# Studying Discrete Mathematics from the perspective of Pure Mathematician

Okay, some necessary information, I am majoring in Pure Mathematics, Physics and Computer Science, and I am hoping to study further in Pure Mathematics.

Currently I am going through a course in Discrete Mathematics as part of my Computer Science branch of modules.

One of the things I've noticed, is that generally Discrete Mathematics courses that fall under Computer Science departments treat a wide variety of different topics in small batches, from elementary Set Theory, Graph Theory, to Algebraic Structures.

So my question is: Is there any sort of discrepancy in the way that the topics covered in a typical Discrete Mathematics course, as compared to a Pure Mathematics course on those same topic (apart from the depth of content)? For example, are certain definitions altered for use in Discrete mathematics as composed to Pure Mathematics?

Furthermore are the usual 'go-to' books for Discrete Mathematics, such as Applied Discrete Stuctures by Al Doerr and Ken Levasseur and Concrete Mathematics by Donald Knuth, vastly different from a Pure Mathematical treatment of the topics covered in the books?

The reason I ask this, is I don't want to end up in a position where I develop the wrong intuitions about topics in Pure Mathematics as a result of going through a course in Discrete Mathematics.

EDIT: Developing the wrong intuition is something I want to avoid as having gone through Physics courses I had to get rid of all the wrong intuitions of vectors I had developed. Any student of both Pure Maths and Physics, would know of the story of how vectors are defined in Physics (as 'things' that have magnitude and direction', and their proper mathematical definition as elements of a vector space)

Is there anyone who has gone through a course in Discrete Mathematics, and has done courses in Pure Mathematics that can offer some opinions/advice on this matter?

Or is my view of a 'divide' between Discrete Mathematics and Pure Mathematics, or of the way they're treated in courses, wrong?

• Why do you fear you would developed a wrong intuition? Getting different perspectives on mathematical objects is a blessing, not a curse. Aug 15, 2016 at 14:16
• @Kyle, see my edit, please Aug 15, 2016 at 14:22
• It's not that vectors are handled differently in mathematics than in physics -- it's just that the two fields use the same word for different concepts! (More precisely, the concept that physics uses the word "vector" for is narrower than the concept that mathematics uses that word for). Aug 15, 2016 at 14:27
• Intuition is mutable. As you learn more mathematics, it changes and it grows. It is not like a journal that, once you have entered an idea, it will be there forever. You can't know what you don't know now but will find out later. When you discover new concepts, your intuition will change accordingly. Even paradigms, those intuitions that everyone is supposed to know, are mutable, just not so much. Aug 15, 2016 at 15:26
• ... mathematics is Edward R. Scheinerman, Mathematics: A Discrete Introduction, which I thought quite good. Aug 15, 2016 at 17:20

"Discrete mathematics" is usually code for "whichever disjointed areas of mathematics computer science students need to know". It is not really intended to be a coherent whole, and often much of the effort in such a course is spent on teaching non-mathematicians the rudiments of how proofs and rigorous arguments look.

At some schools, students who also major in mathematics are not required to take the discrete math course at all (it was that way when I studied, for example).

As a mathematics major, you can mostly expect not to need to panic. Most of the ideas that will trouble the other students in the course are ones you should already have down pat as a pure-math student, so you can relax and focus on absorbing the substance of the theory that falls outside the usual mathematics curriculum -- often "discrete math" goes deeper into areas such as graph theory, formal language theory, possibly some logic and likely some computability theory too.

You may be asked to write down proofs in more excruciating detail than you're used to in math classes, but that is just because non-math CS students often have trouble in that area and need to see all of that detail.

It's not because it's a different kind of mathematics -- and if the standards of the "discrete math" course are different, that is just a concession to the less motivated student base, not because working computer scientists are expected to write their rigorous arguments in a different style than mathematicians.

(There are cultural differences between the fields, of course, but that comes down mostly to different degrees of familiarity with different techniques -- it's not like either of the fields actively frown on use of techniques from the other one).

I know at least one possible difference due to Knuth: reading his book Concrete math I noticed that he took a different value for some combinatorial numbers, if I remember correctly he assumed that if a binomial coefficient have negative upper number it value is infinity or undefined (I dont remember now) but the general convention is not this: the value of these binomial coefficient is defined if the lower number is negative too.

I read in some paper that this general point of view comes from the assumption that a binomial coefficient represent a limit.

I mailed Knuth and he said that his decision was based on computational reasons and that he knew this discrepancy.

So I assume from this topic that it is possible that many other topics have a difference in the way they are treated if you are just using them for computation or from an analytic point of view (aka "pure math" point of view).