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Discrete Mathematics: An Open Introduction [pdf]
> The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones, and work well with screen readers for visually impaired students. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer.

https://discrete.openmathbooks.org/dmoi4/

> The source files for this book are available on GitHub.

https://github.com/oscarlevin/discrete-book/

I love Cliff Stoll's introduction to this topic https://www.youtube.com/watch?v=W18FDEA1jRQ
This is on my todo list for just after https://slc.openlogicproject.org/.
  • giik
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I am yet to find a better introduction than Busby and Kolman's "Introductory Discrete Structures with Applications".

Beautifully written, concise, very accessible with the precise right amount of formalism.

http://books.google.com/books/about/Introductory_Discrete_St...

During my research years, we had to grind on Combinatorial Optimization book by Korte and Bygen for the weekly book reviews. Safe to say, it was not an introductory work. Still it was fun seeing the different examples my colleague would bring up during those meetings.
Two past discussions:

https://news.ycombinator.com/item?id=41267478 - Discussion on the 4th edition from 9 months ago.

https://news.ycombinator.com/item?id=23214961 - Discussion on the 3rd edition from 5 years ago.

Think this would be a great course for high school or even middle school. No plug and chug that makes it a grind, plus a great intro to proofs and deeper mathematical thinking.
I taught mathematics for 30 years at the college level. This is a college level textbook and it is not appropriate for either high school or middle school. Very few students at that level would be able to understand this material.
Not so sure. There are quite a lot of bright high school students that could indeed understand it. Maybe not in general but for a special interest group for sure. The local university had a group covering stuff like this and I found it to be very fun as a high school student, and there were at least 5 people that I went to school with that could easily handle this material (and I didn't go to some special school, either).
Unless you’re talking about an elite private school where 5 student class sizes are the norm, no, a discrete math course is not appropriate for high school students.

I took an intro discrete math course in second year of university (at a school which is easily top 5 in math and engineering in my country) and I along with most of my peers struggled intensely with it, despite all of us having completed the proof-heavy courses in first year.

On the other hand, I routinely work with high school students who are unable to multiply a pair of single digit numbers without a calculator.

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  • McUsr
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You're right I think.

Though some High school kid with interest might grasp the parts he/she is interested in.

There is a big lap from there to "Could be taught in High School".

The sheer amount of work is the main obstacle in addition to the lack of specialization in the courses is one of the obstacles I think, apart from the obvious one which is the lack of foundational skills.

Just for interest.

We covered part of this material to a similar level in high school in Australia in the early 1980s along with Calculus, probability and statistics.

In the math II/III streams ("advanced" for those wanting to do Enginering, Medicine, STEM, hard trades courses that need a grasp of math, etc).

This was a public high school in a remote area BTW, which I attended pretty much straight off an outback cattle station.

It's my understanding the US doesn't tackle any of this, nor Calculus, until undergraduate university.

Other countries address similar material (eg abstract algebra) in high school.

The full text is a bit OTT for high school (as I experienced it in Australia), but a good chunk wouldn't have been out of place, I and many of my peers would have happily read it as an extra curicular interest.

I’m speaking from the perspective of the U.S. I still contend though that the vast majority of people ages 12 - 18 are incapable of learning this book. Note that I’m talking about this book as it is in it’s totality.

This book includes counting principles in it so one can always claim that aspects of this book can be taught to people between ages of 12 - 18. We do teach people how to count at a young age. The subject matter is such that one can introduce concepts from this book in grades 7 - 12 but not at the depth the book covers them.

Calculus is taught in most high schools in the U.S. but very few students take calculus in high school.

> the vast majority of people ages 12 - 18 are incapable of learning this book.

Sure.

That doesn't prevent it being a high school text taught in some high school streams though.

Math I was taught to bulk of upper school students in Australia (in years 11 and 12), Math II/III was taught to those students interested in STEM at a university, Math IV was remedial math to bolster the poorer math students that didn't exactly pick up primary school math.

Again, it's similar in depth to how I was taught math in high school and how my peers were taught in high school, but I guess we were more math orientated than others. *

What makes it not a high school text (for advanced high school students here) is length rather than depth, while a breadth of math subjects were introduced along with concepts such as proof, reasoning, various notations, etc it was uncommon for a single subject such as discrete mathematics to be dealt with at great length, perhaps a third of the material presented here at a similar depth would be more common (again, for some high school students, not all high school students).

* eg: https://profiles.imperial.ac.uk/j.gauntlett , https://mysite.science.uottawa.ca/tschmah/ , https://hpi.uq.edu.au/profile/388/dominic-hyde are three of the people in my very small (12 in total) first year university math class who all had texts not disimilar to this in their senior math streams at their various high schools at the same time I attended high school. The entire group of twelve are not dissimilar.

Length + depth determines the appropriateness of a book at a given level. They go hand in hand and my comment was about this book. Not 1 or 2 chapters of it and not 1 or 2 chapters of it dumbed down a bit.
Again, while part of the book might be presented in a high school courses I took, a number of students that took such courses would read the entire book out of interest.

I'm unsure how you were taught but we in Australia frequently had textbooks that were only partially taught in high school, leaving the rest as untested and for interest.

This, as I'm sure you follow, means it would be fine as a text book despite only part of the text being taught.

Many of my peers were recommended a list of texts to read in high school that were never fomally taught in high school. A couple of people that were in our circle at that time read such things before high school.

It is quite obvious that you have not taught mathematics at a non specialized high school. You think the experience you and 3 other people each of whom got a Ph.D. at a good university in science and/or philosophy is somehow normative. You think this book could be taught to 15 year olds to anything more than a very small number of people is insane. Please go spend time in a classroom as a mathematics teacher before asserting such things. You might be right but your evidence is lacking. You clearly have no idea about the state of things in the average high school.

EDIT: My wife is a physician who dropped out of high school. There are people who think that because she did it this proves that more than an extremely small number of such people could become physicians. They are wrong. She got lucky and the vast majority of high school dropouts can’t become physicians.

People think that their experience going through high is indicative of what it is like on average and they advocate for positions based on their experience. It’s the “I did it therefore you can too” fallacy.

You found 12 people who could learn this book in high school. You found them in university. They didn’t all go to the same high school. Each high school has very, very few students who could understand this book. High schools don’t have the resources to teach a class for just 2 or 3 students.

Twelve, not three, in my first year university math 100 class, I gave three typical examples of the twelve.

It was almost the only university in the state at that time, two others opened in the year or so prior and these were (at the time) free universities for any that got a sufficient TAE score (high school exam for tertiary entrance).

All in all many more people were educated in mathematics or in their particular interests in this small state, and that began in primary and high school.

I attended a non specialized high school in a remote corner of a large state (3x size of Texas) with a small population (less than 2 million at that time).

I don't think that the book can be taught to any 15 year olds, I think it can easily be taught to 15 year old students interested in math, as other similar books have been.

In a not disimilar manner I know others who started olympic level swimming training in high school as part of their high school curriculum. Not for everyone, just for those high school students with aptitude.

I have spent time in a classroom as a mathematics assistant tutor, it was good pick up money during the five years I attended university.

Naturally many people that were awarded Ph.D. attended high school, I'm unsure why you would choose to exclude them from the population of high school students.

The state of things in average state run Western Australia high schools in the 1980s was that most student got mainstream education and students that showed promise in any number of different ares would get moved to specialised stream or invited to subsidised speciality camps; these existed for math, literature, theatre, music, sports, machining, etc.

eg. Heath Ledger got more theatre exposure in his High School years in this state than the majority of other high school students .. the fact that they didn't go on to play the Joker in a major Hollywood production doesn't negate what he did in high school.

You clearly have no idea about the average high school in the place and time to which I refer.

My own son attended a state high school (free public education) that had an aviation course, he and his classmates built an aircraft over two years and then took turns flying it. ( https://www.kentstreetshs.wa.edu.au/aviation )

It's sad that you seem to want to homogenize the high school experience to the least common denominator.

EDIT: “I did it therefore you can too” - not a claim I made, please stop strawmanning.

Again, there is nothing preventing a university level text being taught to high school students with aptitude and this actually happens in some education systems outside your ken.

In parallel other advanced subjects and skills can also be taught to high school students with aptitude .. and some education systems do this.

I'm sorry you apparently have not experienced such a system.

..not a claim I made, please stop strawmanning.

What you’ve been saying is based on your experience (very limited). The essence of your beliefs are that you and 12 other people could have done this book on high school therefore it should be a class in high school. Though you have no evidence that high schools have the resources to give a class that very few people are qualified to take. The distribution of qualified students is such that each individual high school will on average have very few students capable of taking such a class.

You went to high school and university and tutored some people. Therefore you know much more than me on what is appropriate to teach high schoolers. I just have 30 years of experiece teaching college level mathematics. Your arguements are compelling and I now agree with you.

The irony of you making this comment in a thread about math. 12 per year out of a population of 2 million in a remote, thus likely under served, thus likely under performing region.

My own experience is a rate closer to 1 per 35 in a reasonably well off region within the US.

You clearly have not taught mathematics at either the college level or the high school level. But your experience carries a great deal of weight. You are correct. I’ve taught mathematics at Purdue University, Florida State University, University of Kansas, various community colleges and in a prison. I don’t know what I’m talking about. The book in question should be a course at every high school. Each high school will certainly have enough students who can take such a course to justify having the course.

By the way, don’t look up what an outlier is and don’t forget that your anecdote is, in fact, data.

Indeed, I have not. Nor do I need to in order to make simple observations that contradict your claims. You are representing that something which is already widely put into practice is infeasible. A strange hill to die on.

The amusing thing is that if your claims weren't framed as an absolute they'd likely be correct. But when you attempt to make sweeping generalizations about the entire country you will almost invariably be wrong regardless of the topic at hand.

I'll also note that intentionally misattributing claims to me is neither in good faith nor in keeping with HN guidelines.

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I'll chime in as someone from the US who attended a public school and took a calculus class that used a university level text and did exactly as you described. Color me surprised when I arrived at university only to encounter that same book in use for the calculus course sequence.

The person you're engaging with here clearly has an overly generalized view of the US educational system (and is overly confident in it). It's not surprising that there's a bit of variance - the place is rather large after all.

Ninth grade introduced me to algebra and my 'technical' high school program spent a lot of resources teaching calculus and algebra. I also opted into all the other math courses they had, doing discrete math, complex numbers and stochastic math. I didn't have points to put into taking it but the school also offered a course on modern physics, teaching stuff like field equations and quantum mysteries.

One of my adolescence girlfriends left Russia after fifth grade and had an introduction to both algebra and some discrete math there.

You're so sarcastic I have no choice but to believe you.
I need to develop more intuition and maturity to understand a few relatively math-heavy engineering methods and ML/DL papers. Would you have any recommendations for not very bright college students? Perhaps something similar to Calculus Made Easy? Also, have you ever taught math using software like Mathematica or SageMath? (I graduated from college a long while ago and don't really have the bandwidth to solve problem sets by hand. I never enjoyed it or learned much from it.)
Unfortunately the best way to develop intuition is to solve problem sets :). And for ML to make sense, understanding some probability theory might be more important than understanding calculus. In math heavy papers you'll need calculus and linear algebra too, but it's going to be hard to understand them without a fair amount of prior study. I took lots of math classes and wrote out problem sets, and I still can't read many of those papers.
I meant more the subject itself rather than this particular textbook, but I’m curious about your opinion in general.

I came to this opinion after taking it in college and not recalling very much in the way of needed prerequisites, but maybe this is a selective memory…

What are some of the biggest things needed beyond algebra?

  • anthk
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Discrete Math it's far easier than Calculus with infinitesimals, limits and curves everywhere.

As a programmer with Lisp experienc but not HS-er, I'd say that any kid learning Python would be at home with Discrete Math, or most Elementary kids playing RPG's/JRPG's at home.

Here’s a sample problem from discrete math when I took it in university:

For any integer n ≥ 0, let Cn be the set of all integer compositions of n with odd number of parts, and each part is congruent to 1 modulo 3. Prove that:

    |Cn| = [x^n] (x - x^4)/(1 - x^2 - 2x^3 + x^6)
Where [x^n] indicates the coefficient of the x^n term in the formal power series generated by the rational function (rational representation of the ordinary generating function).

I doubt many elementary school students would be able to solve problems like this.

Why not? All that is really required is knowing 1/(1-x) = 1+x+x^2+... and a bit of algebraic manipulation.
And the idea of a formal power series. And integer compositions. And combinatorial enumeration (counting sets in different ways for a proof). And a bit of set theory (cardinality of sets).

There is a whole lot of background stuff here that elementary school students do not have. Way more than what you’ve stated.

You definitely don't need to know any of that background to be able to arrive at the answer. To fully understand everything maybe, but all it takes is:

a = x^1 + x^4 + x^7 + ... = x(1 + x^3 + x^6 + ...) = x/(1-x^3)

a + a^3 + a^5 + ... = a(1 + a^2 + a^4 + ...) = a/(1-a^2)

Substitute + simplify. I don't think this is beyond a (fairly smart) elementary school student.

The question doesn’t ask for an answer, it asks for a proof. You can’t just write a bunch of algebra and call it a day. You have to justify all of your arguments.
There aren't really any complicated arguments being made, so I don't think a proof would be that involved.
You obviously have not taught mathematics to high school students.
On the other hand, all the analysis really boils down to exploiting continuity and smoothness of functions. Once you get that, the epsilon-delta formulation becomes really obvious. And then you just keep building on top of it, adding layers and layers of abstraction, just like with programming.

With discrete math, there are really no unifying themes.

  • anthk
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I think of these with analogues with pixel rendering in order to understand integration and diferentiation on an intuitive way.

Once you 'see' how triangles/slopes are drawn on a GB/GBA, you begin to understand limits.

derivative of x^2 = 2x and a neglibile pixel/point that shouldn't be there but it 'exists' to show a changing factor.

I would expect people to be more comfortable with discrete math, because we are more used to thinking of separate things as opposed to things without a boundry, so to say. There are exceptions to the latter, of course, like air, warmth, rain, etc.
Having just done a discrete math course the best resource has been Kimberley brehms videos on YouTube which follows very closely to a textbook though I can't recall the author at the moment
This is already brilliant! I feel like giving myself a discrete math refresher.
  • anthk
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Another CC book on discrete Math it's Gentle Introduction to the Art of Mathematics.
60 year old. still having trouble with math. Thank for this topic and both simple pdf.
where is Polish translations?