What is Discrete mathematics: Definition and 106 Discussions

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. V9999

    I Discrete mathematics--An easy doubt on the notations of sums

    I have a doubt about the notation and alternative ways to represent the terms involved in sums. Suppose that we have the following multivariable function, $$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$. Now, let ##\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}##. In the light of the foregoing, is...
  2. V

    Expected Value of Election Results

    I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong? Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house. There are 4 cases: winning the senate and house (##S \cap H##), winning just...
  3. The Bill

    Intro Math What were the first modern Discrete Mathematics and Precalculus texts?

    What was the first textbook for the modern syllabus of precaclulus which had "precalculus" in the title or subtitle? What was the first textbook for the modern syllabus of discrete mathematics which had "discrete," "discrete mathematics" in the title or subtitle? If you have personal...
  4. Magnetons

    False. The statement does not logically follow from the given information.

    I think it is "True" because the hypothesis is true and the conclusion is False. :cry::cry:But in the answer sheet, the answer is " This is False. The hypothesis is true, but the conclusion is false:## -1^2=-1## , not1."
  5. C

    I Cardinality of decreasing functions from N to N

    Problem: Find the cardinality of the set ## A = \{f \in \Bbb N \to \Bbb N. \forall n\leq m .f(n) \geq f (m) \} ##. I know that ## A \subseteq P(\Bbb N \times \Bbb N) ## implies ## |A| \leq |P(\Bbb N \times \Bbb N)| = | P(\Bbb N) | = \aleph ##. So I have a feeling that ## \aleph \leq |A| ##...
  6. V

    I Translate compound proposition p → q (implication) to p↓q question

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  7. F

    Upper bound height and lower bound height of a 3-ary ordered tree

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  8. F

    I Classify the isomorphism of a graph

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  9. matqkks

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    I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this. How can I inspire them to love essential kind of mathematics? They love doing mathematical techniques...
  10. H

    MHB Discrete Mathematics - Define a relation R on S of at least four order pairs

    Let S = {1,2,5,6 } Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)
  11. Aaron Buckley

    I Help understanding Big O notation

    First, I don't know if this is the right place so if not, please direct me. Thank you. As for the question, I am in a discrete mathematics class online. The instructor is practically non-existent when asking for help simply saying to "refer to the book for clarification". I have scoured google...
  12. Sarina3003

    Generating functions, binomial coefficients

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  13. Sarina3003

    Counting Sequences with Repetition Using Stars and Bars Method

    Homework Statement The question is counting how many sequence length 10 with 1,2,3 if a) increasing from left to right with repetition allowed b) increase from left to right with each number appear at least once (still with repetition allowed) Homework Equations It is the stars and bars...
  14. P

    Probability of drawing a kind from a deck of poker

    Homework Statement Find the probability that a hand of five cards in poker contains four cards of one kind. Homework EquationsThe Attempt at a Solution Solution given in the book:[/B] By the product rule, the number of hands of five cards with four cards of one kind is the product of the...
  15. M

    Propositional function problems

    1. Suppose P(x) and Q(x) are propositional functions and D is their domain. Let A = {x ∈ D: P(x) is true}, B = {x ∈ D: Q(x) is true} (a) Give an example for a domain D and functions P(x) and Q(x) such that A∩B = {} (b) Give an example for a domain D and functions P(x) and Q(x) such that A ⊆ B...
  16. 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...
  17. 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...
  18. 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]##...
  19. Euler2718

    Is This Logical Argument Valid?

    Homework Statement Determine whether the following is valid: p \rightarrow \neg q , r \rightarrow q , r, \vdash \neg p Homework Equations Modus Ponens, disjunctive syllogism, double negation. The Attempt at a Solution I've boiled it down to p \rightarrow \neg q , q, \vdash \neg p...
  20. Avatrin

    Math Applications of discrete mathematics minus software

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  21. M

    Ordered set proof review request

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  22. M

    MHB Combinatorics problem. Discrete Mathematics II

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  23. a255c

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  24. squelch

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  25. T

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  26. Dewgale

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  27. logico

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  28. G

    Is This Discrete Mathematics Argument Valid?

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  29. K

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    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...
  30. B

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  31. M

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

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  32. S

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  33. S

    A discrete mathematics question about logic?

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  34. B

    Discrete Seeking Recommendation on Discrete Mathematics textbook

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  35. H

    Can Induction Prove 3^n ≥ n2^n for All n ≥ 0?

    Homework Statement The question asks me to prove inductively that 3n ≥ n2n for all n ≥ 0. Homework EquationsThe Attempt at a Solution I believe the base case is when n = 0, in which case this is true. However, I cannot for the life of me prove n = k+1 when n=k is true. I start with: 3^k ≥...
  36. 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...
  37. N

    One-to-One Function: Definition & Examples

    Hey I was reading Susanna Discrete book and I came across her definition of One-to-One function: Let F be a function from a set X to a set Y. F is one-to-one (or injective) if, and only if, for all elements x1 and x2 in X, if F(x1 ) = F(x2 ),then x1 = x2 , or, equivalently, if x1 ≠...
  38. R

    Modulus & Division: Last Digit of Numbers Explained

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  39. T

    MHB Learning calculus through discrete mathematics

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  40. T

    Not sure to take Methods of Discrete Mathematics after Calculus 1

    I am a math major and I need to take Methods of Discrete Mathematics. What is methods of discrete mathematics? Should I take it after My calculus series( including linear/ diff. equations)? Is it easy enough to take with Calculus 2? Thanks
  41. J

    MHB How Does Strong Induction Prove Consistency in the Pile Splitting Problem?

    To give you a sense of strong induction and the relationship between mathematical induction and recursion (next session), let's do the pile splitting problem: Take a bunch of beads, rocks, coins, or any kind of chips. Ten is a good number. Split the pile into 2 smaller piles and multiply their...
  42. J

    MHB Truth Table in Discrete Mathematics

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  43. J

    MHB Discrete Mathematics Binary Search

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  44. J

    MHB Discrete Mathematics Vcomparisons

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  45. B

    Computer Science Discrete Mathematics Proof problem

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  46. C

    Help with discrete Mathematics

    Hi! I have an argument and I have to prove the validity of all possible ways. I have proved by logical implication, tautology, contradiction and contrapositive, but the problem is reduced to prove the hypothesis by logical equivalences and implications. The reasoning is as follows...
  47. I

    Discrete mathematics, bijections between disjoint unions

    Hi, So I am trying to show the following: ##(A \cup B)\sqcup(A \cap B) \leftrightarrow A \sqcup B## The proof that I am trying to understand starts with: ##A \leftrightarrow (A \backslash B) \sqcup (A\cup B) \qquad (1)##, and ##A \cup B \leftrightarrow (A\backslash B)\sqcup B \qquad...
  48. 1

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  49. P

    Overview of Discrete Mathematics

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