Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. You should also read chapters 2 and 3 of the textbook, and look at the exercises at the end of these chapters. A course in discrete structures cornell computer science. In discrete mathematics, we call this map that mary created a graph. Well, that year, you literally can define it as a specific discrete year. Number theory athe math behind the rsa crypto system. Issues about data structures used to represent sets and the computational cost of set operations. Our definition is informal because we do not define what a collection is.
Discrete structures structural induction is l dillig is l dillig, cs243. Discrete and continuous random variables video khan academy. The objects in a set are called the elements, or members, of the set. Discrete definition of discrete by the free dictionary. Discrete variable definition, a variable that may assume only a countable, and usually finite, number of values. Rather than define our own axiom systems and models from scratch, it helps. Read ebook online probability on discrete structures pdf. We say that a is afactorordivisorof b and b is amultipleof a.
Is this a discrete or a continuous random variable. This assignment prepares the student to generate pdf homework and turn it in electronically. A course in discrete structures cornell university. Schmalz in this class, we must express mathematical operations in a rigorous fashion, in order to a understand their structure and function, b predict consequences of changing function or procedure parameters, and c have a unified basis for comparing algorithms. Cs 2233 discrete mathematical structures languages, grammars, and machines 2 phrase structure grammars a phrase structure grammar g consists of. A technique that was commonly applied to verify manual calculations is the. Discrete mathematics and its applications chapter 2 notes 2. A conditional statement is logically equivalent to its contrapositive. The structure of the book supports the spiral method of learning, by first introducing basic information, allowing students to work on the. For example, the number of parts damaged in shipment.
Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. An euler path in a graph is a path that contains each edge exactly once. Many different systems of axioms have been proposed. There are discrete values that this random variable can actually take on. In contrast, continuous mathematics deals with objects that vary continuously, e. In this class, we will examine the discrete structure called a sequence, and will define or analyze various operations over sequences. We define sets and set operations, and compare set theory to type theory.
A matrix with m rows and n columns is called an m x n matrix. Aslam rajmohan rajaraman eric ropiak chris burrows ravi sundaram. We look at some relations that are of interest in mathematics. One of the most common programming languages of the 1980s was pascal. Those are a restricted kind of lattice, called an integer lattice, and they are sublattices of products of chains, so theyre distributi. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. These topics vary depending on the institution or school, course, instructor, and text. A binary relation r on a set x defines a directed graph. 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. Discrete mathematics deals with objects that come in discrete bundles, e. Our discrete mathematics structure tutorial is designed for beginners and professionals both. For integer n and real number x, bxc n i n x set a is said to be a proper subset of b if and only if a b and a b.
Mathematics of discrete structures for computer science. Contrapositive, converse, inversewords that made you tremble in high school geometry. We prove cantors theorem that the power set is always larger than the original set. Methods of proving common mistakes in proofs strategies. Cs381 discrete structures discrete mathematics web course material last update august 2, 2009 note. 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. Two sets are equal if and only if they have the same elements.
Discrete mathematics functions a function assigns to each element of a set, exactly one element of a related set. Discrete mathematics topics, applications and examples. 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. The contrapositive of a conditional statement of the form p. 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. An answer here by amin sakzad talks about subsets of cartesian powers of the integers with certain properties. Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move. Applied discrete structures class notes, section 1.
Discrete structures provide a tool box in computer science. Discrete mathematics deals with the study of mathematical structures. The set of natural numbers or nonnegative integers, denoted by n, is defined. Suppose we want to define a function f that returns an infinite sequence. The ellipsis can also be used to define an infinite set, as in the following.
Think of digital watches versus analog watches ones where the second hand loops around continuously without stopping. Relations may exist between objects of the same set or between objects of two or more sets. It consists of a partially ordered set in which every two elements have a unique supremum also called a least upper bound or join and a unique infimum also called a greatest lower bound or meet. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Outline 1 divisibility and modular arithmetic 2 primes and greatest common divisors 3 solving congruences. Validation and application article pdf available in earthquakes and structures 114. Matrices in discrete mathematics and its applications. Discrete variable definition of discrete variable at.
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. It deals with objects that can have distinct separate values. Pdf discrete element modeling of masonry structures. Discrete mathematicsfunctions and relations wikibooks. Discrete mathematics functions 2846 useful properties of floor and ceiling functions 1. Some people call them series instead of summations ive never figured out why. This article examines the concepts of a function and a relation a relation is any association or link between elements of one set, called the domain or less formally the set of inputs, and another set, called the range or set of outputs.
We denote that a is a proper subset of b with the notation a b. I will answer student questions about the course procedures and material. A x b means r is a set of ordered pairs of the form a, b where a. It is also called decision mathematics or finite mathematics. Real numbers have the property that between any two numbers a third can be found. If you have quantitative data, like a number of workers in a company, could you divide every one of the workers into 2 parts. A graph is a collection of points, called vertices, and lines between those points, called edges.
Functions find their application in various fields like representation of the. Summations are simply the sums of the terms in a sequence. Pdf these are notes on discrete mathematics for computer scientists. As opposed to the study of calculus or real numbers which deal with continuous variables, discrete structures deals with graphs and statements in logic which can be enumerated through the implementation of integers. We discuss several definitions and give examples of their applications. Sample problems in discrete mathematics this handout lists some sample problems that you should be able to solve as a prerequisite to design and analysis of algorithms. Examples of structures that are discrete are combinations, graphs, and logical statements. This is a free textbook for an undergraduate course on discrete structures for computer science students, which i have been teaching at carleton university since the fall term of 20. One reason it is difficult to define discrete math is that it is a very broad description which encapsulates a large number of subjects. Whenever sets are being discussed, the relationship between the elements of the sets is the next thing that comes up.
This course will roughly cover the following topics and speci c applications in computer science. Discrete structures refers to the study mathematical structures that are individually separate and distinct rather than continuous. A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. Cmsc 250 discrete structures umd department of computer. A b, is an assignment of exactly one element of b to each element of a. For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, and can be presented as a complete list of those pairs. Number theory and cryptography richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics. A set l on which a partial ordering relation reflexive, antisymmetric, and transitive r is defined is called a partially. Discrete structures lecture notes stanford university. This is a course on discrete mathematics as used in computer science. It could be 1992, or it could be 1985, or it could be 2001.
Logic is an extensive field of study and selected topics are included in discrete structures. This will lead us into the topic of algorithms and complexity in section 1. Attribute data aka discrete data is data that cant be broken down into a smaller unit and add additional meaning. A set a is said to be a proper subset of b if and only if a b and a b. Discrete optimization or combinatorial optimization means searching for an optimal solution in a. Discrete mathematics is the branch of mathematics dealing with objects that can consider only distinct, separated values. Discrete definition, apart or detached from others. Discrete definition is constituting a separate entity. Applied discrete structures has been developed using pretext, a lightweight. Discrete structures are foundational material for computer science. Sets and set operations cs 441 discrete mathematics for cs m. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b.
To introduce the student to the mathematical tools of logic and induction, and to the basic definitions and theorems concerning relations, functions, and sets. Richard mayr university of edinburgh, uk discrete mathematics. Duplicates dont contribute anythi ng new to a set, so remove them. By foundational we mean that relatively few computer scientists will be working primarily on discrete structures, but that many. Basic building block for types of objects in discrete mathematics. Discrete mathematics is in contrast to continuous mathematics, which deals with structures which can range in value over the real numbers, or. Problems on discrete mathematics1 ltex at january 11, 2007. A binary relation from a to b is a subset of a cartesian product a x b.
Think of digital watches versus analog watches ones where the. Discrete mathematics tutorial provides basic and advanced concepts of discrete mathematics. Only a finite number of values is possible, and the values cannot be subdivided meaningfully. Cs381 discrete structuresdiscrete mathematics web course.
In many of the literature on block designs, whilst contemplating the research of experimental effects, its assumed that the anticipated price of the reaction of an experimental unit is the sum of 3 separate parts, a basic suggest parameter, a parameter measuring the impact of the therapy utilized and a. For this reason, we see applied discrete structures as not only aninexpensivealternative,butahighqualityalternative. It wont be able to take on any value between, say, 2000 and 2001. To expose you to some of the variation, we use two main resources, as well as include supplementary resources and. He is a mathematician, and is sometimes a little strange. Matrices in discrete mathematics and its applications 1. An element x of x is a direct predecessor of an element y of x if and only if xry. Discrete mathematics books in this subject area deal with discrete mathematics. A function is a mapping from one set to another set or, perhaps, from one set to itself. I this is why bijections are also calledinvertible functions instructor. Applied discrete structures al doerr university of massachusetts lowell ken levasseur university of massachusetts lowell may 12, 2019. Turgut uyar, aysegul gencata, emre harmanci created date. To expose you to some of the variation, we use two main resources, as well as include supplementary resources and our own original content. Aslam rajmohan rajaraman eric ropiak chris burrows ravi sundaram college of computer and information science northeastern university.
Continuous variables if a variable can take on any value between two specified values, it is called a continuous variable. The material is o ered as the secondyear course comp 2804 discrete structures ii. If such a path is also a circuit, it is called an euler circuit. In computer science, directed graphs are used to represent knowledge e. This course focuses on problemsolving, supporting both abstraction and modeling providing the foundation needed for programming. Definition of matrix a matrix is a rectangular array of numbers. The order of the elements in a set doesnt contribute. We conclude with several exercises from our text that are similar to those that are assigned. Updated to align to the latest 20 acmieee computer science curricula, discrete structures, logic, and computability, fourth edition is designed for the one to twoterm discrete mathematics course. Paces mathematics of discrete structures for computer science is a delightful book, written to be read and enjoyed as it moves from justifying the central place of maths in computing, through the usual set theory and logic, on to numbers, types and the halting problem. It is the study of mathematical structures that are fundamentally discrete in nature and it does not require the notion of continuity. Computers treat time as a series of discrete moments rather than a continuous flow.