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. S

    Discrete Math Help: Proving Injectivity of f & g

    Homework Statement f: B => C and g: A => B 1. If f of g is injective, then f is injective. 2. If f of g is injective, then g is injective.Homework Equations The Attempt at a SolutionI know that 1 is true and 2 is false because I found those as properties, but I am not exactly sure why, and...
  2. V

    DISCRETE MATH: Use rules of inference to show that

    Homework Statement Use rules of inference to show that if \forall\,x\,(P(x)\,\vee\,Q(x)) and \forall\,x\,((\neg\,P(x)\,\wedge\,Q(x))\,\longrightarrow\,R(x)) are true, then \forall\,x\,(\neg\,R(x)\,\longrightarrow\,P(x)) is true. Homework Equations Universal instantiation, Disjunctive...
  3. V

    DISCRETE MATH: Determine whether an argument is correct or not

    Homework Statement Determine whether the argument is correct or incorrect and explain why. A) Everyone enrolled in the university has lived in a dormitory. Mis has never lived in a dormitory. Therefore, Mia is not enrolled in the university. B) A convertible car is fun to drive. Isaac's car...
  4. V

    DISCRETE MATH: Determine if two statements are logically equivalent

    Homework Statement Determine whether \forall\,x\,(P(x)\,\longleftrightarrow\,Q(x)) and \forall\,x\,P(x)\,\longleftrightarrow\,\forall\,x\,Q(x) are logically equivalent. Justify your answer. Homework Equations P\,\longleftrightarrow\,Q is only TRUE when both P and Q are TRUE or...
  5. V

    DISCRETE MATH: Are these system specifications consistent?

    Homework Statement Are these system specifications consistent? "(A)Whenever the system software is being upgraded, users cannot access the file system. (B)If users can access the file system, then they can save new files. (C)If users cannot save new files, then the system software is not...
  6. C

    Discrete math-counting permutation

    Can someone help me understand this one? The problem is: Four beads-red,blue,yellow, and green-are arranged on a string to make a simple necklace as shown in the figure. How many arrangements are possible? The answer in the book is 3, but I don't get it. I thought it woud be a permutation...
  7. D

    Question about momentum transfer from fluids to discrete particles

    Hi, I am having trouble find information on the following topic, I think mostly in part because I don't know the correct terminology. Basically, I have a number of particles that are falling at their terminal velocities within a gaseous fluid, and turbulent velocity fluctuations in the fluid...
  8. A

    Understanding C(n,1) in Discrete Math: Solving the Mystery of C(n,1) = 1

    C(n,1) ... I know that C(n,0) =1 But have no clue how to figure out C(n,1) :cry:
  9. D

    EM Spectrum Continuous or Discrete?

    I have a question about the EM Spectrum. Is it existing in strips or is it completely continuos? Can a photon have a frequency of any value or does it have to have specific wavelengths according to the space between 2 electron orbits? I understand photons are generated when an electron drops...
  10. S

    Discrete Weighted Transform - A Primitive Nth Root of Unity

    Hello, I got pointed to this forum by OfficeShreder. I have a question I've been puzzling myself over for a while. I am currently trying to implement the "Discrete Weighted Transform". I have reached a step where I need to determine "a primitive Nth root of unity in the appropriate domain"...
  11. N

    Suppose x is a discrete, binomial random variable

    How do I do this p(x<1) this sign has a _ under the < n=6 p=0.1 Suppose x is a discrete, binomial random variable. Calculate P(x > 2), given trails n = 8, success probability p = 0.3 [Hint: P(x > value) = 1 – P(x <= value) <= is a < with a _ under it (tell me the number...
  12. C

    Card Game for my Discrete Structures class

    So my teacher is making us write a paper on various games he proposed to us. This is the one I choose. You get 5 random cards from a standard deck. You place 4 of them in any order you like face up, and the last one face down. Knowing your strategy of placing the first four cards, someone...
  13. R

    HELP unit-pulse response for the discrete time system problem

    HELP!unit-pulse response for the discrete time system problem Please help me solve these problems. Thank you so much for all your help. Compute the unit-pulse response for the discrete time system 1) y[n + 2] + 1/2 y[n+1]+1/4y[n] = x[n+1]-x[n] (for n = 0, 1, 2) For number 1) the options...
  14. R

    HELP Unit-Impulse Response for Discrete System problem

    HELP!Unit-Impulse Response for Discrete System problem Please help me solve these problems. Thank you so much for all your help. Compute the unit-pulse response for the discrete time system 1) y[n + 2] + 1/2 y[n+1]+1/4y[n] = x[n+1]-x[n] (for n = 0, 1, 2) For number 1) the options for...
  15. B

    Can Set Theory Prove Equality and Intersection Properties?

    Hi, I would like some help for the following problems. please bear with me with my special notation: I- intersection U- union S- universal set ~- complement I need to prove that: let be A and B two sets. prove (A U B) I (A I (~B))=A what I did is: (A U B) I (A I (~B)) =[(A I B)...
  16. B

    Proving the Uniqueness of Solutions for Linear Equations with Real Coefficients

    Hi, Please can someone help me with this problem. show that a,b,c are real numbers and a#0, then there is a unique solution of the equation ax+b=c. the uniqueness of the solution is my problem. Thank you B
  17. I

    Conditional expectation (discrete + continuous)

    I need help in solving the following problem: Let X be uniformly distributed over [0,1]. And for some c in (0,1), define Y = 1 if X>= c and Y = 0 if X < c. Find E[X|Y]. My main problem is that I am having difficulty solving for f(X|Y) since X is continuous (uniform continuous over [0,1])...
  18. L

    How Does Discretizing a Smooth Manifold Affect Its Geometric Properties?

    "Discrete" Geommetry... -What can you do if you have a "Smooth" Manifold..but it's very hard to work with?..perhaps you could "discretize" :rolleyes: :rolleyes: the surface splitting it into "triangles" and take the basic coordinates to be the angles (3) of every triangle..but my question is...
  19. E

    Solving Discrete Math Questions - Does Integer Set Include 0?

    I am in discrete math class right now and trying to get the sets of numbers straight. So, does the set of integers include 0? Is it ok to use 0 in proofs, that makes finding a counter-example a lot easier and disprove a statement about all integers. Was just wondering if that is legal...
  20. T

    Geometry and Discrete Mathematics notes? Resources?

    Geometry and Discrete Mathematics notes?? Resources? Hello everyone, I'm going to take Geometry and Discrete Mathematics, next year in high school (grade 12). So this summer I'm planning to read some books that would help me out next year, to bulit up my basic skills. So anyone know any sites...
  21. I

    What is the Distribution of Tossing a Die Until a 6 Appears?

    I got really really confused by this supposedly easy discrete probability problem: The problem asks: a)toss a die until a "6" appears. Find the probability distribution of X where X is the number of tosses neded to obtain the first six. b) Prove that the summation of P(x) from x = 1 to n...
  22. Y

    How can a DCT help identify the obvious frequency of a discrete wave?

    Hello, Suppose I have a discrete function of a perfect cosine wave. So if I will do a DCT on this function I will get one non zero coefficient which corresponds to the perfect cosine wave, and the rest will be zero. Now I have a pass filter, which filters out anything with a frequency which...
  23. Loren Booda

    How can continuous thought realize discrete processes?

    How can the mind objectify discontinuous phenomena if the mind itself is continuous?
  24. S

    What Is the Probability of Guessing Correctly on a Mixed Multiple Choice Test?

    A multiple choice test contains 12 questions, 8 of which have 4 answers each to choose from and 4 of which have 5 answers to choose from. If a student randomly guesses all of his answers, what is the probability that he will get exactly 2 of the 4 answer questions correct and at least 3 of the 5...
  25. Y

    Discrete Derivatives: Types & Uses

    What types of discrete derative are there? Thanks in advance.
  26. A

    Question on Discrete Parameter Markov Chains

    I am required to find a formula expressing the probability of return to some state in a Markov chain at time n in terms of the probability of return to that state at time n - k and the probability of first return at time k. I cannot find this in my notes, and I have tried looking at several...
  27. arivero

    Pauli to Sommerfeld on Discrete Derivative

    Resders of the forum could enjoy this pdf http://documents.cern.ch//archive/electronic/other/pauli_vol3//sommerfeld_0463-2.pdf from http://doc.cern.ch/cgi-bin/setlink?base=pauli&categ=&id=sommerfeld_0463-2 In page 3 of the letter (1 of the transcripcion) you can read Pauli sentence of the...
  28. N

    Finding the Right Book for Self-Studying Discrete Math

    I am planning on self studyint Discrete Math. What would be a good book for this?
  29. N

    Discrete Math Knowledge & Computer Programming

    How useful is the knowledge of discrete math for computer programmers?
  30. N

    Discrete Math and its Usefulness to Electrical Engineers, IC Designers

    How useful is discrete math to an electrical engineer; particularly how useful is it to an ic designer?
  31. P

    Discrete Mathematics - Problems with Languages

    Let \Sigma = { \beta,x,y,z} where \beta denotes a blank, so x\beta \neq x, \beta \beta \neq \beta, and x\betay \neq xy but x \lambday = xy. Compute each of the following: 1: \parallel \lambda \parallel 2: \parallel \lambda \lambda \parallel 3: \parallel \beta \parallel 4...
  32. happyg1

    Calculating the PMF of Min. Observation of Discrete Distribution Sample Size 5

    Hi, I know this isn't hard, but I have a block on it. Here's the question: Let f(x)=1/6, x=1,2,3,4,5,6 zero elsewhere be the pmf of a distribution of the discrete type. Show that the pmf of the smallest observation of a random sample of size 5 from this distribution is...
  33. electronic engineer

    Fourier transform for Discrete signal

    let us asuume this discrete signal: f(n)=a^n * u(n) ; where u(n) is unit step function ; u(n)=1 where n>=0 u(n)=0 where n<0 ;0=<a<1 and the foruier transform for discrete signals is defined as : F(i)=sum (...
  34. I

    Why the energy eigen values for negative energies are always discrete

    Why the energy eigen values for negetive energies are always discrete while that for positive energies are always continuous? Also what is oscillation theorem?
  35. H

    What is the Least Natural Number n for Which √(x² + x³ + 3) is O(xⁿ)?

    I'm having some trouble with this discrete question: Find the least natural number n such that √(x² + x³ + 3) is O(xⁿ). With the value of n that you have found, is it true that xⁿ is O( √(x² + x³ + 3) )? Can anyone help?
  36. H

    Geometry and Discrete, prove this?

    The question asks: In triangle ABC, with vertices A(0,a), B(0,0), and C(b,c), prove that the right bisecors of the sides meet at a common point. Ok, this question is really getting to me. I know that I could find the equations of all three right bisectors and then solve for x and y through...
  37. S

    Discrete Math Proving some power sets

    Ok so I need to prove (by contradiction) that... if the power set(A) is a subset of power set(B), then A is a subset of B. I was given a hint to use proof by contradiction, but in general I'm lost as to what to do... I know the power set of (A) is {B|B subset A} and the powerset of (B) is...
  38. P

    Why Are w[n] = exp(j*pi*n) and w[n] = (−1)^n Equivalent?

    w[n] = exp(j*pi*n) w[n] = (−1)^n Hi, can anyone off hand verify why these two exressions are equal? Thanks
  39. J

    Learn About Discrete Algorithms for Project

    I have a project to do on algorithms, and as far as I can tell, the project is due before we cover the topic :p. My textbook only makes a slight reference to what an algorithm is :(. Can anyone point me in the direction of a website or book that tells me all there is to know about algorithms?
  40. T

    Discrete modes of vibration for sonometers

    Hi, I have a few questions about Sonometers which I've attempted to think through - though I may be missing something. Perhaps someone can fill in any gaps for me? If I was to be asked why only 'discrete modes' of vibration can exist for a sonometer, is it enough merely to talk about the...
  41. A

    What Is the Smallest Value of k for Postage Using Only 4-Cent and 9-Cent Stamps?

    Hi, This is one of the question from my hw, i don't even understand what it's asking? Please shed some light on it.. thx what is the smallest value of k such that any integer postage greater than k cents can be formed by using only 4-cent and 9-cent stamps? Show that k cents in postage...
  42. Loren Booda

    Quantum mechanics holds only at discrete points in spacetime?

    A unified theory of physics has been evasive, I believe, because physicists have only considered a continuum of unification. In other words, we have attempted to relate all spacetime to all quantum dynamics - that they are inclusively connected. Has the argument arisen that quantum measurement...
  43. S

    Proving (A U B) x (A U B) = (A x A) U (B x B) with Discrete Math

    Hello, How do I proove : (A U B) x (A U B) = (A x A) U (B x B) if and only if : (A C B) or (B C A) ? Please Advice, Dimitry Haritonov
  44. S

    Proving Lattice Property in Dual Posets: A Discrete Math Question

    I have a question from hw, the question is stated "Show that if the poset (S,R) is a lattice then the dual poset (S,R^-1) is also a lattice" I know by Rosen theory that the dual of a Poset is also a poset but how can i prove that it is also a lattice, what def. am i missing. Any help would be...
  45. R

    Solve 2^27841 mod 34 by Hand: Discrete Math Problem Solution

    could someone show me how u would solve 2^27841 mod 34 by hand? I know what theorm to use, I am just having trouble using it? Thanks
  46. B

    How Do You Calculate the Correct Heading for an Airplane in Wind?

    The wind is blowing N 75 degrees East at 96 km/h. A plane is flying with air speed 320 km/h. Find the heading the plane must have in order to get from A to B, if B is S40 degrees W of A. Include a diagram in your solution. Alright, i am totally lost. Could someone please maybe put on some...
  47. J

    Discrete Mathematics - Logics Puzzles

    I ran into this logic puzzle and have been working on it for couple hours now but i can't seem to explain clearly why the answer i came up with is the answer. Heres the question: The police have three suspects for the murder of Mr. Cooper: Smith, Jones, and Williams. Smith, Jones, and...
  48. R

    Is f(x) = 2x Onto, One-to-One, or a Bijection for Different Domains?

    Hey if anyone could help me with this I would be sooo grateful. I am trying to grasp the idea of onto, one-to-one and bijection(both) functions. A sample problem is: If f(x) = 2x . What is f(Z), all integers. What is f(N), all naturals. What is f(R), all real. These are 3 different problems...
  49. C

    Why is ~(P v Q) equivalent to (~P & ~Q)?

    Hi, For one of the questions in my Discrete Mathematics course, I have to find what property of a formula makes its dual formula also its negative. With a dual formula, the logical operators of "^" and "v" are reversed, the former meaning "and" and the latter meaning "or". With its negative...
  50. Q

    How to Design a High-Gain Discrete Op-Amp?

    Hi, my name is Quan and I am new to this forum. I am trying to build a discrete operational amplifier but I am very weak with circuit designing since my concentration is in communication. Anybody here have a lot of experience building analog op-amp circuits? If you are in the area then that's...
Back
Top