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.

View More On Wikipedia.org
Recent contents Top users
  1. Jayalk97

    Solving for the impulse response of a discrete time system?

    Hey guys I was just studying for finals and I came across something in my textbook that either wasn't explained properly or something I just don't get. So this page explains that the coefficient of y_n[k] is A_0, which is represented by b_0/a_0, I can see that a_0 is determined by multiplying...
  2. S

    How Can a Function from Real Numbers to a Discrete Space Be Continuous?

    Homework Statement Find ##f:R \to X##, f-continuous, where X is the discrete space. Homework EquationsThe Attempt at a Solution f is continuous at p if for any ##\epsilon > 0## there is ##\delta >0## such that ##d(f(x),f(p))<\epsilon## for all x such that ##d(x,p)<\delta##. Let ##\epsilon =...
  3. S

    How Does Increasing 'n' Affect the Cut-off Frequency 'k_cut' in DFT Analysis?

    Homework Statement I have this function ##f(\theta)=cos(n \ sin(\frac{\theta}{2})\pi)## and I need to take the discrete Fourier transform (DFT) numerically. I did so and I attached the result for ##\theta \in [0,2\pi)## and n =2,4,8,16,32, together with the function for a given n. I need to...
  4. S

    Python Discrete Fourier Transform in Python

    Homework Statement I need to calculate the derivative of a function using discrete Fourier transform (DFT). Below is a simplified version of my code (just for sin function) in python Homework Equations from __future__ import division import numpy as np from pylab import * pi = np.pi def...
  5. L

    Understanding the discrete time Fourier transformation

    Let's consider a signal which is continuous in both time and amplitude. Now we consider the amplitude of this signal at specific time instants only. This is my understanding of sampling a signal in time domain. When performing a Fourier transform on a time discrete signal, we have to apply the...
  6. kaniello

    A Help with Discrete Sine Transform

    Hi, I am a neophyte in Discrete Fourier Transform and I am procticing with discrete Sine-transform. Specifically I want to calculate $$ \mathcal{F}_s \lbrace x \cdot e^{-x^2} \rbrace = \int\limits_{0}^{\infty } x \cdot e^{-x^2} \cdot \sin\left(x\right)\, \mathrm{d}x = \frac{1}{2} \pi^2 \omega...
  7. T

    A Asking about lattice translation as a discrete symmetry

    I've read Modern Quantum Mechanics J.J. Sakurai 2nd Edition. In topic about lattice translation as a discrete sysmmetry(page 287) , they discuss only in case of ground state. If I want to study more in case of including excited states, where can I find this information? thanks.
  8. S

    Correction to the field energy due to the existence of discrete charges

    In the classical electromagnetic field theory, the field density of energy is given by: $$u = (\epsilon/2)E^2 + (\mu/2)H^2$$ One of the differences between the classical electromagnetic theory and the real world is that in classical EM all charge and current density, (ρ(r), J(r)), is...
  9. M

    B Why did Max Planck assume discrete energy values?

    I've read several articles discussing how Max Planck decided to assume that the energy radiated by oscillators in a black-body came only in discrete increments of En = nhf, where n is an integer. Using this concept, he determined that the average energy of an oscillator was given by...
  10. Telemachus

    Fortran Discrete Fast Fourier transform with FFTW in FORTRAN77

    Hi, this thread is an extension of this one: https://www.physicsforums.com/posts/5829265/ As I've realized that the problem is that I don't know how to properly use FFTW, from http://www.fftw.org. I am trying to calculate a derivative using FFTW. I have ##u(x)=e^{\sin(x)}##, so...
  11. Telemachus

    Doubt about a discrete Fourier Transform

    Hi. I was checking the library for the discrete Fourier transform, fftw. So, I was using a functition ##f(x)=sin(kx)##, which when transformed must give a delta function in k. When I transform, and then transform back, I effectively recover the function, so I think I am doing something right...
  12. R

    Courses Which course should I take: Discrete Math or Bridge to Advanced Mathematics?

    Hi, I am currently an undergraduate student and I plan on taking advanced math courses such as Abstract Algebra, Real Analysis, Complex Analysis, etc. There are two courses which I think could help me prepare for the courses above as they are proof intensive: discrete math and bridge to advance...
  13. J

    A Discrete measurement operator definition

    Consider the Gaussian position measurement operators $$\hat{A}_y = \int_{-\infty}^{\infty}ae^{\frac{-(x-y)^2}{2c^2}}|x \rangle \langle x|dx$$ where ##|x \rangle## are position eigenstates. I can show that this satisfies the required property of measurement operators...
  14. M

    Discrete LQR regulator and Bellman's principle

    Hello everyone. Iam studying the LQR regulator in optimal control theory right now but Iam having some issues in understanding the approach of Bellmans principle. As far as I have understood, in Bellmans dynamical programming approach, one goes backward in time to find the optimal Gains K...
  15. R

    Matching Discrete Fourier Transform (DFT) Pairs

    Homework Statement [/B] I am trying to match each of the following 28-point discrete-time signals with its DFT: Set #1: Set #2: Homework EquationsThe Attempt at a Solution Set #1 We have already established (here) that: ##Signal 1 \leftrightarrow DFT3## ##Signal 4 \leftrightarrow...
  16. C

    Response of LTI discrete time causal system

    Homework Statement Discrete time causal LTI system has impulse response h(n) = (-1/2)^n, n≥0. a)Find transfer function of given system. b)Find frequency characteristics of system. c) Find response for u(n) = μ(n)-μ(n-4) The Attempt at a Solution a) Either from definition, or from table...
  17. R

    Discrete Fourier Transform (DFT) Matching

    Homework Statement Match each discrete-time signal with its DFT: Homework EquationsThe Attempt at a Solution I am mainly confused about Signal 7 and Signal 8. Signal 1 is the discrete equivalent to a constant function, therefore its DFT is an impulse (Dirac ##\delta##), so it corresponds...
  18. Rohan Patil

    I Are continous Spectra actually discrete?

    If we assume that there is a fixed charged particle and another charged particle is spiraling down towards it, it emits electromagnetic waves as it is accelerated motion. We get a continuous spectrum. Now, if I allow the emitted photons to fall on a very photosensitive material, whose efficiency...
  19. Eclair_de_XII

    Courses Taking continuous probability over discrete probability?

    I'm considering taking the upper-level probability course at my school over the elementary course offered because of time constraints. The latter is not a prerequisite for the former. Do you think I will be alright taking the more advanced probability course over the elementary course? Any input...
  20. I

    I Normalizing a Discrete Sum: Is the Wavefunction Fully Normalized?

    Say you have two energy eigenstates ##\phi_1## and ##\phi_2##, corresponding to energies ##E_1## and ##E_2##. The particle has a 50% chance of having each energy. The wavefunction would thus be ##\psi=\frac{\phi_1}{\sqrt{2}}+\frac{\phi_2}{\sqrt{2}}## Even though the coefficients are normalized...
  21. R

    Discrete Math - Strong Induction Question

    Homework Statement Prove by Strong Mathematical Induction[/B] Homework Equations N/A The Attempt at a Solution The steps to solving this problem are shown below. I understand all steps of the problem until the part where it says 44/49 becomes 49/49 since 44 < 49. Can someone please explain...
  22. O

    Other A in Discrete Math 1, Lost in Discrete 2.

    Hey everyone, Pardon the novel of a post. Short story: Got an 'A' in Discrete 1 (proof methods) using Epp's book at community college with an easier professor (the only offering for that course), and I'm understanding nothing to very little amount of information in Discrete 2 (computational...
  23. binbagsss

    Q about a Proof -- periods meromorphic function form discrete set

    Homework Statement Hi, As part of the proof that : the set of periods ##\Omega_f ## of periods of a meromorphic ##f: U \to \hat{C} ##, ##U## an open set and ##\hat{C}=C \cup \infty ##, ##C## the complex plane, form a discrete set of ##C## when ##f## is a non-constant a step taken in the...
  24. binbagsss

    I Definition of discrete Subgroup quick q

    Hello, Just a really quick question on definition of discrete subgroup. This is for an elliptic functions course, I have not done any courses on topology nor is it needed, and most of the stuff I can see online refer to topology alot, so I thought I'd ask here. I need it in the complex plane...
  25. S

    I Discrete Mathematics Function Topic

    I am currently taking a course in discrete mathematics. The literature used is "Discrete Mathematics And Its Applications by Kenneth H. Rosen" 6th ed., or 7th ed. I have encountered most of the topics from that book. I.e. Logic, naive set theory, &c. What I have encountered also is the...
  26. B

    Discrete Valuation Homework: Prove Valuation Ring and Unit Description

    Homework Statement Let ##p## and let ##\nu : \mathbb{Q}^\times \rightarrow \mathbb{Z}## be defined by ##\nu (\frac{a}{b}) = \alpha##, where ##\frac{a}{b} = p^\alpha \frac{c}{d}##, where ##p## divides neither ##c## nor ##d##. Prove that the corresponding valuation ring ##R := \{x \in...
  27. D

    How many slits for a discrete spectrum?

    Homework Statement I watched two videos on KhanAcademy, one was about light interference with 2 slits and the other was with 4 slits. The video with 2 slits got a continuous spectrum whereas the one with 4 slits got a discrete spectrum. So my question is: how many slits does it take to get a...
  28. U

    Discrete Mathematics logic questions

    Homework Statement 1. Why is the statement: " Vicky is not clever" Not a mathematical proposition? Provide examples please 2. Why is the statement: "a^2+b^2=c^2 an indeterminate proposition?" 3. Why is the negation of " If a triangle has two equal angles it is isosceles" = "Not all triangles...
  29. R

    Probabilistic Analysis of Drawing Cards and Dice Rolls for Success Conditions

    I am looking for a way to think about this type of problem and this is not a coursework problem 1. Homework Statement How many cards from a 52 card deck of ordinary playing cards would you have to draw to have either a flush or 4 of a kind If you have a 30 6 sided dice, how many dice rolls...
  30. T

    Courses Discrete Structures or Probability and Statistics Engineers

    Hi, I was wondering which course will be more beneficial to take first for a first year student majoring in computer science? Note: I tend to dislike proofs and theories. Discrete Structures - An introduction to the basic concepts of statistical analysis with special emphasis on engineering...
  31. G

    I Discrete vs Continuous Spectra in Blackbody Radiation?

    I was reading this article which talks about the theoretical model behind blackbody spectra: http://www.cv.nrao.edu/course/astr534/BlackBodyRad.html At the start, it mentions standing waves in a cavity. Standing waves in this model consist of an integer number of wavelengths. The standing waves...
  32. Domenico94

    Continuous and discrete spectra

    Is there any way to convert a continuous, aperiodic spectrum, to a discrete spectrum, in a signal? If so, would part of he energy of this signal be lost, I am this process of conversion, or would it be " distributed" amomg the various frequencies?
  33. M

    I Is Discrete Mathematics Hard and Is It Pure or Applied?

    Is Discrete Mathematics hard? Is it pure or applied math?
  34. M

    I Sum principle proof: discrete mathematics

    Theorem: Let ##A_1, A_2, ..., A_k## be finite, disjunct sets. Then ##|A_1 \cup A_2 \cup \dots \cup A_k| = |A_1| + |A_2| + \dots + |A_k|## I will give the proof my book provides, I don't understand several parts of it. Proof: We have bijections ##f_i: [n_i] \rightarrow A_i## for ##i \in [k]##...
  35. I

    Discrete Math Proof: Necessary Condition for Divisibility by 6

    Homework Statement We have JUST started writing proofs recently, and I am a little bit doubtful in my abilities in doing this, so I just want to verify that my proof actually works. I was expecting this one to be a lot longer since the previous 2 were. I don't see any glaring flaws in it, but...
  36. Brian T

    Computational Looking for resources on Discrete Exterior Calculus and FEEC

    Does anyone have recommendations for reading/resources on Discrete Exterior Calculus and/or Finite Element Exterior Calculus? In particular, I want to learn the topics to use in a project for a course and so would like to learn how to implement these methods (specifically geared toward...
  37. Jezza

    Domain of a discrete fourier transform

    Homework Statement The (computing) task at hand is to take a function f(x) defined at 2N discrete points, and use the Discrete Fourier Transform (DFT) to produce F(u), a plot of the amplitudes of the frequencies required to produce f(x). I have an array for each function holding the value of...
  38. Avatrin

    Math Applications of discrete mathematics minus software

    Hi All applications of discrete mathematics I know of seem to be in computer science. I want to know if there is somewhere discrete mathematics are applied outside of software. What can I work as if I like discrete mathematics but do not want to program? (outside of academia, of course)
  39. Jezza

    How Does Slit Height Affect the Discrete Fourier Transform?

    Homework Statement [/B] This is a computing coursework problem. (There is a reasonably long theory preamble). Create a single slit centred on the origin (the centre of your array) width 10 and height 1. The array containing the imaginary parts will be zero and the array containing the real...
  40. Stephanus

    B Discrete definitions of Physics Standard Units

    Dear PF Forum, Perhaps it's not a question, just a light discussion. Time From this, we get the standard time https://en.wikipedia.org/wiki/Caesium_standard, it's 9,192,631,770 ticks per second. I think this number should be discreet. No need to tune it to some figures after decimal point. Is...
  41. I

    I How do you move floors and ceilings in discrete math?

    The title more accurately should have been "How do you cancel floors and ceilings in discrete functions" For instance, ##\frac{log{\frac{3x}{-6(z)}}}{8t} < 1## If I wanted to get rid of the log, I'd just raise the expression by base 10. ##\frac{(\frac{3x}{-6(z)})}{10^{8t}} < 10^1## But what...
  42. FeDeX_LaTeX

    I Discrete Convolution of Continuous Fourier Coefficients

    Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...
  43. OrangutanLife

    Discrete Math: Understanding Sets and Elements

    Hi, My classes don't start until next week and I am trying to get a head start in my linear algebra, discrete math and calc 3 class! I am using Discrete Math by Epp 4th edition. 1) I know 4 =/ {4} but why? 4 is a symbol representing the number 4, and {4} is a set with only one element which...
  44. lsepolis123

    In discrete Math Adv Counting Techniques - see picture - h

    how to solve exercise (34) in discrete Math Adv Counting Techniques - see picture===> how apply the formula?
  45. M

    A Does continuous mass distribution implies finite propagation

    speed? This question emerged in my mind while studying a discrete and continuous mathematical model of a falling slinky. In the discrete model, we suppose an instantaneous interaction between mass points at a distance, so the action propagates through the chain of mass points with infinite...
  46. T

    I Smoothness of Discrete Functions

    Hi Physics Forums Is there a specific technique to measure how smooth a discrete function is? By smooth I mean that if you change the input by a minimum amount then you know that the objective function result will not have a big jump. For example The Closest String Problem is completely...
  47. R

    Elementary Set Theory (Discrete)

    Homework Statement Suppose A⊂B⊂C. What is A/B, A/C, and A∪B Homework EquationsThe Attempt at a Solution This isn't really a homework question, I am just trying to get some exposure to discrete math before I take it in the fall. The set differences A/B and A/C are both empty sets and the 'or'...
  48. O

    Discrete Differential Amplifier

    Homework Statement Calculating the open loop output Impedance at 1kHz and the open loop differential gain The Attempt at a Solution Here is the approach. INPUT STAGE: Q111 and Q112 Av1 = Rtot/2re, where Rtot is the total resistance of R117 and the input resistance of Q78 which is Rin. So...
  49. R

    Courses Thermodynamics/Statistical Mechanics and Discrete questions

    I just switched my major from chemistry to physics because I wasn't happy with the amount of math I was encountering as a physical chemist. So now I am going into a new major program and I am afraid of some of the courses I am about to take. Next semester I will be taking a math methods course...
  50. P

    [Discrete] Prove that |nZ| = |Z| for any postive integer n

    I have been studying discrete mathematics for fun and I am kind of stuck on this bijection problem. 1. Homework Statement I wanted to apologize in advance if i put this homework question in the wrong part of the forums. Discrete Math and much logic math is a computer science type math of...
Back
Top