Discrete Mathematics: An Open Introduction, 3rd edition
discrete.openmathbooks.org
discrete.openmathbooks.org
Personally having a bunch of fancy tools like that makes it harder to keep a flow of (try problem, get it wrong, figure out why, repeat) but I guess I’ve never tried that with a math book. In college I always found it was best to study math with a dumb pdf and a notebook.
Just start reading the book (any discrete mathematics book!) :)
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I'm about 70% through this book. I've done every exercise and have meticulously journaled about each concept as I encounter them into my Zettelkasten.
Best part: you can read it online and then determine if you want the physical book (which I prefer for math).
Those are the things that I found most helpful on my recet foray into the Zettelkasten world. Could save you some time.
One thing my notebooks don’t have is structure and a system of referral. This makes it hard to reference previous ideas and concepts. In fact, I don’t because it’s so difficult - it’s just one big stream of consciousness. And it seems Zettelkasten does that!
That other comment gave a link to that "how to take smart notes" blog post, the HN comments were here: https://news.ycombinator.com/item?id=22341518
A thing to keep in mind: there are two schools of thought. Niklas Luhmann was an extremely productive German sociologist who used his index cards system in a particular way. One school of thought tries to use his system exactly, like explained in the book that blog post is about. Others think that Luhmann had to work like that because he was restricted to physical cards, now with digital tools we can make other choices and get similar effects. Both ways are referred to as "Zettelkasten method".
I also realized that the system really works for producing content, things like producing science papers. You need to put work in how you formulate notes, and in finding connections between them. Without a good reason to keep working with your Zettelkasten, there is a risk it'll just be a pile of notes.
If it’s not possible, let me at least give you a couple of questions:
1. What app are you using? (And why not Roam / why not Obsidian?
2. What do you put on one Zettel? Just theorem? Theorem+proof? Some bigger mass of knowledge?
3. Do you rewrite the proofs in detail, or just explain them using your own words without mathematical rigor?
Discrete and combinatorial mathematics by Ralph Grimaldi
Wow.
However, I found the chapter on generating functions a little frustrating: it gives a very good explanation of what they are, tells you they're super useful, and gives no example of an actual problem where they're used.
Good to see they're at least here online, though it's also shameful that I didn't find this on Google when I searched for it a year or so ago. Tried to buy a calculus textbook to teach a friend - only overpriced latest editions by and large were available.
> Pinter, "Abstract Algebra" > Tao, "Introduction to Measure Theory" (and "Epsilon of Room, Vol. 1")
In a typical discrete maths course, the main "arbitrage" you get from algebra is the "method" of generating functions, which is a technique for dealing with integer sequences and counting problems using polynomials. If you like matrices, you can encode polynomials as Toeplitz matrices, so you can view products of polynomials as matrix products. But generating functions are not a panacea for counting, and you usually have to seed the ground with some combinatorial results before you can water it with generating functions. Also, there are some neat uses of matrices in understanding Pascal's triangle ( http://www-math.mit.edu/~gs/papers/pascal-work.pdf ) and other sequences.
In graph theory, the matrix-tree theorem lets you count spanning trees using a determinant, and the Dijkstra algorithm for path finding can be regarded as a matrix power over the max-plus semiring. But that's about all you get out of linear algebra in a typical first discrete maths course. Matrices become more useful if you go deeper into graphs or into counting, but never take center stage.
[And before the next question comes: I haven't seen many monads used in discrete maths either.]