Recomendations for how to improve my backgroud for advanced topics

[la6ki]

Hello everybody.

For the past couple of months I've been trying to study abstract algebra by myself. I got some solid books and tried following some online courses. As I was studying things like groups, rings, and fields, I found studying those topics much tougher than studying Calculus 1, 2, 3, linear algebra, diff equations, and so on (at least at the level taught to non-mathematicians, like physicists I guess).

I decided that I probably don't have a proper background, so got some books on set theory alone. I got Keith Devlin's The Joy of Sets: Fundamentals of Contemporary Set Theory (because it was recommended in forums) but realized that something in my background was lacking even to read that book. For instance, the author gives a few simple rules of inference and than gives some exercises asking you to prove identities, without having gone through a single more complicated example.

For instance, he gives this definition of an ordered pair:

(a, b) == {{a}, {a, b}}.

And then the following exercise:

Exercise 1.4.1. Show that the above definition does define an ordered-pair operation; i. e. prove that for any a, b, a', b'

(a, b) = (a', b') <-> (a = a' /\ b = b').

(Don't forget the case a = b.)

So, I'm stuck as to how I should go about proving this (nor do I have an idea as to what he means by "Don't forget the case a = b")

I'm guessing that even this book is intended for people who already study mathematics at a university level. The problem is I really don't know what book to get next in order to improve my background.

I would highly appreciated if you could suggest useful readings.

 

[verty]

Proving this may be easier if I translate it into English for you:

The ordered pair (a, b) is the ordered pair (a', b') if and only if (exactly when) a is a' and b is b'.

Think about extensionality (your book should have mentioned this or look it up on Wikipedia) and think how one could show this proposition to be false. What would need to happen for it to be false?

Granted, this may still be highly difficult to solve. I just wanted to give you a test that would measure what level of logic book would be most useful. If it looks utterly impenetrable, I recommend starting with a book of logic puzzles, this one for instance. If it looks easier but you still do not know why it must be true, one of the popular books on logic will do, perhaps this one. If you can work out why it is true but aren't sure how to write the proof, you can, to save time, skip straight to this book. I still like a book like this one, it may seem verbose but learning logic from a logician is really the best way. For example, why did I say "is" and not "is equal to" in my translation? A simplified book will probably not explain this.

There are also books on Mathematical Logic but these may well be too advanced, they assume one already knows logic and some set theory. Have a look for the notes by Simpson for a taster.

Books mentioned:
Smullyan - What is the name of this book?
Bennett - Logic Made Easy
Polya - How to Prove It
Tarski - Introduction to Logic

Disclaimer: this was a very simplified test and should not be taken too seriously. Pick whichever book you like best (not necessarily one listed here of course).

 

[la6ki]

Thank you for you recommendations!

Okay, so I need a better background in logic. I did suspect that. I'm guessing all undergraduate math students go through a logic course before they proceed with set theory and other abstract topics?

As I was investigating how to improve my background, I came across proof theory. I'm not sure if it's covered in the above logic books or if it's a more advanced topic. To what extent is it necessary for my background, given my intention to study more advanced topics, like abstract algebra, non-Euclidian and differential geometry, and advanced probability theory courses?

And also, do you think there are other prerequisites for these topics besides logic? As I said in the first post, I have a good understanding of calculus and linear algebra, and a decent understanding of differential equations.

 

[verty]

Proof theory is something I'm not too familiar with, but it relates to intuitionistic logic, which is logic with a different definition of truth: something is true only if I can prove it is true. So they translate "if A then B" as "if I have a proof of A, then I can convert it into a proof of B". And they define "if <variable> then <variable>" as a function from proofs to proofs; given a proof of A, it constructs a proof of B. So this is also called constructive logic.

One of the problems it has is that a proof by contradiction does not work because proving ~~A just (then) means that I have proved that it can't be proved that I can't prove A, but I do not yet have a proof of A.

This is more complex than standard logic but it may still be interesting to read about.

 

[Stephen Tashi]

I decided that I probably don't have a proper background, so got some books on set theory alone.

There are some people who are interested in the details of set theory, but I think most math graduate students don't go into the technicalities of set theory that you are describing. Their knowledge of set theory is less formal.

My guess is that your problem with abstract algebra isn't due to a deep knowledge of set theory. It might be due to an unfamiliarity with the straightforward facts of set theory like DeMorgan's laws. Or perhaps your problem is that your have never dealt with proofs.

You should define your objective and be clear about whether it involves making "due progress". A person studying as a hobby has the leisure to make a detailed study of logic and set theory before tackling "modern algebra". People in graduate programs can't spend years becoming experts at set theory or logic before they take their graduate algebar courses. If you don't have any schedule in mind and you are the type of person who wants to know all details of topic A before going to topic B then it's fine for you to spend a long time on set theory and logic , especially if that interests you. But you should be aware that this is not tne path that most math graduate students follow.

 

[la6ki]

There are some people who are interested in the details of set theory, but I think most math graduate students don't go into the technicalities of set theory that you are describing. Their knowledge of set theory is less formal.

My guess is that your problem with abstract algebra isn't due to a deep knowledge of set theory. It might be due to an unfamiliarity with the straightforward facts of set theory like DeMorgan's laws. Or perhaps your problem is that your have never dealt with proofs.

You should define your objective and be clear about whether it involves making "due progress". A person studying as a hobby has the leisure to make a detailed study of logic and set theory before tackling "modern algebra". People in graduate programs can't spend years becoming experts at set theory or logic before they take their graduate algebar courses. If you don't have any schedule in mind and you are the type of person who wants to know all details of topic A before going to topic B then it's fine for you to spend a long time on set theory and logic , especially if that interests you. But you should be aware that this is not tne path that most math graduate students follow.

I wouldn't say I study those things as a hobby. I am studying neuroscience and want to have a very solid background in physics and decided that in order to achieve that, I will also need a very solid background in math. Also, advanced probability theory that I'm also trying to learn (this is an actual requirement for my goals), involves much terminology, notation, and concepts which I don't know or understand.

I got the book Probability Theory: A Comprehensive Course by Achim Klenke and I want to grasp all the concepts in it. In the very first page of the first chapter I already encountered this:

and didn't know what the notation meant (e.g., the inverse U with the infinity sign in the second definition). I skipped through the pages and realized that it is impossible to make sense even of the first chapter, let alone when it starts getting more complicated. So, my quest now is to understand what I need to learn before I can make sense of this book :)

 

[Stephen Tashi]

I want to grasp all the concepts in it. In the very first page of the first chapter I already encountered this:

I'd infer from that page (which looks like it has to do with "sigma algebras of sets") that your approach to studying math is to pick the most advanced books you can find on a topic. The typical math student takes probabilty courses where probability theory isn't done correctly by the standards of an advanced point of view. Later the student may take a course in "measure theory" that uses the approach that employs the set theory on that page. Most math students don't become familiar with the set theory on that page from a course on set theory. They become familiar with it from courses on "analysis" and "point set topology". They gradually learn about the behavior of finite and infinite intersections of open intervals on the real number line. (Point Set Topology is where the terms "open set " and "closed set" are treated with some abstraction). This motivates the definitions that you see on the page of that book. Even with a background in analysis and topology, some students need explanations of the set theory on that page when they are introduced to "measure theory".

It isn't reliable to use the principal "There is material I don't understand that involves topic X, therefore I should study a book devoted to topic X". A better approach is to get a course catalog from a university and note the order of the courses and the textbooks that are used.

As to approaching probability theory from the viewpoint of measure theory - for a person in applied mathematics, this is somewhat useful for clearing away misconceptions about theoretical situations, but a book on measure theory doesn't teach anything about applying probability theory. It's actually hard to make the correspondence between the common situations in applied probability and the material in the measure theoretic approach to probability unless you are an expert in the applied situations.

 

[la6ki]

I'd infer from that page (which looks like it has to do with "sigma algebras of sets") that your approach to studying math is to pick the most advanced books you can find on a topic. The typical math student takes probabilty courses where probability theory isn't done correctly by the standards of an advanced point of view. Later the student may take a course in "measure theory" that uses the approach that employs the set theory on that page. Most math students don't become familiar with the set theory on that page from a course on set theory. They become familiar with it from a coursess on "analysis" and "point set topology". They gradually learn about the behavior of finite and infinite intersections of open intervals on the real number line. (Point Set Topology is where the terms "open set " and "closed set" are treated with some abstraction). This motivates the definitions that you see on the page of that book. Even with a background in analysis and topology, some students need explanations of the set theory on that page when they are introduced to "measure theory".

It isn't reliable to use the principal "There is material I don't understand that involves topic X, therefore I should study a book devoted to topic X". A better approach is to get a course catalog from a university and note the order of the courses and the textbooks that are used.

As to approaching probability theory from the viewpoint of measure theory - for a person in applied mathematics, this is somewhat useful for clearing away misconceptions about theoretical situations, but a book on measure theory doesn't teach anything about applying probability theory. It's actually hard to make the correspondence between the common situations in applied probability and the material in the measure theoretic approach to probability unless you are an expert in the applied situations.

Thank you for your detailed response.

I must say that I already have a decent background in probability theory, so I don't know how much I should work on it before I move on. Would you actually recommend to read a book on measure theory as a next step, or try to find a book on probability theory which is more advanced than my current background but still not as advanced as the book I cited above?

 

[Stephen Tashi]

Would you actually recommend to read a book on measure theory as a next step, or try to find a book on probability theory which is more advanced than my current background but still not as advanced as the book I cited above?

My guess is that you should find a more advanced book on applied probabilty in preference to studying measure theory. I can make a better guess if can we get an idea of the type of problems that you want to tackle with probability theory. (After all, there is the possiblity that current mathematics doesn't know how to solve these problems.) Perhaps there are examples of journal articles in neuroscience that use mathematics that you don't understand yet.

Some clusters of advanced topics in applied probabilty theory that come to my mind:
1. Continuous stochastic proceses: stationary random functins, stochastic calculus, brownian motion, wiener process, Ito's integral, stochastic differential equations
2. Statiistical decision theory: optimal statistical decisions, Bayesian inference
3. Discrete probability models: discrete markov chains, Box-Jenkins models (auto-regressive moving average models)

There are "probably" others that I don't know about!

 

[la6ki]

Well, I don't only have specific end goals in mind and I would like to have a deep theoretical understanding of the field as well. So, one of the end goals would be, as I said, to reach the level at which I can understand the book I cited from. Otherwise, stochastic differential equations, discrete Markov chains, and Bayesian inference are definitely things I want to learn. Also, having a good intuition about the properties of different distributions (Gaussian, Poisson, etc.), Monte Carlo methods...

Let me ask a question to make sure I understand correctly.

The book I cited from is too formal because it's not about applied probability theory and if I find a book about applied probability theory, the topics it covers will be in a more understandable language?

And if so, it's a much better idea for me to begin with books on applied probability theory and only after I have mastered the topics there to move to more abstract approaches?

(If I understood correctly, could you recommend a book on applied probability theory I could start with?)

 

[MarneMath]

Thank you for your detailed response.

I must say that I already have a decent background in probability theory, so I don't know how much I should work on it before I move on. Would you actually recommend to read a book on measure theory as a next step, or try to find a book on probability theory which is more advanced than my current background but still not as advanced as the book I cited above?

When you say 'decent background in probability' theory, do you mean to say that you can derive moment generating functions, multivariate distributions, co-variance and correlation? If the notation on that page is confusing you, then perhaps an introduction to mathematical statistics like DeGroot's book "Probability and Statistics" is more in-line with your knowledge. The book you have now is a graduate level book, and it would definitely behoove you to know the basics of a real analysis class with measure theory before you begin to read such book.

The set theory book you are reading is not intended for non-math students, and it assumes a certain familiarity with mathematics. If you find that book to hard to read, then I suggest Rosen's Discrete mathematics because it offers a gentle approach to logic and proofs.

 

[Stephen Tashi]

Let me ask a question to make sure I understand correctly.

The book I cited from is too formal because it's not about applied probability theory and if I find a book about applied probability theory, the topics it covers will be in a more understandable language?

Yes.

And if so, it's a much better idea for me to begin with books on applied probability theory and only after I have mastered the topics there to move to more abstract approaches?

Yes.

(If I understood correctly, could you recommend a book on applied probability theory I could start with?)

Like MarneMath, I don't understand the scope of your current knowledge. Another handicap I have is that I haven't been in school recently and I'm unfamiliar with recent texts.

On the forum, several people have recommended the book "Understanding Probability" by Henk Tijms. On the basis of that, I bought a copy of the book. I've only looked at it superficially and "it looks good", but I haven't read it in detail. It gives a brief overview of important topics in probability theory. You might be able to find a preview of it on the web and see if you already know the material.

The Schaum's Outline math books that I've seen are good for teaching how to work specific problems. The desciption of the newest edition of the Schaum's Easy Outline of Probability And Statistics says it is "pared down" and cuter. That suggests that other editions (whose title doesn't include the word "easy") might be better books.

(By the way, you do need to understand enough basic set theory to know the meaning of an upside down "U".)

 

[la6ki]

Thanks a lot for all the recommendation, guys!

Well, now I realize that the word "decent" can mean very different things for a mathematician and for a non-mathematician :) I guess what I actually meant was that I have the rough basics of probability theory (like having some experience with concepts like sample space, Bayes' theorem, the binomial coefficient...)

I did look at the books you recommended and my first impression is that I will be able to understand them, so I'll give one of them a try.

One final question: is it possible to gain access to the curriculum of first year undergraduate math students so I can see the topics they cover, the depth at which they go into those topics, and the order? Perhaps that will give me a better idea as to how to proceed when I encounter similar problems in the future.

 

[MarneMath]

You can look at any university math department and find the program requirement to graduate with a math degree. That should give you a rough idea what a certain university feels is required and the order it should be learned. However, typically, after three sequences of calculus and linear algebra, students have a wide ability to take many classes, the real issue is getting the current textbook at your level. Right now, you're obviously trying to read at too high of a level.