Discrete Mathematics video, Book recommendation and learning experience

Source: Internet
Author: User

In a twinkling of an eye, I started school for two months. In the middle, I had to spend a month or so on learning discrete mathematics because of some disgusting things, summarize the learning content, methods, and books.
First, what exactly does discrete mathematics contain? Traditional lecture methods include mathematical logic, set theory, algebra system, and graph theory. Some books also include combined mathematics, number theory, and even discrete probability theory. In fact, each part of the course can only be taught separately. Therefore, it is impossible to express them perfectly in a book. It is indeed a big question how to learn which book to use. My learning method is basically video + books. I have read two complete videos: one is the Discrete Mathematics of the Chinese Emy of Sciences, and the other is the video of Liu xuhua from Jilin University. Their corresponding books are Qu wanling's Discrete Mathematics (04-year edition) and Liu xuhua's discrete mathematics. I also flipped through the famous discrete mathematics and its application (the sixth edition and the seventh edition of Chinese will come out at the end of this year ). The following is a one-to-one explanation:
The lectures delivered by Mr. Gao suixiang are based on their own textbooks. Then, I will explain some unintelligible things and I have no understanding or theoretical system. The lecture covers mathematical logic, set theory, algebra system, and graph theory. The other part is quite understandable, and the algebra system is very poor. That is, you don't understand what it is. I have almost understood all the theorem and examples, but I still cannot understand why the three major problems of ancient mathematics:
1. Tri-equi-angle problem: divide any given sub-point.
2. Cubic product problem: Calculate the length of the cube to make the cube twice the size of the known cube.
3. The problem of turning the circle into a square: make a square so that its area is equal to the area of the known circle.
Why can't I do it.
However, this is not all a strange teacher, because of the mathematical logic, set theory, algebra system, these three parts, a total of nine days of class, graph theory content also only three days. One afternoon every day! It may be that the Chinese Emy of Sciences's talented teenagers are really super intelligent and able to keep up with them, but I feel very hard for people like me. The lecture content is closely related to the examination syllabus, the content of the examination, and the content of the examination. For example, the content of Boolean algebra is not mentioned (the reason is that there is no requirement in the outline ). However, after listening to these questions, I can do most of my homework. As to the usage of these questions, I have no time to teach them.

Let's talk about Liu xuhua's class. Great! First, he will introduce the content, development history, and details of each part, which will be of great interest. Secondly, he has a deep understanding of Discrete Mathematics, for example (the following content is careless ):
"Mathematics is the abstraction of specific problems, and mathematical logic is the abstraction of reasoning problems. He does not care which proposition is correct, but whether this reasoning method is correct ."
"Using a one-to-one correspondence to define equality is a very simple idea. For example, when I was a child, I would not divide the total number by two, but you would give me one"
"Predicates are like functions ."
"Abstract algebra is an abstraction of algebraic operations. It only cares about the nature of operations, regardless of the specific operation results ."
In the course of explanation, there is a lot of mathematical flavor: definitions, theorems, proofs, very strict. Even some seemingly inconspicuous things, such as when the left unit element is equal to the right unit element, are all proved. As the teacher said at the end of the last lesson (here is the general idea): "I may have learned thousands of theorems in my life, and many of them have never been used in my life, but they cultivate my precise mind, which is necessary by a computer scientist." Unfortunately, because it is a course of TVU, predicate reasoning in mathematical logic is not discussed. The content of the algebra system only talks about groups. For the ring, domain, the concept of lagam, and the Business Group, not to mention, the content in Boolean algebra is also missing. In addition, due to a long history (more than nine years), the video recording performance is not good, and the pictures and sounds of some chapters are not synchronized (you can download the video and enable two players at the same time ).
I did not comment on other videos because I have not carefully watched them.
Let's talk about books. Qu wanling's discrete mathematics: this should be a classic version. I can only say that in general, a new version was released in, and the preliminary contents of combined mathematics and number theory were added, which cut down the contents of the algebra system. The general feeling is that there is no characteristic. First of all, it wants to hook up with the application, but there are not only a few examples in the book, but some may even be confusing. As a result, the previous content requires the knowledge of the following chapters. In addition, the lecture in the book follows the style of traditional textbooks-simple and various "obvious", which is very painful for self-taught children. To learn mathematics, you must "Speak Chinese" and describe the meaning of each theorem and the reason for derivation of formulas in natural language to better understand the problem. The mathematical problem refers to the abstraction (symbolic) of the actual problem. Only by speaking Chinese can the abstract formulas be transformed back to reflect the original nature of the abstract formula. At the same time, we will introduce the background of this knowledge and how it is used. However, this book has many exercises and answers, so it is an excellent teaching material for exam-oriented education. This book also has a notable problem, that is, the introduced symbols are not very good, so sometimes I have to flip the book for a while to find out where the symbol is. The terms, terms, and symbol table below the book are basically useless. When I don't know which one I encounter, it is definitely useless to look up the table. Of course, I personally feel that apart from the part of the algebra system, the explanations of the Set Theory and mathematical logic are very clear, and there are occasional highlights, for example, in the mathematical logic, it is enough to use a complete set to explain why the circuit only uses non-gate or non-gate. It clearly proves why the anyway method is effective. There is also the solution method (but I didn't understand it) and so on.
Liu xuhua's "Discrete Mathematics": This book is not well-known and does not exist in the school library. If the content in the book is consistent with the content in his class, it is definitely an excellent entry-level teaching material. (This book has been published for 93 years and can still be bought now. It can be seen that it has been recognized by the market ). Another notable feature of this book is that there are very few symbols used, and the combination of functions is to use ordinary multiplication. Binary relationships are used (x, y) instead of <X, y> emphasize order and so on. The most unfortunate thing is that it is said that Mr. Liu has become a plant (the news comes from the network and cannot be verified), so this book has never been released. You can't help but get bored.
Finally, let's talk about Rosen's "discrete mathematics and Its Application". I only read the mathematical logic section, and I 'd like to explain my feelings. First, the audience and functions are clearly defined, that is, the basic content of discrete mathematics. Although the knowledge is wide, the difficulty is very low. For example, there is no "paradigm" in mathematical logic, in the algebra system, we only talk about Boolean algebra (not from the perspective of lattice). Second, we stick to the application, and many examples are related to computers. Third, we will introduce you to a large number of scientists related to mathematical logic. Fourth, the book has a large number of questions, and there are answers and books that can be used to facilitate self-study.

Finally, some learning experiences about discrete mathematics are for reference only.
Not all mathematics related to computers, or mathematics that can be implemented by computers should be put in discrete mathematics. Otherwise, numerical analysis (Calculation Method) will also be pulled in. As we have already said, it is best to open a single course for each content. Otherwise, it will certainly be difficult to learn. If it is not allowed, I think the knowledge of set theory, mathematical logic, algebra system, and graph theory is necessary.
Set Theory is the foundation of the entire Mathematics (whether it is discrete mathematics or continuous mathematics), so if you have not studied it specifically, it is quite appropriate to appear in discrete mathematics. The binary relationships and functions derived from the set are also taken for granted.

Mathematical Logic is an eye-catching thing. For the first time, I found that some complicated reasoning problems can be solved through the "computing" method. This undoubtedly provides good news for stupid people like me. As for the content of the entire mathematical logic, it depends on the situation. As for the solution method mentioned above (the true value of the proposition formula for programming), I personally think it is very important, but it is a little difficult, therefore, it is estimated that there will be fewer people (including teachers ).

Apart from Boolean algebra, an important application in the computer and communication fields is coding. However, it is necessary to use the knowledge of galova domains to encode complex codes, it would be better to talk about galova. However, if the galova domain is not mentioned, the whole explanation is a little tricky. Boolean algebra can be well described without the abstract algebra, however, it is impossible to present the important point of view that the set algebra and the proposition algebra are actually two different interpretations of the Boolean algebra.

Graph theory is a very useful learning. It abstracts the "relationship" that is hard to describe, and expresses it in the form of a matrix. Tree and graph are widely used, such as the Hoffmann encoding and shortest path. I personally think that if I have learned the data structure and a little understanding of the application, I will have a better understanding of these abstract problems.
It is difficult to comment on the combination of mathematics because I have never learned it. The most common application of number theory is cryptography. If you are not familiar with this aspect, it doesn't matter if you are not familiar with number theory.
As for what to learn first, I personally think it would be better to explain mathematical logic first. After all, although set theory is the foundation of the entire mathematics, logical reasoning can be considered as the foundation of mathematics. First, streamline the logic and use logical symbols later to make other theories more concise.
Finally, it is very effective to check whether your understanding of the knowledge is in place: General difficult questions. If you do, you will basically understand this knowledge point, if you don't know the answer, but you can understand it, it means you still have a lack of knowledge about how to use it. If you don't even know the answer, you don't understand the knowledge, that's why the book is really not written. There is no harm in doing more questions.
PS: I have never studied discrete mathematics before, but when I read introduction to algorithms, I found that mathematics is really around me. It guides us to design algorithms, so I took a while. The learning content of discrete mathematics has not been written into notes for sharing, mainly because: 1. there are too many formulas, which may be difficult to write. 2. I do not have a deep understanding of this course. It is basically the content of the book. In addition, the contents in the mathematics book are hard to be "COMPRESSED", so I will not be ugly.

As for what discrete mathematics has brought to me, it is estimated that after reading the book "discrete mathematics and Its Application", I will have some practical experience. I will not talk about it here.
That's all. I haven't updated my blog for two months. I'm sorry, but I will continue to share my learning experience with you later.

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