this very good property. The unit operation derived from inverse elements basically is inverse operations. So again, let me just repeat it. Unary operations derived from two different inverse elements inverse in terms of binary are inverse to each other unary operations because B and G would result in an identity in this case. All right, so this is F and G, that's, did that make sense, let me see. Okay, right, just as an example, I don't know, it might actually work just a little bit better if I do kind of examples of this particular case. So let's consider again multiplication. So instead of this number, pound, sign, whatever, I use plain multiplication. Then my unit element is one. Okay, now my unary operation is a multiplication by some number B. Another unit operation is a multiplication by some number C. Now, if B times C is equal to one, it means B and C are inverse to each other, then obviously my F composition with G would be, first, I multiply, I jumped over one step here. I did not mention that multiplication is associated with operation and that's why consecutive application of first G and then F, let me put it here, F of G of X, what is it? Now, G of X is X multiplied by C. And now the F application is multiplication by B. But since we have a multiplication and multiplication is associated, if you have three different numbers multiplied by each other, it doesn't really matter what the order is. That's why it's X times C times B. So that's why I have written this. It's only because my multiplication is associated with operation. But now if B times C is equal to one, or C times B equal to one, this is X, which means F of G, F and G, or F composition is G, is an identity operation. And that's why we're talking about F and G being inverse to each other in terms of unary operations. All right, there are certain additional properties, a little bit more, maybe, I would say less trivial, more difficult, it's probably too stronger word. And I might actually go through exercises on the website and some other lectures. Basically, I wanted to give a general description of operations as just one small step into the abstract world from something which we all used to have, like operations of multiplication, addition, rotation, negation, whatever else. So again, the purpose of this is to prepare you for abstract thinking, to prepare you to create certain objects just in your head, completely unrelated to the real world. That's where the pure creativity is. And that's why I call mathematics some kind of an ultimate creative art, if you wish. Okay, that's it for today. You can always find this lecture and many others on unizord.com, which is a very, very convenient website, not only for the students who want to learn a little bit more deeper or more interesting math, but also it's very useful for the parents who would like to supervise the educational process of their students, their children. Teachers might use it as an additional educational material because it's easy in a way that students can basically study the theory using my lectures and then go through the exams. And the purpose of the teacher in this case is just to basically check how the exams are going. That's it, thank you very much.