Notes on Discrete Mathematics (2017) [pdf]
cs.yale.edu
cs.yale.edu
The common answer I get is solutions "rob" the student of learning. Why should the fear of the lazy student cheat those that actually attempt their work and want to see if their solution is correct?
Additionally, I heard the argument that "you should know" if your solutions are correct. A beginner is probably prone to subtle mistakes and may think their reasoning is rational. Thus they are probably unaware of any subtle logic errors and could easily fool themselves into thinking their proofs are correct.
Is it partially laziness from the authors in creating new problems for homeworks/exams? Or do people think a literal beginner in math can be their own proof checker?
I wouldn't want a beginner programmer writing "production ready" code without code review, so why should we expect a beginner math student to write error free proofs? What is the point in not allowing solutions?
As great as these text may be, I couldn't and wouldn't recommend them for the beginner. These text seem to be designed solely for a classroom where students will get assistance and feedback from TAs and professors. They seem useless for a beginner to learn on their own without outside proof checking.
If you study math for some time, you'll realize that people in math community (especially students) like to repeat math related cliches (not necessarily true) over and over again. This is that sort of folk lore.
If I were to guess the whys of not including the solutions, I'd say it is to save space as detailed answers to even half the exercises take up considerable space; also "laziness" because writing detailed, explanatory answers is no different from writing the rest of the book. Who wants to do double the work ? :
I offer a couple of Free texts, one of which is pretty popular (http://joshua.smcvt.edu/linearalgebra/). The answers, completely step-by-step solved, are available. I can give you two reasons.
1) Preparing good questions and giving answers is about half the work. On my page you'll see that the book is 500 pages and the answers are 400 pages.
And it is not the half that is fun, or that gets you credit. I find that compiling the answers greatly increases the quality of the book because I keep going back and adjusting the presentation, etc. But others don't think that. Rather the opposite; I have been told that it is work only appropriate for grad students.
2) I have gotten a fair number of emails along the lines of "I'd like to adopt your book but since anyone can download the answers, I cannot teach out of it."
It has not in the past proven to be a fruitful strategy for me to inform the email writer that the first thing any 2018 undergrad does on getting a text asignment is to google for the answers pdf, and that those students always succeed.
So one reason to not provide answers is to get more adoptions. This ties in with (1) because my five year reviews are not impressed that random self-learners find the text useful. They ask about adoptions.
I have gotten a fair number of emails along the lines of "I'd like to adopt your book but since anyone can download the answers, I cannot teach out of it."
When we had an introductory course in linear algebra, we were not given any home assignments. Instead, we were told that if we want to train and exercise some concepts, we should find third-party materials and self-study -- and so, I used your book. I really like that approach, it allows more capable students to do whatever they like instead, and to less capable -- work on it, if needed. Thus, I don't understand the rationale of not teaching out of something that contains solved exercises, because there is no reason to assign mandatory exercises in the first place. It's enough to have tests.
And of course, such books are extremely valuable for non-mainstream (that is, not enrolled into a high school or wherever) audience, which can be smaller, but has more impact regarding feedback.
It's quite common to look down on "lesser" work like this (janitorial work, writing solutions, whaterver) as only useful for those "lower" than you, but the amount of times I asked for help from a professor only to have them struggle to comprehend how to solve the problem (from their own assignment) was too damn high. Maybe if they tried to eat their own dog food, they'd see where they could improve.
>They ask about adoptions.
Of course, because the metrics we used to use to get a gauge of where we were progressing and where we needed work turned directly into a measure of our worth as a laborer. Your worth as a professor is only proportional to how much good PR you bring to your organization, they don't care about you actually teaching students, especially not students who aren't paying for your services via your university tuition.
Publishers reps seem to me to rely on paper mail, and an address like "Math Dept, St Michael's College" does provide some kind of proof. But I sometimes get emails like "I am a student in a poor country and the only way ..." and I simply am not saying no to that.
That last sentence also answers why I don't erase half of the answers.
Free texts are a different model and there are a lot of kinks.
The folks who wrote the book this thread is based on have done something great. (I teach Discrete and I had a look to see if I could use it.) But including answers is a problem to which I am unaware of a solution. Or maybe, it is just that expectations have to change.
This does call for a rather more misanthropic attitude than you seem to have, though.
So no, I'm not doing that. :-)
How much time did you put into writing? I have been wishing to write something similar (that is, math texts about linear algebra, analysis, number theory, etc.) for a long time, although I can only hope it to be of this quality. Do you have any tips for an aspiring math textbook writer? I have a graduate degree in math, but I unfortunately I don't use math in my day job, so I am afraid that the material I produce will be too low in quality to be useful for people...
But wow, that is a pretty deep investment! I was expecting a little less, but I guess it makes sense since writing is not your full-time occupation.
> I have a graduate degree in math, but I unfortunately I don't use math in my day job, so I am afraid that the material I produce will be too low in quality to be useful for people...
It is good to have feedback from real people. I work through material with my students in classes, and sometimes I've had what I thought was a great approach and it would just not work in practice.
I don't know if there are online groups of people working on writng stuff like this, either at the college level or at high school or earlier levels, but if you could find such a group, it'd be good to get in with them. (And tell me. :-) )
> Additionally, I heard the argument that "you should know" if your solutions are correct.
Anyone who believes this has forgotten their own early struggles or never watched beginning students struggle with something comparatively basic, like induction. In fact, introductory (proof-based) math courses are particularly dangerous for not knowing whether or not your proof is sound, because you've not yet built the mathematical maturity to check your own work. In a similar vein, beginning students often have a high level mental model of why a statement is true and which "tools" are needed, but no idea how to concretely put these together in a sound proof.
This is more of an opinion that doesn't reflect reality:
> The common answer I get is solutions "rob" the student of learning.
If the presence of the solution means the student will simply refer to it instead of e.g. completing homework on their own, then the student was probably never really engaged with the material in the first place. It's one thing to laboriously complete the proof first and then use the solution to check it, but it's another thing entirely if the student is just skipping all the work. This is especially true nowadays, because students who want to get easy answers can just go on math.stackexchange for virtually any homework assignment.
These answers are parroted because there is a deep seated "trial by fire" culture in advanced math. But this doesn't have anything to do with why the solutions don't tend to be in the books. To begin with, you have to check your assumptions:
> How can a self-learner without access to a university or professor check their work?
These textbooks are (for better or worse) not designed to be used by self-learners. In fact there isn't really any mainstream publication effort in higher mathematics targeting autodidacts. Therefore that criticism - while valid - isn't really applicable to math textbooks. Ostensibly students are using these textbooks under the instruction of a professional mathematician in a classroom setting.
The other reason mostly follows from the first one. The proof of any given statement is typically significantly longer than the statement itself. As it stands, math textbooks are usually filled with sequences of axioms, definitions, propositions, theorems and lemmas. Each chapter tends to have at least six or so of those sequences, but potentially dozens. The proofs of theorems are included, which means that all but the most trivial of chapter exercises would significantly lengthen the entire enterprise. That's a lot more writing, grammatical editing, mathematical proof-reading, page binding, etc for the whole publication process.
They're to sell a crapload of books indirectly through University professors.
As for the "you should know" angle, I do believe that a lot of problems chosen for math classes are chosen because they are asymmetric. That is, getting the answer is much harder than checking the answer. For fractions or systems of equations you absolutely can check your work. It's frustrating for me that nobody appears to be teaching kids that.
If you want to get an A on a college level math test your best use of time left over at the end of the test is to check your work.
For math specifically, they all state theorems and prove them. It's good practice to try to prove a stated theorems without looking at its proof first.
Their "work" means: Find a solution, AND then check it. So, not checking it by yourself is being lazy too.
Also, if you are learning; software like wolframalpha can help you too to check your solution. [Thanks, throwawaymath for recall this basic answer].
> I wouldn't want a beginner programmer writing "production ready" code without code review, so why should we expect a beginner math student to write error free proofs?
"Code Reviews" are for checking "how did the programmer solved that problem". On the other side; you know if some code is giving good or bad outputs by checking it with "Tests" (In the worst scenario; manually). So, you have to learn to "test" your solution. The creativity is in how you managed to get the solution and teachers should give insights about your way to solve it and present you easier or better ways to achieve the same. (Just change "teachers" for "senior devs" and now, that is code review).
> These text seem to be designed solely for a classroom where students will get assistance and feedback from TAs and professors.
Probably that's true. You have to consider those types of books as "slides"; just like a guide for the teacher and memory refresh for students.
"people learn more when they discover something for themselves than they do if someone tells them about it."
The whole idea of IBL is apparently just even communicating properly what problem you're having is a better learning experience than looking at an answer. As an anecdote a few times when I've asked math.stackexchange I found the answer just by thinking about my question.
Characteristic polynomials, rook polynomials, derangements, counting, recurrence relations, elementary number theory, elementary p-adic numbers, logic, geometric series, abstract algebra.
I have never needed any of it, and never faced a situation where knowledge or intuition in those topics helped me to think differently about a problem, never faced any type of computational complexity questions that I didn’t solve just by looking various things up as-needed.
The only thing discrete math did for me was help me boost my GRE math score to get into a good math grad program, where I switched to machine learning and MCMC topics.
Very occasionally you might encounter a simple permutation or combination counting argument in a paper or something, but it’s rare.
I enjoy discrete math, but really felt misled about it’s actual usefulness in almost any corporate software / research / engineering job.
If you don’t know this stuff but you can grok most basic counting arguments after looking a few things up on Wikipedia, you’ll be fine. Deeper command of these topics really doesn’t yield economic returns to your effort in almost all cases, unfortunately.
In general, if you can model a distributed or concurrent system in terms of a monoid (or better, a commutative monoid) you're onto a winner.
Anyway, part of the value isn't in the material itself, but in the practice of learning hard novel ideas, not just the mechanical fundamentals that you could coast through because you are smart, but then hit a wall when you face hard problems later I life.
In the spirit of a well-rounded college math education, sure, taking one or even two classes on this topic would be fine, for basic literacy.
I’m only saying that it doesn’t come up as a needed skillset in applied math or computer scuence jobs. Virtually never. So to whatever extent people say it helps develop systems thinking, algorithmic thinking, complexity thinking, it’s just not true.
Beyond that, I think it’s more important to learn how to solve problems generally and how to self-teach in order to pliably move into a topic you are less familiar with, grok the details sufficiently well to figure out a plan of attack, and figure things out well enough to know what to research or where / how to ask for help effectively.
A good, well-rounded curriculum is worthwhile, whether in college or self-taught. But there’s no magic set of things that always cover an adequate basis of knowledge for everbody’s career. So learning how to adapt and learn is more beneficial in general.
As far as economic returns go, I’d say first find a particular technology or problem area that is in demand, and begin working that way and using time, work projects, and other resources to expand your set of expertise from there.
Being fluent in statistics/linear algebra & discrete math/category theory -- makes you 'head and shoulders' above others, when it comes to assessing industry trends, and deciding if some are applicable to your work.
Having those skills, does not mean, of course, that they would compensate for the lack of creativity, hard work, risk taking, constant year-after-year-self study, or inter-personal & presentation skills.
Some examples:
Being able to understand word-vectors and related techniques, for chat boats, or compliance -- is very helpful.
Being able to reason about degrees of security of data published on 'immutable databases' or 'data streams' -- is useful, again in compliance (as in making records tamper-proof).
Being able to design protocol/interaction across systems with context-free grammars -- is helpful, in protected against certain types of attacks. I would say this is reasonably relevant for systems that are exposed on public internet.
Being able to read reasonably recent (or 20-30 year old) papers on Differential privacy/homomorphic encryption -- allows you to reason how effective your data protection strategy could be (again relevant for compliance).
Obviously, understanding financial service models, performance models of distributed queuing systems -- require understanding of the maths I mention above.
Being able to recognize where in your, even basic business workflow or database/state management -- there is an appropriate place to introduce finite-state-automate (to reduce number on 'unanticipated conditions') -- is also helpful.
Being able to recognize database mapping through the lens of category theory, improves how you could design your data models (because right at the time of designing your data layout you are also thinking about transformations between them -- which is basic need for non-trivial business system integration).
Opposite to the above post, I always feel that I need more math background that I have. And I would love to go back in time and significantly beef-up my education in that area.
(Note: you lump linear algebra in to the discussion but it is dramatically different than what is typically called “discrete math.”)
Additionally, many of my colleagues on machine learning and intense distributed scientific computing applications that I’ve worked on have had zero discrete math education or self-teaching.
I am more effective at engineering in these applications than some of these people, and less effective than others. Prior exposure to discrete math seems to have no effect.
On the first day of class he told us (paraphrasing) “I never went to class in undergrad, so I don’t expect you to. But the lectures will help you on the problem sets, so I encourage you to attend.”
For those of us who may have had... less than thorough attendance... these notes were a real godsend!
He also has some for his randomized algorithms class: http://www.cs.yale.edu/homes/aspnes/classes/469/notes.pdf
And also for his other classes which you can find here: http://www.cs.yale.edu/homes/aspnes/
(Professor Aspnes is now the DUS of the CS department at Yale)
(I just finished Discrete I & II)
A single source is not better than many.
We can argue if this method, where not every class on the same subject is equal, is good or bad, but till this is what is widely used, it's good for both students and professor to have a reference.
Opinions also differ on what is valuable and should be taught (you can't teach everything in a finite time), if the teacher can skip some partial result he doesn't find interesting he can opt for a different proof altogether.
Thus there'll always be some treasure left for the most ambitious student by tackling various such works.
To answer the question more seriously, there are a number of reasons for this: 1) terminology and notation changes over time, 2) there is not enough space in a standard textbook to include all material on any given subject in mathematics, even at an undergraduate level 3) there have been such standard textbook projects, e.g. the Bourbaki textbooks, but in order to cover everything in complete generality, they had to work at such a high level that the material is all but useless to a beginning student (many French professors may disagree with me on this point). 4) It is not uncommon for textbooks to start with different assumptions, to target the material for certain research applications, to give a different perspective on the material, to promote a favourite notation, to give new proofs or even new theorems, in lecture notes written by professors.
What this doesn't answer however, is why there is such a dearth of technological solutions to the problem of disseminating the 40 odd thousand pages of undergraduate mathematics taught around the world today. There are lots of videos at a very superficial level, very few in depth, and piles and piles of books and pdfs. There are virtually zero really well-produced, animated, scripted videos that cover mathematics in great depth.
The only reason I can come up with for not having such material is lack of time. Academics, especially professors, who would have to write and edit such material, would have to invest so much time, they would not have time to have families, or do research.
And re: in-depth, this is a long tail problem. The profit is in the mainstream, not in the tail. Producing really good in-depth work isn’t that rewarding because the market is small.
On the other hand, technology offers new and interesting ways to disseminate this knowledge, but I see very little at the same depth as a standard undergraduate course, which is not just a video of someone doing the equivalent of a chalk-and-talk.
You can forget lack of time altogether and simply think of it as a lack of resources or money. The problem is to date, lack of an adequate self-sustaining model to provide these resources.
In this case by “model”, I mean either a business model where the work was incentivized by profit, or a well organized community that is able to motivate its members to collaborate and volunteer as has been shown to be possible in other areas.
There is, for one, far more material in any given subject than an introductory course or textbook can (or should) cover, so the author/instructor must choose what to include.
Plus, the order of presentation matters. For example, here are two standard ways of introducing the real numbers:
#1 (Dedekind cut). Picture a square of side length 1. The length of the diagonal, √2, cannot be represented by a ratio of integers, so we need a new number system to represent it. These numbers "in between" rational numbers are called irrational numbers, and together they form the real numbers.
#2 (Cauchy completion). Non-repeating decimals, such as π ≈ 3.141592, cannot be represented by a ratio of integers. We call such decimal numbers irrational numbers. Any number representable by a (finite or infinite) decimal is called a real number.
You can deduce #2 from #1, and vice versa. It's entirely up to the author/instructor to decide which one to start with.
Lastly, there is always a better way to explain the same material.
1) Gilbert Strang's "Introduction to Linear Algebra" was great because Gilbert goes straight to intuitions, the proofs are simple, most exercises have answers, but it does not cover advanced material. I used this book for self-teaching. You could probably learn from it with just high-school level maths. Good for engineers.
2) Hoffman and Kunze's "Linear Algebra" was given as a textbook for my first LA course. While it covered some topics that weren't found in many other textbooks and are not really "standard curricula" in many other universities for (jordan normal form, rational canonical form). I found it more similar to a reference than a textbook; it is intended for math majors. The proofs are imho a bit obtuse and it usually introduces topics without much justification. Determinants are introduced early.
3) Axler's "Linear Algebra Done Right" OTOH covered many of the topics in Hoffman&Kunze but the organization and the proofs were (imho) mucho more clear and motivated. Also intended for math majors. No determinants until the end.
And the reals were introduced to me by the completeness axiom phrased like this: "If M is a point set and there is a point to the right of every point of M, then there is either a right-most point of M or a first point to the right of M."
The material isn't quite the same because your ideas of what a first-year student needs to learn about probability may differ from someone else's. Not just because your and their prejudices are different: it'll be influenced by what else is in other courses. Imagine two universities with first-year probability courses for their mathematicians. One of them also has a statistics course that everyone takes, the other doesn't. They may make different choices about what goes in the probability course. And it'll be influenced by the students they get. Imagine one of the universities is a very prestigious highly-selective one that lots of good mathematicians apply to, and the other ... isn't. The first will likely want more material in their course.
Now, think again about those last two universities. As well as possibly presenting different material, they may well want it presented differently. Fancypants University will want extra-rigorous proofs, connections to other fields of mathematics (measure theory, probabilistic methods in combinatorics, ...), more challenging problems. Slowcoach Polytechnic will want a very clear and straightforward presentation that doesn't leave anything unexplained and doesn't distract students with irrelevancies. So even when they're presenting the same material, the approaches required will be different.
Different students will want different things, too. Besides the sort of differences in the previous paragraph (which of course apply between students as much as they do between universities), some students will be interested in the foundations of the subject (what is a probability, anyway?) and some won't; some will be interested in particular kinds of applications; some will like thinking visually and benefit from a lot of digrams, some will find them distracting; etc., etc., etc.
Now, with all that in view, imagine you're a professor who's teaching Probability 101 for the first time next year. No matter what, you're going to need to do a lot of thinking about what material you put in your lectures, how you divide it up, how you want to present it, etc. You're going to need some notes for your benefit. (The benefit comes both from making them and from having them.) The extra effort required to polish them up and make them usable by students too is actually pretty small. The fact that other people have presented other similar courses and made similar notes is neither here nor there. You can't just take their notes and avoid all the effort, because their course isn't identical to yours, and their students aren't identical to yours, and their style of presentation isn't identical to yours, and much of the benefit of making the notes comes from actually making them. So, you make your own notes, and then you might as well put them on the internet. And here we are.
In Germany, for example, calculus courses will recommend Heuser. Linear algebra will recommend Beutelspacher. Every EE owns a copy of Tietze & Schenk.
https://www.theonion.com/author-of-introduction-to-algebra-r...
He wrote similarly thorough notes for his Randomized Algorithms class.
It also led me to Concrete Mathematics by Knuth, et al ...
https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathemat...
> [Basic mathematics on the real numbers] Why: You need to be able to understand, write, and prove equations and inequalities involving real numbers
If logic is the assembly, real numbers must be the Enterprise Java frameworks :)
But really, aren't they from non-discrete (continuous?) mathematics? How are they useful for computer science? Like, sure we have approximations of them on computers (floats, rationals) but aren't they mostly used for those pesky boring real-world-ish calculations? Isn't CS mostly about integers?
Reals are only needed for calculus on infintesimals, which are not relevant to much of CS. Everything you can do with formal Taylor expansions doesn't need reals. In fact knowing calculus on the reals often causes confusion in students learning formal expansions for CS.