 # What is Discrete Mathematics?

Another teacher asked me this question today (one of his students asked him, and he passed the buck), and the best I could do was give him the statistics definition of discrete (versus continuous) data. Now it bugs me. What should I have told him?

It’s a branch of maths (like pure maths, statistics, mechanics etc.) that deals with several areas (can’t remeber all of them though), particularly codes (bar codes are an absolute favourite).

I do have a discrete maths textbook somewhere (I’ll see if I can find it) as I started a discrete maths As level several years ago in my sixth form college, but gave it up to play football instead.

I have no idea, but I would be very interested to know! I have a vague impression that continuous mathematics deals with time dependant numbers ie musical intervals…

The mathematics of computing and also general logic are areas covered by discrete maths. IIRC Logic gates are another favourite.

Unable to find my text book, but from Durham universtity, here are a few discrete maths questions:

1. (a) How many 4-letter words are there with vowels as the first and last letters? (vowels are
A, E, I, O, U.)?
(b) How many 4-letter words consist only of vowels?

2. (a) How many 5 letter words are there where alternate letters are always the same?
(b) How many 5 letter words are there where alternate letters are always distinct?

3. 4 cards are dealt from a standard 52 card deck of playing cards and are placed one each at
the 4 corners of a square table.
(a) How many ways are there of doing this so that all 4 cards have the save denomination
(e.g. all aces)?
(b) How many ways are there of doing this so that all 4 cards have the same suit (e.g. all

4. X is the set of all 6 digit numbers in ordinary decimal notation (â€™ordinary decimal notationâ€™
means that â€™leading zerosâ€™ are not allowed.) Compute:
(a) jXj.
(b) jfn j n 2 X, 5 exactly divides n gj.
© jfn j n 2 X, n has only one digit a 5gj.
(d) jfn j n 2 X, n is a palindromegj (a palindrome is something which reads the same
backwards as forwards; e.g. â€™was it a cat I sawâ€™).

http://fourier.dur.ac.uk:8000/~dma0nm/pdf/discq1-2001.pdf

whoever can answer those questions earns qudos as they are from a module on a degree course.

I’ll start you off with the first one (btw I don’t have the answers so I may well be wrong)

1. a) 26^2 * 5^2 = 16900

The problems MC Master of Cermonies describes are all combinatorics (i.e. counting) problems, which is one kind of discrete mathematics. Other kinds of discrete mathematics include graph theory, coding theory (as MC has already mentioned), and recurrence equations.

It’s kind of a catch-all term to refer to mathematics which deals with discrete entities instead of continuous ones. Combinatorial problems, for example, are discrete mathematics because they deal with integer quantities instead of real numbers. Recurrence equations deal with discrete units of time, as opposed to differential equations which deal with continuous time, so recurrence equations fall under the “discrete mathematics” label as well.

2a. if the question means words like ABABA, KEKEK, etc, then the answer is 676, since it effectively reduces to a two-letter word.

2b. 26x26x25x25x24 = 10140000 if the first letter must be distinct from the third and fifth letters. 26x26x25x25x25 = 10562500 otherwise.

3a. There are 24 (4!) different ways 4 cards can be arranged at the corners of the table, times 13 denominations = 312.

3b. Tricky. There are 715 (13/4 x 12/3 x 11/2 x 10/1) ways to draw 4 cards from a set of 13, times 24 ways they can be arranged on the table, times 4 suits = 68640.

Does discrete mathematics also encompass the field of number theory?

It should, as there’s no notion of continuity in number theory, but I don’t think that it does in common parlance.

Not checked throughly, but looks right enough, so you earn the qudos I think no.4 lost something when it was cut and paste, so I’m not sure if it’s answerable from that layout.

Yeah, that was pretty much what I figured it was. Stats is the only place where I really have the occasion to use the word discrete. I’m pleased to see that it means the same thing in every other context, too. It does seem odd that number theory is not automatically lumped in with the others here, but, whatever.

I’m taking a course in Graph Theory this summer. Sounds like fun. …gee and I thought its when the numbers go to dark corner when they multiply…

To Orbifold’s list I would add: set theory, logic (not just Boolean), and sums/series as well as recurences. Certain types of algebras may or may not fit in such a course: modern algebra (group theory and all), matrices and other “large” systems.

There’s even a discrete calculus! Although, I don’t know how useful it is…

I took two discrete math courses last year, and although IANAMathematician, I can tell you that we studied the following things:
Number theory (esp. different radices; binary, hex)
Set theory
Graph theory
Matrices
Finite-state machines
Basic cryptography (i.e. Caesar Cypher)
Formal Logic (i.e. A and B, C not D)
Combinatorics
And a handful of other things I can’t remember offhand.

Both of my professors admitted that discrete math was more or less an umbrella term for various subfields of math that were fairly new on the scene (I know at this point someone will mention that some of these have been around for hundreds of years; I can’t remember which ones at present) and were useful for computer majors (as my major is Information Technology)

Just goes to show how approaches can be different.

I just calculated it this way :

For 1st card, it can be any of 52 (since denomination is not yet chosen)
For 2nd card, any of 3
For 3rd card, any of 2
For last card, only 1

Hence, n = 52x3x2x1 = 312.

Propositional and predicate calculus come under the banner of discete maths and are used in formal specification (of software requirements). Try looking up a tutorial on formal methods for examples of how they’re used. Z is an example of a formal spec language (developed in the UK and so pronounced Zed)

Achernar’s example is something quite different. It’s a discrete operator with properties analogous to the differentiation operator. And one would assume that there’s an inverse operator that acts a bit like integration…

Well I taught a course in discrete math some years ago and we did no set theory, no number theory, and no logic. I am not claiming these subjects are not discrete, but they are not usually covered in a DM course. What did we do? There was some counting, some graph theory, a lot of difference equations and summations (which are the discrete analog of differential equations and integrals), a fair bit of work with generating functions, some fun things like the Fibonacci numbers (actually the solution to a simple difference equation) and a bit of boolean algebra. Actually, DM seems to be mainly a catchall term for stuff that is clearly discrete but not covered by any other rubric. Number might look discrete but the main methods are purely analytic such as the connection between the Riemann zeta function (continuous) and the distribution of primes (discrete). Then there is the Langlands program that basically holds that there are very deep connections between the discrete and continuous at all levels.

It is very useful to people who design and analyze digital filters and robotic control stuff.