Discrete Definition and 897 Threads

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
In university curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect.The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

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  1. I

    Understanding Discrete Symmetries in Quantum Field Theory

    I don't quite understand the treatment of discrete symmetries, for example, in Peskin's QFT book: Because by definition time reversal symmetry should flip the spin and momentum, so he defined an operation to flip the spin state of a two-component spinor, i.e. \xi^{-s} \equiv...
  2. N

    Is it Possible to Find the Domain of a Discretized Function for Minimization?

    Hello! I'm not quite sure where to put this. I'm programming, but my question should be strictly mathematical. However, just for your information, I'm programming with Mathematica. I have a function that will give me points. If I give it inputs, I can get a value out of it. I can take a...
  3. pellman

    Discrete vs continuous eigenvalues

    What determines whether an operator has discrete or continuous eigenvalues? Energy and momentum sometimes have discrete eigenvalues, sometimes continuous. Position is always continuous (isnt it?) Spin is always discrete (isn't it?) Why?
  4. N

    I have a frequency response diagram(frequency vs head) of discrete

    I have a frequency response diagram(frequency vs head) of discrete values. I have to perform Fourier analysis on these discrete values. The resultant frequency and head should be in FREQUENCY DOMAIN. how I can do this... by using FFT or any other ... using Nyquist frequency... please tell me...
  5. J

    Anyone recognize this single parameter discrete probability distribution?

    I have a single parameter discrete probability distribution defined over the domain of non-negative integers with pmf in k of: Pr(k;L) = \frac{L^{k}}{k! * k! * I_{0}(2*\sqrt{L})} Where I_{0}() is the modified Bessel function of the first kind with order 0. I do know that E(k^{2}) = L...
  6. M

    Discrete choice probability equation help (in pre-algebraic terms please)?

    As it says in the title, will anyone please give me the equation (in pre-algebraic terms) for discrete choice probability? I can see that the equation is P = Prob( Person n chooses Alternative i ) = G(xni, xnj(forall j inequal to i), sn, β), where xni is a vector of attributes of...
  7. pellman

    Integrals and continuum vs discrete

    One the things which always bugged me about the definition of the integral is that limit of a Reimann sums consists of a countably infinite number of terms, yet it is supposed to be giving the under area under a curve which varies in a continuous manner. Has anyone else thought seriously about...
  8. D

    A couple of discrete structure questions

    1. Explain how to find a shortest path between any given pair of vertices, given the length of edges connecting the vertices and the shortest distance between vertices (after doing the shortest-path calculation) So I am given a two matrices, as said one with length of edges connecting and the...
  9. D

    How to Optimize Accounts Receivable Using Discrete Response DOE?

    Hi Everyone, I am looking for some assistance on a problem that I have been working on for the last few weeks. I would like to optimise the performance of an accounts receivable department. In this dept there are several tools available to obtain a commitment to pay such as letters...
  10. N

    Lattice discrete Fourier transforms

    Homework Statement Hi Say I have a 5x5 lattice, where each entry (or we can call it site) contains the number 1. Now, on the lattice we have a function g(R), which is equal to the number on the site. In this case g(R)=1 for all sites (here R is a vector from the point (3,3), which denotes the...
  11. M

    Solving Linear System of Equations with Discrete Components

    Hi community, I have a linear systems of equations (8x8), but the matriz's components are discrete. In this moment, I make values' all of components and I write them in 64 files of 150x6000. My question is How do you solve a linear systems of equations discrete?, I mean, How do I work with...
  12. M

    I got a C in discrete math My life is over

    I think I've screwed up my future. I am in my third year, double major in physics and math (math major more to supplement my understanding of physics), and this semester was just horrid. I made the mistake of moving off campus, working two jobs, and in the time I had to study, could not focus...
  13. L

    Photoelectric effect, discrete values of the tangent

    It puzzles me. In Einstein's paper on the photoelectric effect he proposed that photons with E = nhf were the explanation. Wouldn't a more elegant explanation be that the tangent of the electromagentic wave must take on discrete values because of the boundary conditions between the emitter...
  14. L

    Arrivals and departures in discrete time

    Hello I have a problem. im looking at discrete event arrivals of items. the x-array represents the time of arrival and if one number occurs more than once it obviousely means that more than one item arrives at the exact same time. for instance: x=[0 0 0 0 3 3 3 3 6 6 6]; this means...
  15. N

    Mathematica Mathematica: Discrete Fourier Transform

    Hi all I have a function F, which depends on a discrete variable x, and I need to Fourier Transform it. I have put all the values of F in a table. Then I have used the command "Fourier" on the table, which - according to http://reference.wolfram.com/mathematica/ref/Fourier.html - results...
  16. F

    Understanding Discrete Quantities in Electrical Charge

    Homework Statement "The electrical charge exists in discrete quantities, which are integral multipuls of the electronic charge, 1.6022 e -19 C" What does "discrete quantities" mean? and Why is electrical charge an integral multipul of the electronic charge instead of just a multipul of...
  17. mnb96

    Discrete derivatives with finite-differences

    Hello, I have a function in discrete domain f:\mathbb{Z}\rightarrow \mathbb{R}, and I assume that f is an approximation of another differentiable function g:\mathbb{R}\rightarrow \mathbb{R}. In other words f(n)=g(n), n\in \mathbb{Z}. When one wants to approximate the first derivative of g...
  18. Pythagorean

    Normalization: discrete vs. continuous

    So, I'm taking an EE class and my teacher is terribly handwavy. She couldn't really explain this to me (not homework, lecture). I detect a fundamental problem in the math, coming from a science background, but it could just be my ignorance: Here's her lecture: physical setup: a...
  19. L

    [PROBABILITY] Joint probability function for two DISCRETE variables

    Homework Statement X1 and X2 represent the values of two honest dice throws (independent of each other). Find the joint probability function of U and V when: a) U = min{X1,X2}, V = X1 + X2 The Attempt at a Solution This is what I thought: P(U = u,V = v) = P(\min \left\{...
  20. mnb96

    What is the result of a discrete Gaussian summation?

    Hello, If we are given a gaussian function which is continuous in x we know that: \int_{-\infty}^{+\infty}e^{-x^2}dx=\sqrt{\pi} What if the gaussian function is discrete in x? What is the result of \sum_{x=-\infty}^{+\infty}e^{-x^2} = \\? where x\in \mathbb{Z}
  21. C

    Discrete Math: Number Segments and Common Parts | Homework Problem

    Homework Statement The statement that S is a number segment means that there are numbers a and b and S is the set to which x belongs only in case x is a number between a and b.Problem 1) If each S1 and S2 is a number segment and there is a number common to S1 and S2 then is the common part a...
  22. H_man

    Putting a discrete sum of cosines in closed form

    Homework Statement I've just found what I think is the Green's function for a source between two ideal conducting planes at x = 0 and x = l:Homework Equations G(x,x') = \Sigma \frac{icos(\pi n x/l)}{(\pi n /l)} The Attempt at a Solution The question then wants me to put...
  23. I

    Solved: Discrete Functions: One-to-One, Onto Properties

    Homework Statement Suppose A, B, C are sets and http://latex.codecogs.com/gif.latex?f:A\to%20B,%20\text{%20and,%20}%20g:B\to%20C If f and g are one-to-one so is http://latex.codecogs.com/gif.latex?g\circ%20f .[/URL] If f and g are onto, so is...
  24. Q

    Exploring Discrete Space: Planck Length and Beyond

    Discrete space has been proposed at Planck length. But has any other distance been proposed(studied), like near proton width.
  25. S

    Discrete math - links to biology?

    Discrete math -- links to biology? Hey, I'm in a math and biology program in college and I've recently become more and more into the discrete side of math. I was wondering if anybody knew of any areas of research that integrate discrete mathematics and biology, as there doesn't seem to be...
  26. S

    Probability function of a discrete random variable problem

    Homework Statement Ten cards are face down in a row on a table. Exactly one of them is an ace. You turn the cards over oen at a time, moving from left to right. Let X be the random variable for the number of cards turned before the ace is turned over. What is the probability function for...
  27. S

    The Normal Distribution and Discrete Populations

    Homework Statement One sixth of the male freshman entering a large state school are out of state students. If the students are assigned at random to the formitories, 180 to a building. 1) What approximately is the probability that in a given dormitory at least one fifth of the students are...
  28. G

    Discrete Math Problem : Mathematical Induction

    Homework Statement Prove that H1 +H2 +...+Hn = (n +1)(Hn-n)? Homework Equations Hn denotes the nth harmonic number. The nth harmonic number is the sum of 1+1/2+...1/n, which is n / n +1. I'm not really sure if Hn = (1/ n) . Prove by Mathematical Induction Hn denotes the...
  29. C

    Discrete Fourier Transform Frequency

    Hi everybody, I'm in the process of writing a discrete Fourier transform program using the algorithm on the DFT wikipedia page. When I throw in functions that I know the frequency domain signal of it gives the predicted shape but I have absolutely know idea how to generate a frequency axis...
  30. F

    Discrete and continuous signal processing

    First, I'm not an engineer, so I don't know this topic very well. Anyway, we were covering Fourier Transforms in one of my analytical methods class (chem major; NMR was the topic) and the phrase "discrete signal processing" came up. In our particular case, we collect individual points on...
  31. D

    MATLAB Calculating Discrete Fourier Transform Coefficient with Matlab

    I've been asked to write a function (.m file) in Matlab to calculate the discrete Fourier transform coefficient for an arbitrary function x. So far this is what I've done: function a = mydft(x,N) %MYDFT Calculates the discrete Fourier transform %usage: %[a]=mydft(x) %x=[ x[0] x[1] ... x[N-1] ]...
  32. S

    Solving Explicity Formula for Sum of i from 1 to n

    Can anyone help me out with this one? I need to find an explicit formula for: n 2 ∑ i i=1 I was already asked to find the explicit polynomial formula for the above equation which is n·(n + 1)·(2·n + 1) _________________ 6 I'm not sure on what the difference...
  33. G

    Mathenatucak Induction Problems in discrete math

    Homework Statement Prove that 3 divides n3 + 2n whenever n is a positive integer. Homework Equations The Attempt at a Solution Basis Step : P(1) : [13 + 2(1) ] /3 [1+2] /3 [3]/3 1 Since 3/3 =1, P(1) is true Inductive Step: [ k3 + 2k...
  34. I

    Standard topology and discrete topology

    How to compare the topology on R generated by the subbasis S={[x,y)|x,y are rational}U{(x,y]|x,y rational} to the discrete topology on R?
  35. B

    What's the best discrete mathematics textbook?

    Apparently everyone uses either Discrete Mathematics and Its Applications by Kenneth H. Rosen or Discrete Mathematics with Applications by Susanna S. Epp. Are these really the best ones? Both are very long texts which make me think they're not rigorous and they're descriptive like Stewart’s...
  36. S

    Proving a Discrete math problem

    Another one of my homework asks is this true or false and prove it: For all sets A, B, and C if A U C is a subset of B U C then A is a subset of B Please help!
  37. S

    Discrete Math Problem: Proving Subset Relationships in Sets A, B, and C

    One of my homework problems says is this true or false and prove your answer: For all sets A, B, C if A n C is a subset of B n C then A is a subset of B. I believe the answer is true but i have no idea please help!
  38. H

    Finding the Convolution of Two Discrete Signals

    Homework Statement x(n)=u(n); h(n)=(0.5)^n; y(n)=x(n)*h(n); find y(n) Homework Equations The Attempt at a Solution thank you
  39. G

    Discrete mathematics and its application 2.4 problem 26

    Homework Statement Find a formula for when m \sum k=0 the flooring function of[k1/3 ] ,m is a positive integer. Homework Equations n\prod j=m aj The Attempt at a Solution the flooring function of[k1/3] = K the summation of K is \frac{m(m+1)}{2} There's a table of...
  40. K

    Convolution in Discrete time of a function with Impulse with delay?

    Here is convolution: c[k]= (0.5)^k * delta(k-1) What do I do about delta(k-1)? I know if it is c[k]= (0.5)^k * delta(k), then it just equal (0.5)^k But what do I do with delta(k-1)?
  41. M

    Help Max Get Started on a Discrete Math Proof for Sum of Consecutive Integers

    I'm completely stumped on how to begin a discrete math proof, and I'm looking for a little advice on what might be a good way to approach this. In a previous problem I did a proof by contradiction to show that at least one of the real numbers a1, a2, ... an is greater than or equal to the...
  42. C

    Discrete Mathematics theory problem

    Homework Statement I'm supposed to prove the following. I assume it means that (w,v) and (v,w) don't both belong to f. If they do, then f certainly isn't a single function. For instance take f= x^2. The point (2,4) certainly belongs to f, but the point (4,2) does not. It in fact belongs to...
  43. E

    Discrete Random Variables and Probability Distributions

    Homework Statement Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying...
  44. Z

    Finding an unknown charge in a system of discrete point charges.

    A point particle that has a charge of 15.0 µC is located at x = 0, y = 0 (we'll call it q0 and a point particle that has a charge q is located at x = 12.0 cm, y = 0 (we'll call it q1. The electric force on a point particle that has a charge of 6.0 µC at x = 24.0 cm, y = 0 is -(19.7) N \hat{i}...
  45. W

    Discrete Math - equivalence laws

    I need to show that P<->Q is logically equivalent to ( P ^ Q ) v ( ~P ^ ~Q) So far I have P <-> Q is equivalent to ( ~P v Q ) ^ ( ~Q v P ) by a example I have no idea where to go from here
  46. C

    Proper Subsets in Discrete Math

    Discrete Math "Proper Subsets" Hey everyone, I am confused on part of this. Any input would be much appreciated! X has ten members. How many members does ~P(X) have? (~P is the set of all subsets) How many proper subsets does X have? Well the number of members of ~P is 2^10 or 1024...
  47. E

    Courses Discrete math/combinatorial course

    Hi, I'm currently taking a seminar course , it's like a "small group" course (like a lecture as a tutorial) in mathematics. It's open for anybody, most people there actually aren't math students. It's basically a discrete math/combinatorial/"problem solving" driven course but the thing is, I...
  48. S

    Question about a Discrete Structures class

    I'm in a bit of a hairy situation right now. Spring 2010 will be my last semester at my current school studying Forensic Science, and in the fall I'll be transferring to a physics program at another school. I'm trying to get some classes in that will help me in th fall before I leave but there's...
  49. R

    Quantum Mechanics: Discrete Energy Changes Explained

    According to quantum mechanics,the energy changes are not continuous but discrete. What does it really means?
  50. N

    Is space continuous or discrete?

    Dear All, I’ve been wondering about the “Is space continuous or discrete?”-debate recently. My question is the following: as far as I know, Heisenberg’s uncertainty principle and quantum mechanics are the main reasons why we believe it is discrete. Are these the only theories which predict...
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