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. Lolligirl

    Discrete Math: Proving Injectivity/Surjectivity of g°f

    1. Show by example that it is possible for g°f(x) to be surjective while f(x) is not I am confused by the general pattern of injectivity (one-to-one) and surjectivity (onto). I know the following by looking through my book: If f and g are surjective, then g°f is surjective. If f is...
  2. T

    Discrete Relations: can't understand relation definition

    Homework Statement Let Z be the set of all integers. Then, S is a relation on the set Z x Z defined by: for (a1, a2), (b1, b2) belong to Z x Z, (a1, a2)S(b1, b2) <-> a1b2 = a2b1. Homework Equations The Attempt at a Solution The actual problem is about symmetry...
  3. S

    Analog and Digital; Continuous and Discrete

    Is it possible to have the following signal: 1) Discrete and analog 2) Discrete and digital (I believe this one is true and very common in man made product) 3) Continuous and analog (I believe this one is also true and very common in nature) 4) Continuous and digital If a picture...
  4. P

    What are the discrete subrings of the real set?

    Homework Statement Problem from Artin's Algebra, find all discrete subrings of the real set. The Attempt at a Solution Clearly, Zn = {...,-2n,-n,0,n,...} is a portion. But having trouble proving that this forms *all* of the discrete subgroups.
  5. H

    Numerical integration of discrete data

    Hi, I'm searching for days for a numerical integration methode for discrete data given at non-equidistant nodes. The simple Simpson rule can only be used for equidistant nodes so I'm looking for methode which I can solve my problem. Any suggestion is welcome! Thanks in advance!
  6. J

    Discrete random variable cumulative distrub. function

    Homework Statement http://s359.photobucket.com/albums/oo40/jsmith613/?action=view&current=MathQUUU.pngHomework Equations The Attempt at a Solution So I know that k = 1 But if F(>3) = 1 then why does F(3) also equal 1 Thanks
  7. Y

    Prove Pascal's Triangle-type Function - Discrete Mathematics

    Homework Statement For all n ∈ Z+, the function Pn of i variables is defined recursively as follows: Pn(x1,...,xn) = Pn-1(x1 + x2, x2 + x3,...,xn-1 + xn) and P1(x1) = x1. Find a closed formula for Pn. Homework Equations Pn(x1,...,xn) = Pn-1(x1 + x2, x2 + x3,...,xn-1 + xn) and P1(x1)...
  8. P

    Is Light Truly Discrete or Merely a Manifestation of Matter's Nature?

    We never actually directly detect light, only its interaction with matter. Why do we have to consider this "wave particle duality" with light? Isn't it enough to say that light IS continuous, and it is the discrete nature of mater that gives rise to things like the photoelectric effect? Is...
  9. R

    AI search strategies using discrete mathematics

    Homework Statement The question is to provide a new composite heuristic h3 that is admissible and dominates h1 and h2. Then show the cost of each of the states through H according to h3. Homework Equations States: A B C D E F G H h1: 10,12,12,6,7,7,2,10 h2: 8, 9, 14,4,9,7,1,9...
  10. L

    Designing Comparator with Discrete BJTs

    Just wondering how would one design a comparator using just discrete BJTs.
  11. C

    Hydrogen atom with discrete nonlinear Schrödinger equation

    Hi everyone, How can I solve hydrogen atom with discrete nonlinear schrödinger equation? Could you help me with the mathematics of that, please?
  12. F

    Two independent Poisson processes (one discrete, one continuous)

    Hi Guys, I've used this forum as a great resource for a while now and it's always helped me out. Now I'm really stuck on something and was hoping you guys could help out. It's a pretty long question, but if you guys can just give me a general direction of what to do, I can go ahead and work it...
  13. M

    What is mathematical analysis and/or discrete mathematics used for?

    I am starting a maths major and I will going to go into pure maths. I am going to specialize in either analysis or discrete maths. I understand that mathematical analysis has a very strong connection to calculus and that discrete mathematics is used mainly in the cryptography and security...
  14. S

    Using a discrete Monte-Carlo technique in a multi-variable model

    If I have a large amount of data I can sample, with a several discrete variables, and I need to get an average of some function of that data, but it's too computationally intensive to do exhaustively... I want to do some sampling of the possible outcomes. I guess random sampling (Monte-Carlo...
  15. V

    Discrete Math: Is "Next Year Interest Rates Will Rise" a Statement?

    Homework Statement Is the following a statement: "Next year interest rates will rise" Homework Equations Sort of obvious, but a statement is defined as something which is true or false. The Attempt at a Solution I'm guessing that it is a statement, even if it isn't known whether it...
  16. E

    Troubleshooting DFT of Discrete Signal in C

    Hello to everyone :) Since I did not notice a presentation section I put some info about myself before the question,. My name is Enrico, I am Italian (form Modena) and I am 18. I am studying Electronics&Telecommunications in High School and I have applied for UCSD/UCLA/Berkeley (I hope they...
  17. radou

    Discrete T1 space vs. locally finite basis

    Homework Statement The formulation of the problem confused me a little, so just to check. No T1 space has a locally finite space unless it is discrete. The Attempt at a Solution This means that, if X is a discrete T1 space, it has a locally finite basis, right? Btw, for the...
  18. O

    Discrete LTI filter impulse response

    If I have the unit impulse response function for a discrete-time LTI system (Unit sequence response?), h[n], how can I calculate the time taken for the output to fall below 1% of its initial value, after a unit impulse is applied to the input? In particular, I have: h[n]=(\alpha...
  19. inflector

    Discrete Surface and Volume Integrals

    In another forum, I have been challenged to prove mathematically that a certain idea which consists of fields of discrete elements will satisfy http://en.wikipedia.org/wiki/Divergence_theorem" . The fields are not expressible in terms of a differentiable function but rather consist of discrete...
  20. P

    Discrete Math: Binary Relations

    Homework Statement A = {0, 1, 2, 3, 4 ,5} Let R be a binary relation on set A such that: R = {(0,1), (1,0), (1,3), (2,2), 2,1), 2,5), (4,4)} a. Make a Directed Graph for the relation R on A b. What must be added to R to make it reflexive/symmetric?
  21. J

    Stability conditions of discrete system

    Homework Statement Consider a discrete-time system, driven by: x[k+1] = Ax[k] for non-zero inital conditions x[0] a) write the closed-form solution of x[k]. If the system is asymptotically stable how should x(k) behave? b) what is condition for asymptotic stability? Homework Equations...
  22. R

    Matlab graphing using discrete fourier transform

    Homework Statement Compute the discrete Fourier transform of the ecg signal, graph the amplitude and phase response. The problem gives data in the form of a ecg.mat file. Contained are two double variables: voltage data for the ecg and a time vector for the ecg. both the voltage and time...
  23. C

    Discrete Probability - error with question?

    Homework Statement On a long train journey, a statistician is invited by a gambler to play a dice game. The game uses 2 unbiased ordinary dice which the statistician is to throw. If the total score is 12, the statistician is paid 3 dollars by the gambler. However,if both or either dice show a...
  24. Z

    Energy: Discrete or Continuous

    First let me pose the assumptions that I am making (because this is not something I am an expert in): 1.) Energy and Mass are equivalent 2.) Quantum mechanics discretizes just about everything, or that a discrete element can be found for everything. 3.) Mass is discrete via the Higgs Boson...
  25. I

    How can you prove this discrete math induction statement?

    Homework Statement Homework Equations base case: n=1 The Attempt at a Solution im not sure where to start because the examples that my professor showed us did not have a n(n-1) (n+1) but rather (p+1)P=1+1)(2(p+1)+1) im just very lost in this example
  26. S

    About Discrete Probability~ Help

    Zaki and Ramli play a game in which they take it in turns to toss a tetrahedral dice. They agree that the first man to toss a "2" wins the game. Ramli toesses the dice first. (a) Find the probability that Zaki loses on his first toss. (Ans : 3/16) (b) If x is the probability that Ramli...
  27. H

    Torsion-free modules over a Discrete Valuation Ring

    Let R be a discrete valuation ring with fraction field F. I believe it's straightforward to show that any torsion-free module M with the property that M \otimes_R F is a finite dimensional F-vector space is of the form R^m \oplus F^n. What if M \otimes_R F is infinite dimensional?
  28. P

    Normal Distribution - Discrete or Continuous?

    Suppose that the height of adult females in a population is a normal random variable with a mean of 165 cm and a standard deviation of 12 cm. If heights are measured to the nearest centimetre, what percentage of the adult female population will have a measured height between 150 and 160 cm...
  29. A

    2.2 Set Operations: Discrete Mathematics and its application

    Ex 36, p 147. Let f be a function from the set A to the Set B. Let S and T be the subset of A. Show that b) f(S \cap T) \subseteq f(S) \cap f(T). Thanks.
  30. marcus

    Both continuous and discrete in the same space at the same time (Kempf)

    Kempf gave a talk on this. I'll find the PIRSA link. I remember watching the whole video and being impressed. It may be easier to understand than the paper because communicating a higher proportion of the person-to-person intuition---more beginner level. You can try it either way. Either watch...
  31. C

    Continuous and Discrete Fourier Transform at the Nyquist frequency

    Hi there, A quick question concerning the FFT. Let's say I explicitly know a 2D function \tilde{f}\left(\xi_1,\xi_2 \right) in the frequency domain. If I want to know the values of f\left(x_1,x_2 \right) in the time domain at some specific times, I can calculate \tilde{f} at N_jdiscrete...
  32. E

    How Do Transition Probabilities Determine the Behavior of a Markov Chain?

    Homework Statement Let \left( X_n \right)_{n \geq 0} be a Markov chain on {0,1,...} with transition probabilities given by: p_{01} = 1, p_{i,i+1} + p_{i,i-1} = 1, p_{i,i+1} = \left(\frac{i+1}{i} \right)^2 p_{i,i-1} Show that if X_0 = 0 then the probability that X_n \geq 1 for all n \geq 1 is...
  33. C

    How Can You Prove the Triangle Inequality Using Case Analysis in Discrete Math?

    Discrete Math -- Proof methods Homework Statement Prove |x-y| ≤ |x| + |y| for all real numbers x and y (where |x| represents the absolute value of x, which equals x if x≥0 and equals -x if x<0). prove by cases Homework Equations The Attempt at a Solution
  34. Z

    Proving Existence of a Survivor in a Discrete Math Problem | Odd n Case

    Homework Statement Suppose n > 1 people are positioned in a feld, so that each has a unique nearest neighbour. Suppose further that each person has a ball that is thrown at the nearest neighbour. A survivor is a person that is not hit by a ball. Prove that if n is odd, then there is at least...
  35. S

    Convolution of discrete and continuous time signals

    Not a specific question per se but... Is it possible to convolve a discrete-time signal with a continuous-time one? if you have x(n) and y(t) can you calculate the convolution of x and y (say, by taking y(t) for t in the set of integers or by treating each x(n) as its value multiplied by...
  36. S

    Help with discrete random variables

    Homework Statement 1. Suppose u flip a coin Z = 1 if the coin is heads Z = 3 if the coin is tails W = Z^2 + Z a) what is the probability function of Z? b) what is the probability function of W? 2. Let Z ~ Geometric (theta). Compute P(5<=Z<=9). Homework Equations The Attempt at a Solution...
  37. I

    Convergence of Random Variables on Discrete Prob Spaces

    Well, I thought I understood the difference between (weak) convergence in probability, and almost sure convergence. My prof stated that when dealing with discrete probability spaces, both forms of convergence are the same. That is, not only does A.S. convergence imply weak convergence, as...
  38. C

    Iterative expectation of continuous and discrete distributions

    Homework Statement Suppose X ~ uniform (0,1) and the conditional distribution of Y given X = x is binomial (n, p=x), i.e. P(Y=y|X=x) = nCy x^{y} (1-x)^{n-y} for y = 0, 1,..., n. Homework Equations FInd E(y) and the distribution of Y.The Attempt at a Solution f(x) = \frac{1}{b-a} = \frac{1}{1-0}...
  39. H

    Expectation value of momentum in discrete states

    Is there any way of proving <p> = 0 for a discrete (bound) state given it's wave function? I've seen proofs using the hermitian properties of p but I'm interested in proving that the integral of Psi*(x) Psi'(x) is identically zero regardless of Psi(x) as long as it's a solution of Schroedinger's...
  40. R

    Do Sets in Discrete Topological Spaces Have Boundaries?

    Do sets in a discrete topological space have boundaries?
  41. D

    Convert a discrete time transfer function to a continous time transfer funtion

    Homework Statement I would like to know on how to convert a digital filter transfer function which is in defined in discrete time to continuous time transfer function. Homework Equations The relevant equation is attached...
  42. T

    Two Discrete Mathematic Proofs I Need Help With

    Homework Statement Prove that at least one of 2*10500 + 15 or 2*10500 + 16 is not a perfect square. Can you say specifically which one is not a perfect square? Consider the proof that √2 is irrational. Could you repeat the same proof for √3? What about √4? Homework Equations n/a...
  43. T

    Discrete Mathematics Proof Problem

    Homework Statement Which is larger, square root of 2 or cubed root of 3? Prove one is larger than the other without using decimal approximations for either number. The Attempt at a Solution I attempted to solve this through the contradiction that they were even. If they are not even then...
  44. B

    1st derivative of a discrete function?

    I have solved this problem but still have a question about it (problem and my solution posted below). What I wanted to do was express the problem as a function and use an optimization technique that would require taking the first derivative of the equation. The problem I ran into is: if using...
  45. F

    Is Every Discrete Isotropy Group of an R^n Action Isomorphic to Z/kZ?

    Quick question: Suppose I have a (transitive) R^n action on a manifold M. If the isotropy group of R^n is discrete, does that mean that it is automatically isomorphic to Z/kZ, with 0<=k<=n? Basically, my discrete subgroup is a lattice then, right? Thanks!
  46. inflector

    Starting from Discrete or Continuous?

    I've been looking over quantum gravity threads here for a year or so. One thing keeps puzzling me. It appears to me that difficulty coming up with a viable quantum gravity theory is melding the continuous nature of the general relativity equations with the discrete (i.e. quantized) nature of...
  47. C

    Discrete Math: Subsets and Venn Diagrams Explanation

    Homework Statement Let their be a set A, and let B be the set: {A, {A}} (the set containing the elements A and the set that contains element A) As you know, A is an element of B and {A} is also an element of B. Also, {A} is a subset of B and {{A}} is also a subset of B. However...
  48. F

    MATLAB Discrete Math vs MATLab for Physics: Which is Better?

    Which one is a better course to take? I feel like MATLab is simple enough I can learn on my own if I ever need it. How useful is discrete math in physics? Should I take that now or stick to MATLab?
  49. L

    Charateristic equation of a discrete system

    Calculate value(s) of the root(s) of the charateristic equation of a discrete system that correspond(s) to a time constant of 0.03 seconds when the sample period is 0.02 seconds. I'm not sure if I should use a first order or 2nd order equation. Either/or, why is it? Any ideas on how I should...
  50. T

    How to Calculate Expectation and Variance for a Discrete Random Variable?

    Homework Statement A random variable X takes values 1,2,...,n with equal probabilities. Determine the expectation, R for X and show that the variance, Q^2 is given by 12Q^2=n^2-1. Hence, find P(|X-R|>Q) in the case n=100 Homework Equations The Attempt at a Solution I can show...
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