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. G

    Is This Discrete Mathematics Argument Valid?

    Homework Statement Hey guys I am having a bit of a difficult time with this question, if some one could help me out it would be appreciated, thanks. Consider the following argument. "If the weather is fine, and the train is early, then the dog will sit on the tuckerbox. The train will be...
  2. Atlas3

    Is Zero Discrete? - Math Explained

    Is zero discrete?
  3. S

    MHB Discrete Math: Linear Inhomogeneous Recurrence

    How do I solve this 1. (a) Solve the recurrence relation an =6an−2 +8an−3 +3an−4 +64·3^n−4, n􏰀>=4 where a0 =0,a1 =1,a2 =4 and a3 =33. (b) Write down a closed form of the generating function of the sequence an.
  4. K

    Question regarding counting in discrete mathematics

    Homework Statement Let A = {1, 2, 3, 4} and let F be the set of all functions from A to A. Let R be the relation on F defined by: For all functions f, g that are elements of F, (f, g) are only elements of R if and only if f(i) = g(i) for some i that is an element of A. Let the functions α, β...
  5. evinda

    MHB Computation of discrete logarithm

    Hello! (Wave) The prime number $p=67$ is given. Show that $g=2$ is a generator of the group $\mathbb{Z}_p^{\star}$. Compute the discrete logarithm of $y=3$ as for the base $g$ with Shanks-algorithm. Compute the same discrete logarithm using the Pohlig–Hellman algorithm. That's what I...
  6. Togomori

    Discrete class D amplifier outputs low volume

    Hello everyone, I have a discrete class D project that I need help with but since I'm not an analog guy I got a problem. So far I got everything set up and the discrete class D works perfectly. But only at maximum of 6 volts otherwise the Transistor M3 gets really hot and burned, how do I solve...
  7. P

    Change of variables and discrete derivatives

    Hey I am trying to evaluate d/dx, d/dy and d/dz of a wavefunction defined on a grid. I have the wavefunction defined on equally spaced points along three axes a=x+y-z, b=x-y+z and c=-x+y+z. I can therefore construct the derivative matrices d/da, d/db and d/dc using finite differences but I...
  8. I

    How to represent this discrete function?

    Homework Statement my function: sin\frac{N\pi}{2} , N = 1, 2, 3, ... N = 1 ---> f(N) = 1 N = 2 ---> f(N) = 0 N = 3 ---> f(N) = -1 Homework EquationsThe Attempt at a Solution -\frac{1}{2}(-1)^N obviously it's wrong but just an 'attempt' my book refers to the...
  9. M

    Implementing Trapezoidal Motion Profile Using Discrete Method

    Hi, I'm trying to program an arduino to generate a Trapezoidal Motion Profile to control a DC motor with a quadrature encoder. Essentially, the user will input the desired Target Position, Max Velocity and Acceleration (decel = -accel) and the code will calculate the target position versus...
  10. D

    Show That If Alt Sum of Digits Div By 11, n Is Divisible By 11

    Homework Statement Given a positive integer n written in decimal form, the alternating sum of the digits of n is obtained by starting with the right-most digit, subtracting the digit immediately to its left, adding the next digit to the left, subtracting the next digit and so forth. For...
  11. M

    How Can You Simplify the Set Expression (A ∪ B ∪ C) ∩ ((A ∩ B) ∪ C)?

    Homework Statement The question is, simplify this equation: (A ∪ B ∪ C) ∩ ((A ∩ B) ∪ C) The correct answer is (A ∩ B) ∪ C Homework Equations We have been given the commulative, associative, distributive, identity, complement and idempotent laws and DeMorgan's laws, and I researched the...
  12. T

    Improving Frequency Resolution with Window Functions in FFT Calculations

    Okay I have a question involving calculating the FFT of a signal from a sensor. I have simulated many different scenarios in MATLAB of various noise characteristics involving the signal. I want to take the FFT of a noisy signal. As long as my expected input signal has a higher amplitude than...
  13. S

    MHB Easy question regarding symbols in discrete mathematics

    is the set of symbols that make up strings denoted by the symbol Σ or Σ* , also what is this difference?
  14. S

    A discrete mathematics question about logic?

    Hey guys here is the question i'd appreciate if you could help me with it: it says that no one dies in planet X,some spies of planet Y were captured by planet X's police.They are so professional that won't say anything to police so their inspection will go on forever and their inspection has no...
  15. M

    Discrete Math Proof: Proving Equivalence of 4 Statements

    Homework Statement Prove that the following four statements are equivalent: (a) n2 is odd. (b) 1 − n is even. (c) n2 is odd. (d) n2 + 1 is even. Homework Equations None really, just the use of different proofs ( indirect, etc...) The Attempt at a Solution I'm having trouble with this one...
  16. B

    Discrete Seeking Recommendation on Discrete Mathematics textbook

    Dear Physics Forum mentors, I am an undergraduate sophomore with double majors in mathematics and microbiology. I wrote this email to seek your recommendation on the discrete mathematics textbook that is in-depth, theoretical, proof-based, and also comprehensive. I am currently taking a...
  17. electronic engineer

    Calculating FFT for discrete values

    Homework Statement We have a set of values: f(n)=f(0,1,2)=(1,3,2) so f(0)=1, f(1)=3, f(2)=2, where n=0,1,2the number of values N=3 The question is to calculate the FFT of this signal, the Fourier spectrum the power spectrum and phase spectrum. I'm not sure concerning FFT. And also about...
  18. Z

    Is decay spectrum continuous or discrete?

    For a definite particle,the decay mode is determinant,finite kind,which embody the characteristic of quantum mechanics. But for a specific mode of a definite particle's decay,the decay spectrum,ie,energy of products,continuous,or discrete? Decay is a process which has unique initial...
  19. M

    Basic Discrete Math Question: Understanding Conditional Statements

    Before I make a fool of myself let me just say I just had my first class today and the book/ teacher aren't helpful in my question. And I'm not even sure I'm in the right section, this is just my major 1. Homework Statement "If 1+1=3 then 2+2=4" Homework Equations We just covered conditional...
  20. nomadreid

    Diagonalizabilty versus spread for uncertainty (discrete)

    I have seen two characterizations of the problem in measuring a discrete variable of a state ψ exactly with each of two non-commuting Hermitian operators A and B: (1) that the product of the standard deviations ( = √(<ψ|A2|ψ>-<ψ|A|ψ>2), & ditto for B) ≥ 1 (2) that one cannot simultaneously...
  21. A

    Discrete Probability: Mean Number of Red Bulb Non-Failure Cycles

    Homework Statement A traffic signal operates in the following cyclic regime: amber (A) light for 5 seconds, then red (R) for 30 seconds, then amber again for 5 seconds, then green (G) for 40 seconds (thus making a cycle ARAG), and then in the cyclic manner, i.e. ARAGARAG... . Let us assume...
  22. Ahmad Kishki

    Discrete Fourier Series question

    Why is the summation for the discrete Fourier series from 0 to N-1 (where N is the fundamental period of the signal) wheras it goes from minus infiniti to infiniti for continuous Fourier series...Thank you
  23. Ahmad Kishki

    Are all discrete sinusoids distinct?

    For the discrete complex sinusoid with period N, how many distnict sinusoids are there? And why?
  24. eseefreak

    Discrete math study strategy - Tips and advice

    Hi everyone, I haven't been successful in Discrete Math this semester. I have finished all of the calculus I-III series and I did very well. I want to know if anyone can give me some tips on how to study for my final coming up in a few days. Now, I understand that is a vague question but I am...
  25. N

    Understanding the Function of Set S in Discrete Mathematics

    Hey guys, I was reading Kenneth's Discrete Mathematics and I came across this definition in the function chapter: Let f be a function from A to B and let S be a subset of A.The image of S under the function f is the subset of B that consists of the images of the elements of S.We denote...
  26. H

    Proving the Theorem: p!/[(p-i)! * i] = 1/p for Prime Number p and Integer i

    Prove the following theorem: Theorem For a prime number p and integer i, if 0 < i < p then p!/[(p− i)! * i] * 1/p Not sure how to go about this. I wanted to do a direct proof and this is what I've got so far. let i = p-n then p!/[(p-n)!*(p-n)] but that doesn't exactly prove much.
  27. L

    Uniform Discrete Sample Distribution

    Homework Statement 2. Homework Equations [/B] So the sample mean is 2. the sample variance would be [[(3-1+1)-1]/12]/36 = 8/432. The Attempt at a Solution Is it, P[ (2.1-2)/sqrt(8/432) < z < (2.5-2)/sqrt(8/432)] = 0.232574. The book answer is 0.2312. I just want to be sure.
  28. B

    Solving Discrete Math Question: Proving ∪n=2∞[0,1 - 1/n] = [0,1)

    Homework Statement Show that, ∪n=2∞[0,1 - 1/n] = [0,1) Homework EquationsThe Attempt at a Solution
  29. P

    Proving A - (B ∩ C) = (A - B) ∪ (A - C) in Discrete Math

    Show that A - (B intersection C) = (A - B) union (A - C) I went about this completely around on a test but here is what I have Right Hand Side = ( A - B) union ( A - C) = (A intersection B) union (A intersection C) = A - (B intersection C) ? Easy problem but confused..thanks
  30. K

    Fourier transformation on discrete function

    Hi there, I am reading a material on the application of Fourier transformation in physics. One application is to transform the position-dependent function to k-dependent function, i.e.## F(k) = FFT[f(x)]## We know that the in physics, the wavenumber could be written in momentum as...
  31. T

    MHB Learning calculus through discrete mathematics

    Hello, Are there any free online textbooks or resources that teach calculus through discrete mathematics or does that not exist?
  32. M

    Discrete Fourier Transforms of Signals

    Homework Statement Homework EquationsThe Attempt at a Solution I'd like to see if I have the right line of thinking in my solutions: a. The sampling frequency should be such that no aliasing or folding occurs, so it should be twice the frequency of the original signal. $$x(t) = -17...
  33. C

    Calculating Electric Field Components for Discrete Charge Distribution

    Homework Statement Two test charges are located in the x–y plane. If q1 = -3.50 nC and is located at x = 0.00 m, y = 0.680 m and the second test charge has magnitude of q2 = 3.60 nC and is located at x = 1.00 m, y = 0.650 m, calculate the x and y components, Ex and Ey, of the electric field, ...
  34. S

    MHB Understanding the Markov Property: Discrete and General Cases

    Hi, I have some troubles understanding the definition of the Markov property in the general case since I'm struggling with conditional expectations. Let $(X(t), t \in T)$ be a stochastic process on a filtered probability space $(\Omega, \mathcal{F}, \mathcal{P})$ with adapted filtration...
  35. S

    Linear Algebra and Discrete Math at the same time?

    I am a math major currently in a community college reputed for having an outstanding math department, lucky me :D. I am taking Calculus 2 this semester. Next semester I'll be taking Calculus 3 with linear algebra or discrete math. Can I take all three at the same time or would it be an overkill...
  36. B

    Finding the PMF of a function of a discrete random variable

    The discrete random variable K has the following PMF: p(k) = { 1/6 if k=0 2/6 if k=1 3/6 if k=2 0 otherwise } Let Y = 1/(1+K), find the PMF of Y My attempt: So, I am really confused about what this is asking. I took...
  37. A

    Discrete Spectrum Non-Degeneracy in 1D: How to Prove?

    Homework Statement Prove that in the 1D case all states corresponding to the discrete spectrum are non-degenerate. Homework Equations \hat{H}\psi_n=E_n\psi_n The Attempt at a Solution Okay so, what I am stuck on here is that the question is quite broad. I can think of specific...
  38. A

    Sum of discrete uniform random variables

    Homework Statement Let ##X_k## be iid uniform discrete on ##\{0,...,9\}##. Find the distribution of ##\sum\limits_{k=1}^{\infty} \frac{X_k}{10^k}##Homework Equations The Attempt at a Solution I've tried a lot of things, I've tried decomposing ##X_k## into 10 bernoulli trials, I've tried using...
  39. H

    Discrete logistic equation, restricted growth model

    Homework Statement I need to write \begin{align*} N_{k+1} = \frac{\lambda N_{k} }{1+aN_{k} } \end{align*} in the form \begin{align*} N_{k+1} = N_{k} + R(N_{k})N_{k} \end{align*} Homework Equations As above The Attempt at a Solution I know that \begin{align*} N_{k+1} = N_{k} +...
  40. N

    MHB Compute Discrete Time Fourier Transform

    Hi bros, so I feel like I am very close, but cannot find out how to go further. Q.1 Compute the DTFT of the following signals, either directly or using its properties (below a is a fixed constant |a| < 1): for $x_n = a^n \cos(\lambda_0 n)u_n$ where $\lambda_0 \in (0, \pi)$ and $u_n$ is the...
  41. D

    Uniform discrete probablity distributions

    Homework Statement An electric circuit has 5 components. It is known that one of the componenets is faulty. To detremine which one is faulty, all 5 componenets are tested one by one until the faulty component is found, The random variable X represents the number of test required to determine...
  42. N

    MHB Convolution of two discrete sequence

    Hi, New to this topic, and need some help. My task is to find the convolution between $ y= x ∗ h$ where $x = u_n - u_{n-N}$ and $h_n = u_n - u_{n-M}$ and $M\ge N$ are positive integers My understanding is that in general, $ y= x ∗ h = \sum\limits_{m=-\infty}^\infty x_m h_{n-m} $ so for my...
  43. S

    Discrete probability distributions

    Homework Statement Here's the question: Given X is a discrete random variable. E(X-2) = 1/3 , VAR(X-2) = 20/9 . Detrmine value of E(X^2) the ans is 23/3 . but i ended up getting 3 . why i am wrong? Homework Equations The Attempt at a Solution
  44. W

    Sampling a signal and do the discrete Fourier transform

    When I sample a certain digital signal with increasing sampling frequency, the fast Fourier transform of the sampled signal becomes finer and finer. (the image follows) Previously I thought higher sampling frequency makes the sampled signal more similar to the original one, so the Fourier...
  45. M

    Lead-lag test for discrete variable vs continuous variable

    Let's say I'm applying electrical currents to a certain part of a human test subject and measuring certain deflections in his heart readings during this period. Before I increase the electrical currents, which could be dangerous, I'm interested to see if the changes in electrical currents are...
  46. K

    Discrete fourier transform data of 2 different sampling frequencies

    Hi All, I have a problem I've been thinking about for a while, but I haven't come up with a really satisfactory solution: I want to do a discrete Fourier transform on data that has been sampled at 2 different sampling frequencies. I've attached a picture of what my data will look like...
  47. J

    Vectors and discrete time signals

    How can a discrete time signal can be a vector? i cannot grasp the idea. i know MATLAB uses matrices which denote vectors, but how does a discrete time varying amplitudes be a vector?
  48. K

    Does discrete logic equation ignore death rate during off season?

    Observation of the growth of some beetle population suggests that discrete logistic equation would be more appropriate since these animals have distinct breeding season. discrete logistic equation: Xn+1=Xn+rXn(1-(X2/K)) the result of this equation accounts for population only season to...
  49. T

    Showing convexity of a discrete function

    Suppose we have a function ##f:\mathbb{N}\times\mathbb{N}\to\mathbb{R}_+## that isincreasing: ##f(x+e_i)\geq f(x)## for any ##x\in\mathbb{N}^2## and ##i\in\{1,2\}##;convex: ##f(x+2e_i)-f(x+e_i)\geq f(x+e_i)-f(x)## for any ##x\in\mathbb{N}^2## and ##i\in\{1,2\}##.How could one show that a...
  50. N

    Discrete Control System with Time Delay

    Hello, I am trying to develop a software control algorithm to compensate for oscillator imperfections/frequency drift. I have a NTP server which I can get a pretty good estimation (@1Hz) of the "true" time and compare it to my system's time. I can differentiate the offset-error to calculate...
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