Raising the bar

The evolution of school math, from the Mesozoic era to the present day

To write this post, I looked through all the current math textbooks used in schools in Kazakhstan, a couple of Soviet textbooks and a couple of American ones.

The content has become more advanced

In the Mesozoic era, when the current dinosaurs of Kazakhstani mathematics went to school, limits, derivatives and integrals were an elective. This advancement of school mathematics towards modern mathematics was achieved thanks to the efforts of A.N. Kolmogorov.

Soon these sections, which together make up the basics of analysis, were included in the main program of the 9th and 10th grades. Any innovations were met with strong resistance under the pretext that the existing program was overloaded. Therefore, subsequent additions were made only after 30 years. This is the theory of probability and vectors, and with them accompanying tools such as combinatorics and the scalar product.

Conclusion on the content: it has become more complex, closer to modern mathematics and adequately reflects the basics. The exception is symmetric polynomials: I have been doing mathematics for 50 years, and I have never needed them either in theory or in practice.

Many topics have been shifted down 1-2 grades

The 1st grade textbook assumes that students can read. Can everyone? How justified is it to devote 50 pages to studying the first 10 numbers?

In grade 2, letter symbols and an equation with one unknown appear. Isn’t it too early?

Verbal multi-step problems are introduced in the 3rd grade. This is normal for their age, but they require special teaching methods. Many students have difficulties at this level that do not disappear until they graduate from school. A simple method I used with my children allowed them to overcome these challenges with ease. It also ensured a smooth transition to equations with unknowns. Read on Facebook.

In grade 4, the use of formulas and equations turns from occasional to regular. Is everyone ready for this? Many of the American students I taught had an organic aversion to formulas.

In the 6th grade, technical difficulties increase. The pictures below explain the situation. Imagine that you only have a shovel and you need to climb a high brick wall (picture on the left). Anyone will understand the idea: to make a mound of earth (picture on the right), but not everyone will implement this idea.

Idea vs techniques

To master elementary school mathematics, understanding ideas and working occasionally is enough. Starting from the 6th grade and especially in high school, systematic work is necessary to consolidate the technique. This explains the following paradox: less capable but diligent students often learn better than more capable fidgeters. I often tell my students: “Your knowledge is not the books on your shelves or what you can find on the Internet. It's what you carry in your head."

The textbooks I looked at helped me understand another paradox, which I will explain with an example.

Last academic year I had a student who, by my standards, I considered capable. He quickly grasped my explanations, could reproduce quite complex calculations, but for some reason sometimes he floundered in simple questions. One of them is absolute value. It turns out that it is given in the 6th grade. The explanation for the paradoxical contradiction between person’s abilities and his misunderstanding of simple things, in my opinion, is this: a person who did not understand something at one time tends to ignore this something later. The reasons may be different: associated negative emotions, or the impression that this is well known, or reduced demands on oneself.

The trouble with most high school students is that there are many holes of this kind left in their knowledge from previous classes. In this regard, the illustrations below: on the left there is a slender building of science, without holes or flaws, on the right there is a building with holes that easily collapses under increased load.

Knowledge building

The vaunted pedagogical principle of relying on student interest fails disgracefully in the upper grades. The psychology of people is such that they enjoy learning what they understand. And if understanding is zero, and the technical obstacles leading to understanding are great, then there is zero interest. It is a vicious circle: without interest there is no study, without study there is no interest.

Conclusions on material distribution. Even with the changes described, the program is feasible for good students. On the other hand, for poorly prepared students from about 6-7 grades it becomes hopelessly difficult. At the 6-7 grade level, the vertical structure of mathematics becomes pronounced, which leads to the need to master all previous material, in order to understand new material. Intuitive understanding, which is sufficient in elementary school, is not enough in high school. There is an urgent need to improve the memory and logic of high school students.

Teaching methodology

Primary school students do not have developed analytical skills. The entire methodology is built on the “Do as I do” principle; students are given many standard exercises to reinforce the material.

In high school, the material is much more complex and following the same methodology leads to a contradiction between the theory and exercises. Since the theory is advanced, theoretical level exercises are required to consolidate it. The theory is given, but theoretical thinking is not instilled. Instead, textbook authors provide many exercises of a computational nature. Some exercises have no practical applications and are completely meaningless from a theoretical point of view. Below is a screenshot of one such exercise from a book for grade 11. Giving such exercises is the best way to discourage interest in mathematics.

Example

Republican National Test questions are compiled in accordance with the program and are quite advanced.

In the United States, there is no single program for the entire country. Each state has its own program, and schools have some freedom within that program. There are many competing textbooks, while we only have 1-2 textbooks for each class.

Our textbooks lag far behind American ones in terms of intuitive explanations, illustrations, and exercises, both applied and conceptual. The design of our textbooks is much worse. On the other hand, American textbooks are too verbose (some reach 1500 pages). Such verbosity in itself creates a problem, especially when textbooks in all subjects are like this. I once emphasized that mathematics is a speculative science and excessive verbalization interferes with the study of mathematics.

Many parents can help their children learn languages ​​and literature. When it comes to math, most parents' options are limited to the first few grades. Children are like plasticine; in skillful hands they can reach the pinnacle of science. In high school, this plasticine is completely in the hands of teachers and parents are powerless to change anything.

I am convinced that the transition from a 10-year school to an 11-year school was caused by the inability of teachers to meet 10 years with the existing methodology. I am also convinced that children of average ability, with the right methodology, can do it within 10 years, as my children did.

This post is not an obituary. This is an ode to a living Teacher.

Mukhtarbai Otelbaev

Early years

Mukhtarbai Otelbaev (I will call him MO for short) surprised us at the first meeting. It was a time when we had more strength than experience and knowledge, and everything was still ahead. In the fourth year of our bachelor program at the Kazakh State University, MO came to give an elective. More precisely, he ran into the classroom, followed by several graduate students who immediately sat down at their desks. No greeting nor anything about the course content. He began to tell theorems from functional analysis, five of them, one after another, without proof, saying that they were obvious. He often stood cross-legged by the blackboard and touched his nose and cheeks, smearing them with chalk. I don't know if MO understood from our eyes that we had never seen such teachers before. Other teachers gave the theory in small pieces, and there were some who, during the lecture, peeped into their notes, and no one had such a flight of thought and imagination.

I was then a Lenin Scholar and the best in the cohort, but I concluded for myself that I knew nothing. Functional analysis is what I need to study, and MO is a person to hold on to. When choosing a specialization, I settled on the Department of Differential Equations, because functional analysis was read there and Tulebai Amanov worked there. When I came to the Institute of Mathematics and Mechanics for practice, Amanov took me to his laboratory and asked: who wants to lead this young man? MO replied: I do, and this marked the beginning of my long ascent to real mathematics.

Arriving from Moscow, he was already an unusually strong mathematician. Those who judged by degrees thought: he had just defended his Candidate’s dissertation, he still had to work a lot to mature, and they were wrong. MO had already set his sights on a Doctor’s dissertation. During pre-defense advertising, he continued to prove new theorems publishing and adding them to his dissertation. Therefore, Boris Levitan said: we have to let him defend as soon as possible. Pyotr Lizorkin highly appreciated MO's approach to weighted Sobolev spaces and specially published two papers with him to popularize his method. Lizorkin supported doctoral defenses of several Asians, including me. With this in mind, Sergei Nikolsky later said: Lizorkin is a cosmopolitan.

In 1978 I tried to read all of MO’s articles. It was the first and last attempt of its kind. But I saw how original and profound they were. Sometimes he would let me type his articles. His manuscripts were scribbled, and I got the impression that he never rewrote them. I was especially impressed by his papers on weighted Sobolev spaces. (He wrote one jointly with Vladimir Maz’ya. In it, MO proposed one of the variants of his function q* and learned from Maz’ya the theory of capacities and techniques for working with the Strichartz seminorm. He applied and developed this knowledge further in another work, where he obtained two-sided estimates for the Kolmogorov widths.) I studied these articles in detail and, when it was not clear, I asked MO for explanations. What he explained in a couple of sentences sometimes required a long proof, and he never had any notes.

Peak of creativity

MO, together with Tynysbek Kalmenov, organized a seminar that for many years was the center of attraction for mathematicians in Kazakhstan. The talks were on a variety of topics. The discussion was lively and frank. Later I had the opportunity to compare the atmosphere of this seminar with the seminars in the USA. In the US, people speak very cautiously. Some are afraid to show their ignorance, others do not want to offend the speaker. The value of the seminar is greatly reduced by this. MO, after complex talks, explained the essence in a few phrases. It was very helpful to watch him think. He immediately saw the idea and could intelligibly explain it. In general, in his explanations, he always went straight to the idea and never talked about technical details. To understand him, you need a certain level. I think he's a great teacher for doctoral students, bad for bachelors, and terrible for high school students.

MO never followed the beaten track in his research. Even where he applied his previous inventions, he thought over everything anew. A good example is his q* function. He proposed its first version in his Candidate’s dissertation. Then there were more variants in articles on spectral theory and Sobolev spaces mentioned above. Later, Ryskul Oinarov breathed new life into it by inventing a generalization serving three weights.

The speed of thinking of MO is simply amazing. I remember once thinking about the connection between one fact from the spectral theory and another fact from the theory of functions for three days. Having understood what was the matter, I decided to test MO. When I came to the laboratory, he was playing blitz chess with someone. I asked him: what do you think about this? He thought for three seconds and said, oh, that's obvious, and continued playing. Then I said: if it is obvious, then prove it. He thought for another three seconds and told me the idea of ​​the proof. Three days in my case and six seconds in his, and even playing chess – such a difference can only be explained by a natural gift. Once, in a conversation with me, he said that among the mathematicians with whom he met, only three thought as quickly (Evgenii Nikishin, Shavkat Alimov, and I don’t remember the third).

One of his most important admonitions to me was this: constantly solve new problems. He himself never wrote more than two articles on the same topic (except for the announcements in the Doklady of the Academy of Sciences of the USSR) and never tried to inflate the volume of an article by writing out all sorts of corollaries and applications. Once he told me: if I cannot find a solution to a problem in three days, then I drop it. Some mathematicians master just one theory and then spend their whole lives looking for only such problems where it can be applied. MO did the opposite: he took on completely new problems and developed methods from scratch that aimed specifically at them. When you see this, you get the impression that you are dealing with a higher being. The combination of the generality of the problem statement and the specificity of the approaches was one of his main lessons for me.

The exception to the three-day rule was the Navier-Stokes problem. He took it up in the early 1990s. There was a beautiful operator part and there was a piece where he hoped to get the result he needed by picking numbers. To check his calculations, he sat me down next to him. He counted in his mind, and often called the numbers wrong and still got the correct result. I counted on the calculator and lagged behind him. He gave me the original version of the article to check. To my shame, I lost the manuscript. He has been working on the Navier-Stokes problem for more than 20 years, which is surprising given his impatience. The result was published, caused a strong resonance in the world, because a million-dollar prize was promised for it, and the method of proof was refuted.

A prime example of MO's innovative approach was his paper on the Vekua theory. Ilia Vekua and his students posed the problem of solving an equation, which defines generalized analytic functions, in the spaces Lp. MO posed and solved a more general problem: to find the widest space in which the properties of this equation are valid. The article was published in a modest collection of conference proceedings and is not even listed on the website www.mathnet.ru, which has web pages of all Soviet mathematicians. Despite this, the article was noticed and appreciated in the US and an English translation was published.

MO’s research was the result of a balance between two of his traits: mathematical vanity and impatience. MO and Oinarov have published two excellent papers on the theory of integral operators. Oinarov told me how they wrote the second article. Oinarov obtained some result, showed it to MO and asked: can you do this? MO, who lost interest after the first article, caught fire again, and the second article was born.

Our work on the monograph on weighted Sobolev spaces played a special role in my development as a mathematician. I wrote the whole book, reading the articles of MO and consulting with him as needed. I didn't want to drag out the writing process for several years and set myself the task of writing at least 5 pages a day. This taught me to work 10-12 hours a day, regardless of my physical condition, and developed my imagination, which is indispensable in mathematics. There was more benefit from one year of work on the book than from the previous 5.

Outside math

MO once told me that he physically can't say things that don't make sense. This quality of his is directly related to his directness and honesty. Knowing that he is the best mathematician in Kazakhstan, he could not diplomatically bypass sharp corners in communication with our academicians. When Orymbek Zhautykov told him that it was necessary to support our Izvestia of the Academy of Sciences of the KazSSR with his articles, MO replied: who needs this journal? Naturally, Zhautykov, who put a lot of effort into the creation of our Academy of Sciences and the journal, was offended. During the seminar of Umirzak Sultangazin, a small dispute arose. MO said something on the topic of the report, Sultangazin objected to him, and MO said: in mathematics, I can always argue with you. Sultangazin knew that MO was right, but he held a grudge. MO also managed to offend several other superior comrades. Because of this, he was not allowed to become a corresponding member of the Academy of Sciences for a long time.

MO generously shared ideas, and that’s why he had many students. Sometimes he made the main contribution and simply gave the article to his student, so that the student could check the proofs and publish it as a joint one. There was a time when MO gushed with articles. In one particular year he published 14 papers in the central journals (the equivalent of the current rated journals), more than the rest of the Institute of Mathematics and Mechanics. According to the existing rule, he was supposed to receive a bonus, but Yengvan Kim, whom MO also managed to offend, began to create obstacles. This was considered at a meeting of the trade union committee, I was present there, and in the end the issue was resolved positively.

A couple of times MO held high administrative posts. I believe that an administrator should be a good politician, weigh the pros and cons and be persistent in order to achieve the desired result. MO is prone to one-sided opinions (like most mathematicians), does not know how to maneuver and is not patient and persistent enough. Therefore, I think that he was a bad administrator, and if he succeeded, it was only because he had good assistants.

In the 1990s, he toyed with the idea of ​​purifying the air in Almaty with a giant pipe that he thought should be laid in the Small Almaty Gorge to pull the dirty air up into the mountains. Several factors played a role here: his self-esteem and the continued unrecognition of his talent by officials, his one-sidedness in evaluating his idea and neglect of other factors, such as the exorbitant cost and underdevelopment of the plan, as well as the decreased ability of MO to concentrate on mathematics.

This year 2022, MO turns 80 years old. Mathematicians create beauty that only they can see. I am grateful to fate for allowing me to watch how a real master creates.

Order Relations

Nurlan Abiev

The present post we devote to order relations.  For preliminaries see our previous post  Equivalence Relations.  Recall that $latex \mathbb{R}$ and $latex \mathbb{N}$ are the sets of real and natural numbers respectively.

1 Partially ordered sets

Definition 1.  A partial order on a set $latex X$ is a relation $latex R \subseteq X\times X$ satisfying the following conditions for arbitrary $latex x,y,z \in X$:
i) $latex xRx$  (reflexivity);
ii) If $latex xRy$ and $latex yRx$ then $latex y=x$  (anti-symmetry);
iii) If $latex xRy$ and $latex yRz$ then $latex xRz$  (transitivity).

Respectively, given a set $latex X$ with a chosen partial order $latex R$ on it we call a partially ordered set.

We often denote a partial order $latex R$ by the symbol $latex \preceq$  (do not confuse it with the usual symbol of inequality).  Moreover, we denote $latex x\prec y$ if $latex x\preceq y$ and $latex x\ne y$.

Remark 1. The corresponding  definitions of the concepts  partial order and  equivalence relation are distinguished  each from other only with respect to the symmetricity property (compare Definition 1 here with Definition 4 in Equivalence Relations).

Example 1. Examples of partially ordered sets:

i) We can define a partial order on any set $latex X$  assuming  $latex a\preceq b$ if  $latex a=b$.

ii) Let  $latex X$ be any given set. Assuming $latex A\preceq B$ if $latex A\subseteq B$, where $latex A,B$  are subsets of $latex X$, we obtain a partial order on the power set $latex 2^X$ (the set of all subsets of  $latex X$).

iii) Suppose that  $latex n\preceq m$ means "$latex n$ divides $latex m$", where $latex n,m\in \mathbb{N}$. Then $latex \preceq $ is a partial order on $latex \mathbb{N}$.

iv) For $latex x,y\in \mathbb{R}$ define $latex x\preceq y$ if $latex x\le y$ ($latex x$ is smaller than $latex y$). Consequently we obtain a partial order on $latex \mathbb{R}$.

v) The following is a partial order on the set of all real valued functions defined on $latex \mathbb{R}$:
$latex f\preceq g $ if  $latex f(t)\le g(t)$ for all $latex t \in [a,b]$.

vi) On $latex \mathbb{R}^2$ we can introduce a partial order putting $latex (x_1,x_2)\preceq (x_2,y_2)$ if $latex x_1\le x_2$ and $latex y_1\le y_2$.

vii) The alphabetical order of letters $latex a\preceq \cdots \preceq z$  is a partial order on the English alphabet.

viii) The lexicographical order on the set of English words is a partial order.

Remark 2. Not every relation may be a partial order.

Example 2. On the set of positive integers  assume $latex n\preceq m$ if any prime divisor of $latex n$ also divides $latex m$. Such a relation is not a partial order, since it is not anti-symmetric.
Indeed we have true expressions  $latex 4\preceq 8$ and $latex 8\preceq 4$. However they do not imply $latex 4=8$ at all.  Reflexivity and transitivity of this relation are obvious.

Definition 2. Assume that $latex X$ is a partially ordered set and $latex A\subseteq X$. We call an element

i)   $latex b\in A$  the largest (greatest) in $latex A$, if  $latex x \preceq b$ for every $latex x\in A$ ($latex b$ is greater than any other element of $latex A$).

ii) $latex a\in A$ the smallest (least) in $latex A$, if $latex a \preceq x$ for every $latex x\in A$ ($latex a$ is smaller than any other element of $latex A$).

iii) $latex b\in A$ a maximal  in  $latex A$ if $latex x\in A$ and $latex b \preceq x$ gives $latex x=b$ (there is no element of $latex A$ greater than $latex b$).

iv) $latex a\in A$  a  minimal in $latex A$ if $latex x\in A$ and $latex x \preceq a$ gives $latex x=a$ (there is no element of $latex A$ smaller than $latex a$).

Picture 1

Remark 3. The words "largest" and "maximal" seem to us to be synonyms. Accordingly, we identify the words "smallest" and "minimal". Such an identification, of course, is legal on the set of real numbers, for instance. But in general, there is subtle difference between these concepts. To clarify them let's consider the following interesting example (see also  https://bit.ly/36v3ns8).

Example 3. Let $latex X$ be a set of boxes. In $latex X$ introduce the partial order accepting $latex x\preceq y$ if a box $latex y$  contains a box $latex x$ or if they are the same box. Then under the largest box we should mean the box which contains all other boxes, but to be maximal means the property of a box when it can not be contained in any other box.

Picture 2

Case 1. Assume now that $latex X$ consists of 3 elements (boxes)  as shown in Picture 1. Obviously, the red box is the largest element and the green box is the smallest element.

Case 2. As another case consider $latex X$ consisting of 4 elements (boxes) as in Picture 2. Clearly, there is no largest element. Red and purple boxes are  maximal elements. There is no smallest element. Green and purple boxes are minimal elements.

Case 3. Finally, let $latex X$ be consisting of 4 elements (boxes) (Picture 3). There is no largest element. Red and purple boxes are maximal elements. The green box is the smallest element.

Picture 3

Example 4. On $latex \mathbb{R}^2$ introduce a partial order $latex (x_1,x_2)\preceq (x_2,y_2)$ if $latex x_1\le x_2$ and $latex y_1\le y_2$. Consider the subset $latex A$ of $latex \mathbb{R}^2$ defined by the inequalities $latex 0\le x\le 1$ and $latex 0\le y \le 1-x$.  Then observe that $latex (0,0)$ is the smallest element in $latex A$, but $latex A$ has no largest element.  There is an infinite family of maximal elements $latex (x,1-x)$ in $latex A$ but no distinct pair of them is comparable.

Remark 4. As follows from the definitions and examples above it is quite possible to have more than one maximal (respectively minimal) element of a set, even though the largest (respectively the smallest) element doesn't exist. By the property of anti-symmetry the largest and the smallest elements of a set are both unique (if they exist, of course). Moreover, the largest (respectively, the smallest) element is maximal (respectively, minimal). The maximal element is not always the largest. Likewise, the minimal element need not to be the smallest.

2 Linearly ordered sets. Well ordered sets

Finally, recall some useful definitions.

Definition 3.  A partially ordered set $latex X$ is said to be linearly ordered if any two elements $latex x,y\in X$ are comparable under $latex \preceq$, in other words, one of the following conditions holds: either $latex x\preceq y$ or $latex y\preceq x$.

For example, the set of reals $latex \mathbb{R}$ is linearly ordered under the partial order  $latex \le $  on $latex \mathbb{R}$.

Example 5.  Are the sets considered in Example 1 linearly ordered?

i) No. Any two distinct elements are not comparable.

ii)  No.  Consider $latex X=\{a,b,c\}$ with subsets $latex \{a,b\}$ and $latex \{b,c\}$.

iii) No. It suffices to take elements $latex 4$ and $latex 5$ in $latex \mathbb{N}$.

iv) Yes. Any two elements of  $latex \mathbb{R}$ are comparable under the partial order $latex \le $.

v) No.  Take functions $latex x^2$ and $latex 2-x^2$ defined on the interval $latex [0, 2]$.

vi) No. It suffices to take elements $latex (1,0)$ and $latex (0,1)$ in $latex \mathbb{R}^2$.

vii) Yes. The alphabetical order is linear.

viii) Yes. The lexicographical order is linear as well.

Example 6. As you can observe from Example 3  the set in case 1 is linearly ordered. But the sets in cases 2 and 3 are not.

Definition 4.  A partially ordered set $latex X$ is said to be well ordered if every nonempty subset of $latex X$ has the least element.

Example 7. $latex \mathbb{N}$ is well  ordered under the usual order  $latex \le $  introduced in $latex \mathbb{R}$ (this fact we will prove in the future).

People need real knowledge

Traffic analysis

The number of visits to my website has exceeded 206,000. This number depends on what counts as a visit. An external counter, visible to everyone, writes cookies to the reader's computer and counts many visits from one reader as one. The number of individual readers has reached 23,000. The external counter does not give any more statistics. I will give all the numbers from the internal counter, which is visible only to the site owner.

I have a high percentage of complex content. After reading one post, the reader finds that the answer he is looking for depends on the preliminary material. He starts digging it and then has to go deeper and deeper. Hence the number 206,000, that is, one reader visits the site on average 9 times on different days. Sometimes a visitor from one post goes to another by link on the same day. Hence another figure: 310,000 readings.

I originally wrote simple things about basic statistics. Then I began to write accompanying materials for each advanced course that I taught at Kazakh-British Technical University (KBTU). The shift in the number and level of readership shows that people need deep knowledge, not bait for one-day moths.

For example, my simple post on basic statistics was read 2,300 times. In comparison, the more complex post on the Cobb-Douglas function has been read 7,100 times. This function is widely used in economics to model consumer preferences (utility function) and producer capabilities (production function). In all textbooks it is taught using two-dimensional graphs, as P. Samuelson proposed 85 years ago. In fact, two-dimensional graphs are obtained by projection of a three-dimensional graph, which I show, making everything clear and obvious.

The answer to one of the University of London (UoL) exam problems attracted 14,300 readers. It is so complicated that I split the answer into two parts, and there are links to additional material. On the UoL exam, students have to solve this problem in 20-30 minutes, which even I would not be able to do.

Why my site is unique

My site is unique in several ways. Firstly, I tell the truth about the AP Statistics books. This is a basic statistics course for those who need to interpret tables, graphs and simple statistics. If you have a head on your shoulders, and not a Google search engine, all you need to do is read a small book and look at the solutions. I praise one such book in my reviews. You don't need to attend a two-semester course and read an 800-page book. Moreover, one doesn't need 140 high-quality color photographs that have nothing to do with science and double the price of a book.

Many AP Statistics consumers (that's right, consumers, not students) believe that learning should be fun. Such people are attracted by a book with anecdotes that have no relation to statistics or the life of scientists. In the West, everyone depends on each other, and therefore all the reviews are written in a superlative degree and streamlined. Thank God, I do not depend on the Western labor market, and therefore I tell the truth. Part of my criticism, including the statistics textbook selected for the program "100 Textbooks" of the Ministry of Education and Science of Kazakhstan (MES), is on Facebook.

Secondly, I have the world's only online, free, complete matrix algebra tutorial with all the proofs. Free courses on Udemy, Coursera and edX are not far from AP Statistics in terms of level. Courses at MIT and Khan Academy are also simpler than mine, but have the advantage of being given in video format.

The third distinctive feature is that I help UoL students. It is a huge organization spanning 17 universities and colleges in the UK and with many branches in other parts of the world. The Economics program was developed by the London School of Economics (LSE), one of the world's leading universities.

The problem with LSE courses is that they are very difficult. After the exams, LSE puts out short recommendations on the Internet for solving problems like: here you need to use such and such a theory and such and such an idea. Complete solutions are not given for two reasons: they do not want to help future examinees and sometimes their problems or solutions contain errors (who does not make errors?). But they also delete short recommendations after a year. My site is the only place in the world where there are complete solutions to the most difficult problems of the last few years. It is not for nothing that the solution to one problem noted above attracted 14,000 visits.

Fourthly, my site is unique in terms of the variety of material: statistics, econometrics, algebra, optimization, and finance.

The average number of visits is about 100 per day. When it's time for students to take exams, it jumps to 1-2 thousand. The total amount of materials created in 5 years is equivalent to 5 textbooks. It takes from 2 hours to one day to create one post, depending on the level. After I published this analysis of the site traffic on Facebook, my colleague Nurlan Abiev decided to write posts for the site. I pay for the domain myself, $186 per year. It would be nice to make the site accessible to students and schoolchildren of Kazakhstan, but I don't have time to translate from English.

Once I was looking at the requirements of the MES for approval of electronic textbooks. They want several copies of printouts of all (!) materials and a solid payment for the examination of the site. As a result, all my efforts to create and maintain the site so far have been a personal initiative that does not have any support from the MES and its Committee on Science.

Equivalence Relations

We welcome our new author Nurlan Abiev. He intends to cover topics in Mathematics.

Under a set we mean an undefined term thought intuitively as a collection of objects of common property.

Definition 1. Let $latex A$ and $latex B$ be sets.
i) $latex B$ is said to be  a subset of $latex A$ if each element of $latex B$ belongs to $latex A$. Usually, this fact we denote by $latex B \subseteq A$. Likewise,  $latex A\subseteq A$ for every set $latex A$ .

ii) $latex A$ and $latex B$ are  equal if they consist of the same elements. We say that $latex B$ is  a proper subset of $latex A$,  if $latex B \subseteq A$ and $latex B \ne A$. Consequently, we accept the notation $latex B\subset A$.

iii) Obviously, a set consisting of no elements we  call  the empty set  $latex \emptyset$. Moreover, for any nonempty set $latex A$ we accept the agreement $latex \emptyset\subset A$.

Definition 2. The  product set $latex X\times Y$ of sets $latex X$ and $latex Y$ is defined as a set of all ordered pairs $latex (x,y)$, where $latex x\in X$ and $latex y\in Y$.

Example 1. Let  $latex X=\{a,b,c\}$ and $latex Y=\{0,1\}$. Then we have $latex X \times Y=\{(a,0), (a,1), (b,0), (b,1), (c,0), (c,1)\}$. Further, note that $latex (b,1) \in X \times Y$, but $latex (1,b) \notin X \times Y$.

Definition 3.  Any subset $latex R$ of $latex X \times Y$ we call  a relation between  $latex X$ and $latex Y$, in other words $latex R \subseteq X\times Y$.

Furthermore, we say that $latex x$ is in the relation $latex R$ to $latex y$ if $latex (x,y)\in R$, and denote this fact by $latex xRy$. Respectively, the negation $latex \overline{xRy}$ means $latex (x,y)\notin R$.

Definition 4.  An equivalence relation on a set $latex X$ is a relation $latex R \subseteq X\times X$ satisfying the following conditions for arbitrary $latex x,y,z \in X$:
i) $latex xRx$  (reflexivity);
ii) If $latex xRy$ then $latex yRx$  (symmetry);
iii) If $latex xRy$ and $latex yRz$ then $latex xRz$  (transitivity).

In cases when $latex R$ is an equivalence relation we will use the notation $latex x\sim y$ instead of $latex xRy$.

Example 2. Assume that  $latex X$  is the set of all people in the world. Consider some relations on $latex X$.

i)  Descendant relation. A relation $latex x\sim y$ if $latex x$ is a descendant of $latex y$ is transitive, but not reflexive nor symmetric.

ii)  Blood relation. A relation $latex x\sim y$ if $latex x$ has an ancestor who is also an ancestor of $latex y$ is reflexive and symmetric, but not transitive (I am not in a blood relation to my wife, although our children are).

iii)  Sibling relation. Assuming $latex x\sim y$ if $latex x$ and $latex y$ have the same parents we define an equivalence relation.

Recall that $latex \mathbb{R}$ and $latex \mathbb{Z}$ are the sets of reals and integers respectively.

Example 3. Examples of equivalence (non equivalence) relations:

i) For $latex x,y\in \mathbb{R}$ define a relation $latex x\sim y$ if $latex x-y\in \mathbb{Z}$. This is an equivalence relation on $latex \mathbb{R}$.

ii) Let $latex x\sim y$ if $latex x\le y$. This relation is not symmetric.

iii) For $latex x,y\in \mathbb{Z}$ define $latex x\sim y$ if $latex x-y$ is even. Consequently we obtain an equivalence relation on $latex \mathbb{Z}$.

iv) A relation $latex x\sim y$ if $latex x-y$ is odd can not be an equivalence relation on $latex \mathbb{Z}$ because it is not reflexive and not transitive.

v) Let $latex x,y\in \mathbb{R}$. Clearly, $latex x\sim y$ if $latex |x-y|\le 1$ is not an equivalence relation since it is not transitive.

vi) Any function $latex f\colon X\rightarrow Y$ induces an equivalence relation on $latex X$ setting $latex x\sim y$ if $latex f(x)=f(y)$.

Definition 5.  Assume that $latex R$ is an equivalence relation on a given set $latex X$. We say that $latex R_a:=\left\{x\in X \; | ~\; xRa \right\}$ is the equivalence class determined by $latex a\in X$ under the equivalence relation $latex R$.

The set $latex \{R_a~|~ a\in X\}$ of all equivalence classes we call  the quotient set of $latex X$ over $latex R$, in symbols $latex X/ R$.

Theorem 1. Every equivalence relation on a given set provides a partition of this set to a disjoint union of equivalence classes, in other words, if $latex R$ is an equivalence relation on $latex X$ then
i) $latex X=\bigcup_{x\in X}R_x$;
ii) $latex aRb \Rightarrow R_a=R_b$;
iii) $latex \overline{aRb} ~\Rightarrow~ R_a\cap R_b=\emptyset$.

Thus any two equivalence classes coincide if they admit a nonempty intersection.

Proof.
i) By reflexivity $latex x\in R_x$ for every $latex x\in X$. So $latex X= \bigcup_{x\in X}R_x$.

ii) Let $latex x\in R_a$. Then $latex xRa$ and $latex aRb$ imply $latex xRb \:\Leftrightarrow\: x\in R_b$, showing $latex R_a\subseteq R_b$.
By symmetry we also have $latex bRa$. Then $latex R_b \subseteq R_a$ analogously. Therefore, $latex R_a=R_b$.

iii) Suppose that $latex R_a \cap R_b \ne \emptyset$, and let $latex x$ be their common element. According to symmetry and transitivity it follows then $latex xRa$ and $latex xRb$, which implies $latex aRb$.
Theorem 1 is proved.

The converse of Theorem 1 is also true.

Theorem 2. Every partition of a set $latex X$ to a disjoint union of its subsets defines on $latex X$ an equivalence relation $latex x\sim y$ if $latex x$ and $latex y$ belong to the same subset of $latex X$.