Discrete mathematical structures notes, book, ebook for. Model theory studies these sets and other mathematical structures. The aim of this book is not to cover discrete mathematics in depth it should be clear from the description above that such a task would be illde. The more set theory oriented books below are those of devlin, nissanke and stanatmcallister. Discrete mathematics is the branch of mathematics dealing with objects. Online sources such as wikipedia can also be helpful. An important feature of a set is that its elements are \distinct or \uniquely identi able. It is clear, we hope, from the definition of a subset, that given any seta wehavea a and. R tle a x b means r is a set of ordered pairs of the form a,b. During a 12week term with three hours of classes per week, i cover most of the material in this book, except for chapter2, which has been included. Discrete structures for computer science computational geometry.
Schaums outline of discrete mathematics, revised third. Discrete mathematics and its applications by kenneth h. When expressed in a mathematical context, the word statement is viewed in a. Applied discrete structures is designed for use in a university course in dis.
Paradoxes russells paradox let r be the set of all sets that are not members of themselves. For a nonempty set a, aa is called the cartesian product of a with itself. Two sets are equal if and only if they have the same elements. For example, 0,2,4 x x is an even natural number less than 5 from the definition of identity follows that there exists only one empty set. These will be the only primitive concepts in our system. This study guide is written to accompany discrete structures, logic, and computability, third edition, by james l. Sets amotz barnoy department of computer and information science brooklyn college amotz barnoy brooklyn college discrete structures 1 1. Rather, its a description of a set of branches of math that all have in common the feature that they are discrete rather than continuous. Hauskrecht cardinality of the set union cardinality of the set union. Discrete here is used as the opposite of continuous. For example, the set of integers is a discrete mathematical system be cause even. Richard mayr university of edinburgh, uk discrete mathematics. Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
Basic building block for types of objects in discrete mathematics. Discrete data may be also ordinal or nominal data see our post nominal vs ordinal data. The discrete mathematics notes pdf dm notes pdf book starts with the topics covering logic and proof, strong induction,pigeon hole principle, isolated vertex, directed graph, alebric structers, lattices and boolean algebra, etc. Set theory for computer science university of cambridge. Discrete mathematics functions 3346 cardinality of in nite sets i sets with in nite cardinality are classi ed into two classes.
We went through some examples using the sum and product principles. Set theory studies sets, which are collections of objects. Free set theory books download ebooks online textbooks. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. A course in discrete structures cornell university. Discrete data is graphically displayed by a bar graph. B, to be the sum of capacities of all the edges going from set a to set b. Free discrete mathematics books download ebooks online. Discrete structure cs302 b tech rgpv aicte flexible. Two sets are called disjoint if their intersection is empty. The 2nd edition of this outline is available for free online as a pdf. Informally, an injection has each output mapped to by at most one input, a surjection includes the entire possible range in the output, and a bijection has both conditions be true.
Mathematics plays a key role in computer science, some researchers would consider computers as nothing. Definition of sets, countable and uncountable sets, venn diagrams, proofs of some general identities on sets relation. Most of the learning objectives are statements of the form, be able to. The set theoretic definitions of relations and functions are discussed, along. A binary relation from a to b is a subset of a cartesian product a x b. In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. Definition, types of relation, composition of relations, pictorial representation of relation, equivalence relation, partial ordering relation, jobscheduling problem function. Proposition logic and quantifiers, set theory, mathematical induction, recursive definitions, properties o. Occasionally there is need to discuss the set that contains no elements, namelytheemptyset, whichisdenotedby thissetisalsocalledthenull set. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. Hauskrecht basic discrete structures discrete math study of the discrete structures used to represent discrete objects many discrete structures are built using sets sets collection of objects examples of discrete structures built with the help of sets. Many different systems of axioms have been proposed. Transformation into conjunctive normal form fact for every propositional formula one can construct an equivalent one in conjunctive normal form. Discrete structures lecture notes stanford university.
Although calculus is not necessary to understand the ideas in this book. This concept allows for comparisons between cardinalities of sets, in proofs comparing the. The order of the elements in a set doesnt contribute. Sets of propositions can be used to conclude new relationships. Problems on discrete mathematics1 ltex at january 11, 2007. Functions can be injections onetoone functions, surjections onto functions or bijections both onetoone and onto. Set theory basic building block for types of objects in discrete mathematics. The notion of a set is taken as a primitive and so we will not try to define it. This course and this student manual reflect a collective effort led by your. Aug 11, 2018 a set is defined as an unordered collection of distinct elements of the same type where type is defined by the writer of the set. Discrete mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. 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. The axioms of set theory, ordinal and cardinal arithmetic, the axiom of foundation, relativisation, absoluteness, and reflection, ordinal definable sets and inner models of set theory, the constructible universe l cohens method of forcing, independence. Set definition a set is an unordered collection of different elements.
This is a text that covers the standard topics in a sophomorelevel course in discrete mathematics. Along the way, we recall some notions from discrete. Every subset is open in the discrete topology so that in particular, every singleton subset is an. Our discrete mathematics structure tutorial is designed for beginners and professionals both. Number theory athe math behind the rsa crypto system. Each section contains a representative selection of problems. Today, we began with the building blocks of counting. Set theory and types of set in discrete mathematics. B is a partition of v into subsets a and b such that s 2a and t 2b. Though sometimes, this subject can also be taken up in the third year as well.
A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. Dec 08, 2012 discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. R is transitive x r y and y r z implies x r z, for all x,y,z. I this is why bijections are also calledinvertible functions instructor. Solutions to discrete mathematics and its applications. Nov 25, 2016 suppose that a is the set of sophomores at your school and b is the set of students in discrete mathematics at your school.
Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. I dont recommend this book as a standalone text to learn discrete math. Check our section of free ebooks and guides on discrete mathematics now. For example, the collection of sets with the operations of union, intersection, and complement and their accompanying properties is a discrete mathematical structure. Generally, a set is denoted by a capital symbol and the master or elements of a set are separated by an enclosed in.
Issues about data structures used to represent sets and the computational cost of set operations. Discrete mathematics for computer science skyline university. Problems on discrete mathematics1 chungchih li2 kishan mehrotra3 syracuse university, new york latex at january 11, 2007 part i 1no part of this book can be reproduced without permission from the authors. Cantors diagonal argument to show powerset strictly increases size. Each definition is accompanied by commentary that motivates and explains new concepts.
That is, the discrete space x is free on the set x in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps. An argument is a set of two or more propositions related to each ot. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set. Sets, proof templates, and induction x e a x is an element ofa 1. Duplicates dont contribute anythi ng new to a set, so remove them. Now is the time to make today the first day of the rest of your life. The set fx jpxgor fpxgcontains all the objects x for which px is true. A set is one of the most fundamental object in mathematics. This proposition is defined to be true only when both p and q. A nonempty set of verticesnodes v a set of edges e, each edge being a set of one or two vertices if one vertex, the edge is a selfloop a directed graph g v, e consists of a nonempty set of verticesnodes v a set of edges e, each edge being an ordered pair of vertices the. This course will roughly cover the following topics and speci c applications in computer science. A set is typically expressed by curly braces, fgenclosing its elements.
Chapter 0 preface this booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. Discrete mathematics and algorithms lecture 3 an st cut a. Unlock your discrete mathematics and its applications pdf profound dynamic fulfillment today. A central part of this book is the use of the computer to help lear. Allothersubsetsofa,includingtheemptyset,arecalledproper. The discrete topology is the finest topology that can be given on a set. Introduction to groups, rings and fields ht and tt 2011 h. These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets.
It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. Set theory, relation, function, theorem proving techniques. Outline 1 propositions 2 logical equivalences 3 normal forms richard mayr university of edinburgh, uk discrete mathematics. Discrete mathematics is the branch of mathematics dealing with objects that can consider only distinct, separated values.
Ifa isnonempty,thena iscalledanimproper subset ofa. Boolean functions and computer arithmetic, logic, number theory and cryptography, sets and functions, equivalence and order, induction, sequences and series. When the values of the discrete data fit into one of many categories and there is an order or rank to the values, we have ordinal discrete data. The material examinable is that of the lectures detailed below. B f red, blue, black, white, grey g is a set containing ve colors.
This note is an introduction to the zermelofraenkel set theory with choice zfc. By definition, if a valid argument form consists premises. Elements of discrete mathematics virginia commonwealth university. A book of set theory, first published by dover publications, inc. In this chapter, we will cover the different aspects of set theory. Two sets are identical if and only if 2 they have exactly the same members. In traditional experimental laboratories, there are many different tools that are used to perform various experiments. Lecture notes in discrete mathematics arkansas tech faculty web. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. Sets and set operations cs 441 discrete mathematics for cs m. Search the worlds most comprehensive index of fulltext books.
It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its. A set can be written explicitly by listing its elements using set bracket. I used in discrete math classes and this little book helped me to understand better some procedures and techniques. Set operations a the set of sophomores taking discrete mathematics in your school.
Basic sets, comprehension, indexed sets, unions, intersections, products, disjoint unions, powersets. Apr 01, 2000 discrete mathematics and its applications is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. Spiral workbook for discrete mathematics open textbook library. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Discrete mathematics tutorial provides basic and advanced concepts of discrete mathematics. Inverse functions i every bijection from set a to set b also has aninverse function i the inverse of bijection f, written f 1, is the function that assigns to b 2 b a unique element a 2 a such that fa b i observe. Bijection, injection, and surjection brilliant math. A set is an unordered collection of distinct objects. Chapter 5 algebraic structures discrete mathematics book.
Complex issues arise in set theory more than any other area of pure mathematics. It is a very good tool for improving reasoning and problemsolving capabilities. If ais a set and ais an element of it, we write a2a. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
In this book, we will consider the intuitive or naive view point of sets. Example 1 if we add two natural numbers then resultant is again a natural number. Reasonable efforts have been made to test interactive exercises and tools that have been developed here. This book is designed for a one semester course in discrete mathematics for sophomore. We can list each element or member of a set inside curly brackets like this. The objects in a set are called the elements, or members, of the set. Make and study mathematical constructions by means of. Shed the societal and cultural narratives holding you back and let stepbystep discrete mathematics and its applications textbook solutions reorient your old paradigms. The prolog programming language is the tool used for the experiments in this book.
Mathematics of discrete structures for computer science. Prolog experiments in discrete mathematics, logic, and. Jun 29, 20 discrete math is not the name of a branch of mathematics, like number theory, algebra, calculus, etc. A set is an unordered collection of different elements. It is increasingly being applied in the practical fields of mathematics and computer science. Cs381 discrete structures discrete mathematics web course material last update august 2, 2009 note.
134 1441 1532 804 1283 1287 502 408 562 1426 1732 1498 1206 1782 1276 706 1449 1748 847 1300 273 822 1715 587 1659 965 1230 836 795