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1.1: An Overview of Discrete Mathematics


What is discrete mathematics? Roughly speaking, it is the study of discrete objects. Here, discrete means “containing distinct or unconnected elements.” Examples include:

  • Determining whether a mathematical argument is logically correct.
  • Studying the relationship between finite sets.
  • Counting the number of ways to arrange objects in a certain pattern.
  • Analyzing processes that involve a finite number of steps.

Here are a few reasons why we study discrete mathematics:

  • To develop our ability to understand and create mathematical arguments.
  • To provide the mathematical foundation for advanced mathematics and computer science courses.

In this text, we will cover these five topics:

  1. Logic and Proof Techniques. Logic allows us to determine if a certain argument is valid. We will also learn several basic proof techniques.
  2. Sets. We study the fundamental properties of sets, and we will use the proof techniques we learned to prove important results in set theory.
  3. Basic Number Theory. Number theory is one of the oldest branches of mathematics; it studies properties of integers. Again, we will use the proof techniques we learned to prove some basic facts in number theory.
  4. Relations and Functions. Relations and functions describe the relationship between the elements from two sets. They play a key role in mathematics.
  5. Combinatorics. Combinatorics studies the arrangement of objects. For instance, one may ask, in how many ways can we form a five-letter word. It is used in many disciplines beyond mathematics.

All of these topics are crucial in the development of your mathematical maturity. The importance of some of these concepts may not be apparent at the beginning. As time goes on, you will slowly understand why we cover such topics. In fact, you may not fully appreciate the subjects until you start taking advanced courses in mathematics.

This is a very challenging course partly because of its intensity. We have to cover many topics that appear totally unrelated at first. This is also the first time many students have to study mathematics in depth. You will be asked to write up your mathematical argument clearly, precisely, and rigorously, which is a new experience for most of you.

Learning how to think mathematically is far more important than knowing how to do all the computations. Consequently, the principal objective of this course is to help you develop the analytic skills you need to learn mathematics. To achieve this goal, we will show you the motivation behind the ideas, explain the results, and dissect why some solution methods work while others do not.


Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable. Examples of structures that are discrete are combinations, graphs, and logical statements. Discrete structures can be finite or infinite. Discrete mathematics is in contrast to continuous mathematics, which deals with structures which can range in value over the real numbers, or have some non-separable quality.

Since the time of Isaac Newton and until quite recently, almost the entire emphasis of applied mathematics has been on continuously varying processes, modeled by the mathematical continuum and using methods derived from the differential and integral calculus. In contrast, discrete mathematics concerns itself mainly with finite collections of discrete objects. With the growth of digital devices, especially computers, discrete mathematics has become more and more important.

Discrete structures can be counted, arranged, placed into sets, and put into ratios with one another. Although discrete mathematics is a wide and varied field, there are certain rules that carry over into many topics. The concept of independent events and the rules of product, sum, and PIE are shared among combinatorics, set theory, and probability. In addition, De Morgan's laws are applicable in many fields of discrete mathematics.

Often, what makes discrete mathematics problems interesting and challenging are the restrictions that are placed on them. Although the field of discrete mathematics has many elegant formulas to apply, it is rare that a practical problem will fit perfectly to a specific formula. Part of the joy of discovering discrete mathematics is to learn many different approaches to problem-solving, and then be able to creatively apply disparate strategies towards a solution.

Contents


Syllabus

Comprehensive, book-style, notes (not repackaged overheads). Available in weekly installments during lectures, and online at the end of the corresponding week.

This book has much to commend it, including an enormous number of examples and exercises and a computer science oriented exposition. It is pitched at a somewhat easy level, suitable for supplementing the lecture notes. It is rather expensive (about £50) but there are many copies in Oxford libraries.

Very basic, but easy to read. Only covers the first half of the course. Cheap.

Some of the book is rather advanced, but also covers the basics quite well. Has many good practice questions (some difficult). Earlier editions are equally useful.


Logic and Discrete Mathematics: A Concise Introduction

This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade.

The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual.

  • Suitable for a variety of courses for students in both Mathematics and Computer Science.
  • Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics
  • Concise, clear and uncluttered presentation with numerous examples.
  • Covers some applications including cryptographic systems, discrete probability and network algorithms.

Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study.


Example 1-1: Admissions Data Section

A university offers only two degree programs: English and Computer Science. Admission is competitive and there is a suspicion of discrimination against women in the admission process. Here is a two-way table of all applicants by sex and admission status. These data show an association between the sex of the applicants and their success in obtaining admission.

Male Female Total
Admit 35 20 55
Deny 45 40 85
Total 80 60 140


Poisson distribution Section

Let (Xsim Poisson(lambda)) (this notation means “X has a Poisson distribution with parameter (lambda)”), then the probability distribution is

Note that (E(X) = V(X) = lambda), and the parameter (lambda) must always be positive negative values are not allowed.

The Poisson distribution is an important probability model. It is often used to model discrete events occurring in time or in space.

The Poisson is also limiting case of the binomial. Suppose that (Xsim Bin(n,pi)) and let (n ightarrowinfty) and (pi ightarrow 0) in such a way that (npi ightarrowlambda) where (lambda) is a constant. Then, in the limit, (Xsim Poisson(lambda)). Because the Poisson is limit of the (Bin(n,pi)), it is useful as an approximation to the binomial when (n) is large and (pi) is small. That is, if (n) is large and (pi) is small, then

where (lambda = npi). The right-hand side of (1) is typically less tedious and easier to calculate than the left-hand side.

For example, let (X) be the number of emails arriving at a server in one hour. Suppose that in the long run, the average number of emails arriving per hour is (lambda). Then it may be reasonable to assume (X sim P(lambda)). For the Poisson model to hold, however, the average arrival rate (lambda) must be fairly constant over time i.e., there should be no systematic or predictable changes in the arrival rate. Moreover, the arrivals should be independent of one another i.e., the arrival of one email should not make the arrival of another email more or less likely.

When some of these assumptions are violated, in particular, if there is a presence of overdispersion (e.g., observed variance is greater than what the model assumes), the Negative Binomial distribution can be used instead of Poisson.

Consider, for example, the number of fatalities from auto accidents that occur next week in Centre County, PA. The Poisson distribution assumes that each person has the same probability of dying in an accident. However, it is more realistic to assume that these probabilities vary due to

  • whether the person was wearing a seat belt
  • time spent driving
  • where they drive (urban or rural driving)

Person-to-person variability in causal covariates such as these cause more variability than predicted by the Poisson distribution.

Let (X) be a random variable with conditional variance (V(X|lambda)). Suppose (lambda) is also a random variable with ( heta=E(lambda)). Then (E(X)=E[E(X|lambda)]) and (V(X)=E[V(X|lambda)]+V[E(X|lambda)])

For example, when (X|lambda) has a Poisson distribution, then (E(X)=E[lambda]= heta) (so mean stays the same) but the (V(X)=E[lambda]+V(lambda)= heta+V(lambda) > heta) (the variance is no longer ( heta) but larger).

When (X|pi) is a binomial random variable and (pi sim Beta(alpha, eta)). Then (E(pi)=frac=lambda) and (V(pi)=frac<(alpha+eta)^2(alpha+eta+1)>). Thus, (E(X)=nlambda) (as expected the same) but the variance is larger (V(X)=nlambda(1-lambda)+n(n-1)V(pi) > nlambda(1-lambda)).


Discrete Mathematics MCQs for Software Engineering Students

q)=q Describes:
A. Double negative law
B. Commutative laws
C. implication Laws
D. None of the above

2. A graph G is called a ….. if it is a connected acyclic graph:
A. Cyclic graph
B. Tree
C. Regular graph
D. Not graph

3. An argument is _____ if the conclusion is not true when all the premises are true:
A. invalid
B. False
C. valid
D. None of the above

4. The relation < (1,2), (1,3), (3,1), (1,1), (3,3), (3,2), (1,4), (4,2), (3,4)>is:
A. Reflexive
B. symmetric
C. Transitive
D. None of the above

5. A ∩ (B U C) = (A ∩ B) U(A ∩ C) is called:
A. Distributivity of intersection over union
B. Distributivity of union over intersection
C. None of these
D. Distributivity Law

6. Letters of CHORD taken all at a time can be written in:
A. 500
B. 120
C. 122
D. 135

7. If f(x)=3x+1 then its inverse is:
A. x-1
B. x +2
C. 1/3(x-1)
D. None of the above

8. The number of colours required to properly colour the vertices of every planer graph is:
A.2
B.3
C.4
D.5

9. A number of elements in a set is called:
A. Finite
B. Cardinality
C. Strength
D. None of the above

10. A partial ordered relation is transitive,Antisymmetric and :
A. reflexive
B. bisymmetric
C. anti reflexive
D. none of the above

11. Which of the given statement is correct?
A. Functions cannot be defined recursively
B. Sets cannot be defined recursively
C. A recursive definition has one part: Base
D. The process of defining an object in terms of smaller versions of itself is called recursion


Discrete Mathematics pdf notes – DM notes pdf file

file to download are listed below please check it –

Complete Notes

Note :- These notes are according to the R09 Syllabus book of JNTU.In R13 and R15,8-units of R09 syllabus are combined into 5-units in R13 and R15 syllabus. If you have any doubts please refer to the JNTU Syllabus Book.

Logic and proof, propositions on statement, connectives, basic connectives, truth table for basic connectives,And,Disjunction,conditional state,bi conditional state,tautology,contradiction,fallacy,contigency,logical equialances,idempotent law,associtative law,commutative law,demorgans law,distributive law,complements law,dominance law,identity law.A praposition of on statement is a declarative sentence which either true (or) false not both, connective is an operation

Combinatorics, strong induction,pigeon hole principle, permutation and combination, recurrence relations, linear non homogeneous recurrence relation with constant, the principle of inclusion and exclusion.

Discrete Mathematics Notes pdf – DM notes pdf

Graphs, parllel edges, adjacent edges and vertices,simple graph,isolated vertex,directed graph,undirected graph,mixed graph,multigraph,pseduo graph,degree,in degree and outdegree,therom,regular graph,complete graph,complete bipartite,subgraph,adjecent matrix of a simple graph,incidence matrix,path matrix,graph isomorphism,pths,rechabality and connected path,length of the path,cycle,connected graph,components of a graph,konisberg bridge problem,Euler parh,euler circuit,hamiltonian path,hamiltonian cycle.

Alebric structers,properties,closure,commutativity,associativity,identity,inverse,distributive law,inverse element,notation,semi group,monoid,cycle monoid,morphisms of semigrouphs,morpism of monoids,groups,abelian group,order of group,composition table,properties of groups,subgroups,kernal of a elomorphism,isomorphism,cosets,lagranges therom,normal subgroups,natural homomorphism,rings,field.

lattices and boolean algebra,reflexive,symmetric,transitive,antisymmetric,equivalance relation,poset,hane diagram,propertie of lattices,idempolent law,commutative law,associative law,absorbtion law,boolean algebra.

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Frequently Asked Questions

Q1: What is discrete mathematics?

A1: Study of countable, otherwise distinct and separable mathematical structures are called as Discrete mathematics. It focuses mainly on finite collection of discrete objects. The field has become more and more in demand since computers like digital devices have grown rapidly in current situation.

A2: Combinatorics is the mathematics of arranging and counting. Though a lot of people know how to count, combinatorics uses mathematical operations to count objects/things that are far away from human count in a conventional way. The field also concerned with the way things are arranged which includes rule of sum and rule of product. Permutation and combination come under this topic.

Q3: What are permutations and combinations?

A3: Permutation is an arrangements of things with regards to order where as combination is an arrangement of things without regard to order.

A4: A branch of mathematics concerned with collections of object is called Set theory. The sets could be discrete or continuous which is concerned with the way sets are arranged, counted or combined. A complements of a set A is the set of elements/things/objects which are not in set A. The cardinality of a finite set is the number of elements/things/objects in that set. The way sets can be combined are described by Intersection and Union. Identities for the complements of intersection and union are given by De Morgan’s laws.


Grading

In order to encourage students to experiment with the concepts taught in class, homework assignments will be given on alternate weeks. They will be due in class on Fridays, at the beginning of lecture. Each assignment will consist of four to five challenging problems, for which the proof or justification of each answer is more important than actual numerical answer.

Since homework is a learning activity, students are welcome to discuss ideas with each other, although collaboration in the writing stage is not permitted. In other words, please do not look at the actual document that another student is handing in.


Handwritten Discrete Mathematics Notes PDF Lecture Download

In these “Discrete Mathematics Notes PDF”, we will study the concepts of ordered sets, lattices, sublattices, and homomorphisms between lattices. It also includes an introduction to modular and distributive lattices along with complemented lattices and Boolean algebra. Then some important applications of Boolean algebra are discussed in switching circuits. The second part of this course deals with an introduction to graph theory, paths and circuits, Eulerian circuits, Hamiltonian graphs, and finally some applications of graphs to shortest path algorithms.

We have provided multiple complete Discrete Mathematics Notes PDF for any university student of BCA, MCA, B.Sc, B.Tech CSE, M.Tech branch to enhance more knowledge about the subject and to score better marks in the exam. Students can easily make use of all these Discrete Mathematics Notes PDF by downloading them.

Topics in our Discrete Mathematics Notes PDF

The topics we will cover in these Discrete Mathematics Notes PDF will be taken from the following list:

Ordered Sets: Definitions, Examples and basic properties of ordered sets, Order isomorphism, Hasse diagrams, Dual of an ordered set, Duality principle, Maximal and minimal elements, Building new ordered sets, Maps between ordered sets.

Lattices: Lattices as ordered sets, Lattices as algebraic structures, Sublattices, Products, and homomorphisms Definitions, Examples, and properties of modular and distributive lattices, The M3 – N5 theorem with applications, Complemented lattice, Relatively complemented lattice, Sectionally complemented lattice.

Boolean Algebras and Switching Circuits: Boolean algebras, De Morgan’s laws, Boolean homomorphism, Representation theorem Boolean polynomials, Boolean polynomial functions, Disjunctive normal form and conjunctive normal form, Minimal forms of Boolean polynomial, Quine-McCluskey method, Karnaugh diagrams, Switching circuits and applications of switching circuits.

Graph Theory: Introduction to graphs, Königsberg bridge problem, Instant insanity game Definition, examples and basic properties of graphs, Subgraphs, Pseudographs, Complete graphs, Bipartite graphs, Isomorphism of graphs, Paths, and circuits, Eulerian circuits, Hamiltonian cycles, Adjacency matrix, Weighted graph, Travelling salesman problem, Shortest path, Dijkstra’s algorithm.


Watch the video: Discrete Math - Introduction to Algorithms and Pseudo Code (October 2021).