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