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. E

    Inverse Discrete Fourier Transform proof help

    I'm reading the text Understanding Digital Signal Processing Second Edition and in the text they give the IDFT without any proof and so I tried to do a quick proof, but I have not been able do it here is my attempted steps: X(m)=\sum^{N-1}_{n=0}x(n)e^{-i2\pi mn/N} \\ Considering x(1)...
  2. F

    How Do You Calculate the Charges on Two Repelling Particles?

    Homework Statement Two point particles separated by 0.4 m carry a total charge of 185 µC. (a) If the two particles repel each other with a force of 80 N, what are the charges on each of the two particles? Homework Equations F = k* (q1*q2)/(r^2) The Attempt at a Solution I tried...
  3. I

    Discrete math textbook problem

    Homework Statement find the domain and image of f such that f(x) = {(x,y) \in R \times R \vert x = \sqrt{y+3} and domain and image of g such that g = { (\alpha,\beta) \vert \alpha is a person, \beta is a person, \alpha is the father of \beta Homework Equations the domain and image...
  4. T

    Discrete math - equivalence relation

    Let A be a set. For every set B and total function f:A->B we define a relation R on A by R={(x,y) belonging to A*A:f(x)=f(y)} *belonging to - because i don't know how to make the symbole... Prove that f is one-to-one if and only if the equivalence classes of R are all singletones
  5. hxtasy

    Sliding DFT discrete Fourier transform

    "Sliding DFT" discrete Fourier transform... I was wondering if any of you had had experience with the sliding DFT algorithm. It is somewhat similar to the Goertzel algorithm. I am having some trouble understanding the mathematics of the algorithm, and I also cannot seem to identify a useful...
  6. F

    Discrete mathematics (PMI, composition, onto)

    Homework Statement a.) F = {(1, a), (2, b), (3, a), (4, c)} G = {(b, 1), (a, 2), (c, 3)} i. Find F o G ii. Find G o F b.) A function F: N x N --> N is represented 2(m + n) + 1 for F(m, n) i. Is F one-to-one? ii. Is F onto? c.) Prove by Mathematical Induction...
  7. N

    Discrete time state space model: solving for input

    Preface: This homework assignment was due long ago. At this point I am only trying to understand the problem (or really if the posted solution follows the problem) before my exam. I have no real indication that this problem (or even one like it) will be on my final, but I feel that my inability...
  8. M

    Discrete topology, product topology

    For each n \in \omega, let X_n be the set \{0, 1\}, and let \tau_n be the discrete topology on X_n. For each of the following subsets of \prod_{n \in \omega} X_n, say whether it is open or closed (or neither or both) in the product topology. (a) \{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}...
  9. M

    What Is the Minimum Amount of Money Needed with Different Coin Combinations?

    Homework Statement use combinatorial methods to determine the smallest amount of money that using cents, nickels, dimes, or quarters, requires a) four coins b) five coins c) six coins d) seven coins e) eight coins Hint: Consider the ways to partition sets of those respective...
  10. Z

    What is the 15th Term of the Expansion (X^3 + Y)^25?

    Homework Statement What is the 15th term of (X3 + Y)25? Homework Equations The Attempt at a Solution
  11. Z

    What is the 15th term of (X3 + Y)25?

    Homework Statement What is the 15th term of (X3 + Y)25? Homework Equations The Attempt at a Solution
  12. G

    Discrete Dynamical Systems Proof Help.

    Homework Statement How many points in ΣN are fixed by σkN? Homework Equations σkN is the kth iteration of the shift map σN. The Attempt at a Solution I'm not sure where to start. I probably just need a hint.
  13. S

    Proving there is a fixed point in a discrete group of rotations

    Homework Statement Let G be a discrete group in which every element is orientation-preserving. Prove that the point group G' is a cyclic group of rotations and that there is a point p in the plane such that the set of group elements which fix p is isomorphic to G' The Attempt at a...
  14. S

    Is a Discrete Group of Rotations Cyclic?

    Homework Statement Prove that a discrete group G cosisting of rotations about the origin is cyclic and is generated by \rho_{\theta} where \theta is the smallest angle of rotation in G The Attempt at a Solution since G is by definition a discrete group we know that if \rho is a...
  15. T

    Discrete energy states (explanations in QFT?)

    I've been thinking about one of the postulates about one particle quantum mechanics, it says that whenever we measure an energy value, we get one of those eigenvalues. Firstly, pretty much 99% of the stuffs I know in nonrelativistic QM applies in the realm of electromagnetism. I just don't...
  16. F

    Discrete mathematics induction

    Homework Statement Prove that for all integers a >= 1, a^n - 1 is divisible by a - 1 for all n >= 1. Homework Equations None. The Attempt at a Solution Proof - Let P(n): a^n - 1 is divisible by a - 1, then P(1): a^1 - 1 is divisible by a - 1 is TRUE since a^1 - 1 = a - 1, and...
  17. L

    Discrete Voltage Regulator w/ BJTs

    Hi, My problem is very simple, I have a project in which I have to design a voltage regulator out of discrete BJTs, Zeners and resistors etc. the only limitation is that I cannot use an IC. I would prefer not using a zener because i would like to make the output variable through a voltage...
  18. G

    Help with a proof with discrete dynamical sysmtes / chaos theory.

    Homework Statement Consider the families of iterating functions Fλ(x) = λ(x3 - x). Fλ(x) undergoes a bifurcation at λ=1, about the fixed point x=0. Figure out what ilk of bifurcation is occurring for Fλ(x) and prove your assertion rigorously.Homework Equations My book says this about...
  19. G

    Discrete mathematics: incursion

    Homework Statement a 1= 2, a k+1, 2ak-1 Homework Equations What is the 5th term The Attempt at a Solution a1= 2 a2=2(2)-1= 3 a3=2(3)-1=5 a4=2(4)-1=7 a5=2(5)-1=9 5th term =9?
  20. M

    Proving NTG Relation on S x S: Reflexive, Non-Transitive, and Non-Antisymmetric

    confused:Given the simple LTE (less then equal) relation on S= {1,2,3,4} defined by [less and equal ], we define a complex NTG (not grater then) relation on S x S by (w,x) NTG (y,z) if w[less and equal) y or x [less and equal z. (this or confusing me ) Show that NTG is (R) reflexive, but not...
  21. M

    Can You Prove This Theorem Using Discrete Mathematics Tables?

    :confused: Construct a formal proof of the theorem: If (p-> q), (neg [r] -> s), and (neg [q] V neg [s], then (p-> r). [refer to table of logical equivalences (p62) and the table of logical implication (p62)]The tables are in the textbook Kenneth Ross, 5th edition, Discrete Math's...
  22. F

    Velocity operator inconsistency and discrete particle

    I'm going to mix a couple questions together instead of creating a new topic for each question. I hope you don't mind. I'm an electrical engineer(micro-electronics), so while I got the basics of QM in my studies I had to do most of my more 'in depth' learning on my own by reading books/ocwm...
  23. M

    Logics and Proof - Discrete Mathematics

    Prove or disapprove that the product of two rational numbers is irrational How do you solve this? Thanks
  24. M

    2.2 Set Operations: Discrete Mathematics and its application

    page.130 Ex.20 Ex.20 Show that if A and B are sets, then (A\capB) \bigcup (A\capB) = A. how do u solve this? The Attempt at a Solution
  25. W

    A Problem in Discrete Probability

    I recently came across a page named "Gems of Discrete Probability" - http://www.cse.iitd.ernet.in/~sbaswana/Puzzles/Probability/exercises.html Being a mathematics enthusiast, I tried the first question. Being very rusty in probability, I failed to come up with a satisfying answer: the best I...
  26. E

    Exploring Binomial Theorem through Discrete Math

    Ok, so after a little discussion with my discrete math teacher today, he sent me on a little "quest". Here is how it happened: The topic we were covering was set theory, and as I had been studying very basic combinatorics the night before, I noticed something about the powerset, namely...
  27. E

    Is This Proof that 1=2 Valid or Fallacious?

    Homework Statement Alright here it is: Theorem: if there exists an x belonging to reals such that (x^2)-x-2=(x^2)-4 then 1=2. Remark: note that there is such an x belonging to reals. Proof: 1) by hypothesis assume there exists an X belonging to reals such that (x^2)-x-2=(x^2)-4...
  28. L

    Discrete time signal to continuous time signal

    Homework Statement Suppose that a discrete-time signal x[n] is given by the formula x[n] = 10cos(0.2*PI*n - PI/7) and that it was obtained by sampling a continuous signal at a sampling rate of fs=1000 samples/second. Determine two different continuous-time signals x1(t) and x2(t)...
  29. H

    The discrete self-trapping equation

    Can anybody help me find a paper? The name is "The discrete self-trapping equation", or "J.C. Eilbeck, P.S. Lomdahl, A.C. Scott, Physica D 16 (1985) 318." Thank you very much !
  30. M

    Solve Discrete Math Problem: f(x,y)= 4x+y-4

    I know I have to write an equation to solve the problem down. But I really don't know how to use the given information. I did it by enumeration, but I don't get it how this will be shown by an algebriac argument. Please some one help me at least with an idea. If S = {1,2,3,4}, consider the...
  31. C

    Discrete Math - a modulus proof

    Homework Statement I have to prove the following claim. Claim: For any positive integers m and n, m and n both greater than 1, if n|m and a≡b(mod m), then a≡b(mod n).Homework Equations n/aThe Attempt at a Solution so i first changed each equation (ex: a≡b(mod m)) to a=b+qm and a=b+qn I...
  32. S

    Discrete Mathematics (confused and help wanted)

    Dear all, I have an example taken from the book titled "Discrete Mathematics For Computer Science" by Kenneth Bogart. In the book, page 11, example 1.2-2, it says: Write down all the functions from the two element set {1,2} to the two element set {a,b}. I couldn't understand the...
  33. F

    Simple Discrete Structures problem

    OK this is the first assignment I have in this class and I can't figure out how to negate and simplify the logical structure of W <--> S (bi-conditional implication) I got this so far: ~[(W --> S) ^ (S --> W)] by Definition ~(W --> S) v ~(S --> W) by DeMorgan's Law ~(~W v S) v ~(~S v W)...
  34. A

    Is Space Discrete: Exploring the Rationality of Pi

    I've been reading that it is, there is a smallest volume of space, if this is so then there is also a smallest length. So what i was wondering is that if there is a smallest length than any length could be measures exactly, like the circumfrence of a circle and the diameter, so if...
  35. O

    Discrete Probability and Distribution to understand the topic.

    I'm currently studying this topic at school... I'm a 12th grader. So I just want to ask, does anyone know how to comprehend the topic in an easier way? or probably anyone know a guide somewhere in the net? I am just confused at binomial distribution, and poisson. I understand a little bit...
  36. T

    Where can I find a comprehensive resource for learning discrete mathematics?

    Hi, Im after some advice on what materials to use in order to gain a fairly 'decent' understanding of the following topics: Elementary Set Theory, Subsets, Unions, Intersections, Complements. Logic, Functions, Mappings, Injectivity. Subjectivity. Bijectivity, Permutations, Proof techniques...
  37. A

    Finding Solutions to a Discrete Math Function Problem

    Hi I need some help with the following problem: 1. Find all functions f: Z+ -> Z+ such that for each n Є Z+ we have f(n) > 1 and f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18 2. I've been reading everywhere and I can't seem to find anything like this. I was wondering if anybody knew where to start 3...
  38. S

    Unit Pulse Response for a discrete time system

    Homework Statement Compute the unit-pulse response h[n] for n= 0,1,2,3 for the following discrete time system: y[n+2] + 1/2y[n+1] + 1/4y[n] = x[n=1] - x[n] Homework Equations I think i am supposed to replace the functions of x with delta functions, which are zero at all except n=0...
  39. P

    Equations of Lines: How to Find, Solve, and Verify Intersection Points

    i have an exam in a few days and am certain a question like this is going to pop up but i have no solutions to this question and no idea how to work it out the question is as follows Find the equations of the line L1 through the point with position vector (4,2,1) and parallel to the vector...
  40. P

    Discrete systems possible mistake in answers

    starting with the iterated map derived in the notes for calculating rootp(p>0) Xn+1=1/2(Xn+(p)/(Xn) calculate root 7 starting with x0=1 so ok starting with that i get x0=1 x1=4 and then something strange happens the sub in looks like this X2=1/2(4+2/4)=9/4 i was under the...
  41. W

    Which Major Would Benefit Me More: Applied Math or Discrete? (Link Included)

    I want to choose either one of these as a second major. Problem is, I'm undecided. My current major is pure math; I want another major so that I have a escape door to the corporate job market in case I decide to stir away from academia. Which one of these two disciplines would benefit me the most?
  42. P

    Spectrum Filtering based on discrete convolution

    Hi, there! That's probably the most relevant forum thread where I can consult pros about my problem. Well, I'm a visual programmer and quite far of Spectra Physics to what my issue's dramatically related to. Specifically, I need to implement in C# the frequency-domain signal filtration (with a...
  43. T

    I have the possibility of taking Calculus 1 and Discrete Math next semester.

    From the people I've spoken to, the general consensus is to take the class in separate semesters if possible. What do you guys recommend? I have 3 semesters left before I finish my AA and I want to get as many math courses in as possible... Thanks.
  44. F

    Having some trouble with Discrete Math.

    Hey all, I just started my Junior year at Florida International University this summer and decided to start light by taking Programming I and Discrete Math to kick things off with and get used to the university. Programming class is going fine, but Discrete Math class is really giving me a...
  45. Fra

    Upper bound for K-L divergence on discrete prob. space

    Does anyone know of any analytical expression for the upper bound on the Kullback–Leibler divergence for a discrete random variable? What I am looking for is the bound expressed as 0 <= S_KL <= f(k) Where k is the number of distinguishable outcomes. Ultimately I am also looking for...
  46. F

    Convolution of Two Discrete Signals with Non-Zero Impulse Response

    please help me in this problem: two discrete signals x[nT]={0 0 1 0 0 2 2 2 2} h[nT]=[-1 2 3 3 2 1] find there convolution if the impulse response doesn't start from zero ,use the table or the matrix
  47. N

    Mathematica Discrete Fourier Transform to find phase shift - Mathematica

    If I use the following code in Mathematica f1[t_] := Cos[w t + d1]; f2[t_] := Cos[w t + d2]; data1 = Table[f1[t], {t,1,10000}]; data2 = Table[f2[t], {t,1,10000}]; ft1 = Fourier[data1]; ft2 = Fourier[data2]; To take the Fourier transform of two data sets, how can I use the resulting data...
  48. I

    Discrete probability distribution

    Homework Statement The problem is as shown in the attatchment. Homework Equations The relevant equations are also given in the attatchment. The Attempt at a Solution My problem is how to adapt the given formula in order to find the sum of the function k(40-r) Do i use the...
  49. W

    Discrete vs. Applied Mathematics

    So, I am interested in majoring in math at Georgia Tech starting this summer, and was wondering what the difference between discrete and applied mathematics is. Any information is greatly appreciated. Also, what does anyone think about double majoring in math and physics?
  50. C

    I need advice with my Discrete Structures class

    I need advice with my "Discrete Structures" class Hi everyone, I'm a sophomore undergraduate student at the Polytechnic University Of Puerto Rico, currently majoring in "Computer Engineering", for the record this university works by trimesters. I'm currently getting a bit frustrated with my...
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