This video is an updated version of the original video released over two years ago. Examples of discrete structures built with the help of sets. An assertion involving predicates is valid if it is true for every universe of discourse. To introduce the student to the mathematical tools of logic and induction, and to the basic definitions and theorems concerning relations, functions, and sets. Discrete mathematicsset theory wikibooks, open books for. 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. In this article, we will learn about the introduction of sets and the different types of set which is used in discrete mathematics. Cardinality of a set if a set is not finite, then it is an infinite set examples. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. This is a course on discrete mathematics as used in computer science. A binary relation from a to b is a subset of a cartesian product a x b. Functions find their application in various fields like representation of the. Demorgans law and the distributive law of union over intersection duration. Aslam rajmohan rajaraman eric ropiak chris burrows ravi sundaram college of computer and information science northeastern university boston, massachusetts.
The terms are used so interchangably and are so vague that the people asking you for. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Function terminology examples i what is the range of this function. The union of xand y, denoted by xy, is the set that consists of all elements of xand also all elements of y.
Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. In this article, we will learn about the relations and the properties of relation in the discrete mathematics. Recursively defined sets i we will study an example of a recursively defined set before we give a general definition. The objects in a set are called elements or members of a set. A course in discrete structures cornell computer science. We discuss several definitions and give examples of their applications.
Sets, functions, sequences, sums, and matrices chapter 2 with questionanswer animations. Relation and the properties of relation discrete mathematics. In this course you will learn set theory, relations, functions and mathematical induction discrete math is the real world mathematics. Discrete mathematics discrete mathematics is foundational material for computer science. Issues about data structures used to represent sets and the computational cost of set operations. This page intentionally left blank university of belgrade. Discrete mathematics for computer science some notes.
In discrete optimization the set d is a discrete, countable set. Introduction sets are one of the basic building blocks for the types of objects considered in discrete mathematics important for counting programming languages have set operations set theory is an important branch of mathematics many different systems of axioms have been used to develop set theory here we are not concerned with a formal set of axioms for. In this if a element is present then it is represented by 1 else it is represented by 0. Basic building block for types of objects in discrete mathematics. For structures in category theory, see structure category theory. Discrete mathematics is in contrast to continuous mathematics, which deals with structures which can range in value over the real numbers, or.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete math emphasizes sets, and operations over sets that are or can be expressed in terms of a subset of the integers. Pdf symmetry and asymmetry of solutions in discrete. He had defined a set as a collection of definite and distinguishable objects selected by the mean. A set can be written explicitly by listing its elements using set bracket. Sets, relations and functions, sequences, sums, cardinality of sets richard mayr. Discrete mathematics for computer science some notes jean gallier abstract. The objects in a set are called the elements, or members, of the set. An element x of x is a direct predecessor of an element y of x if and only if xry. Discrete structures structural induction is l dillig is l dillig, cs243.
Today we introduce set theory, elements, and how to build sets. Setis the fundamental discrete structure on which all other discrete structures are built. Examples of structures that are discrete are combinations, graphs, and logical statements. Many areas of computer science require the ability to work with concepts from discrete mathematics, specifically material from such areas as set theory, logic, graph theory, combinatorics, and probability theory. This course will roughly cover the following topics and speci c applications in computer science. To define a set s inductively, we need to give 3 things. Also, there no question that probability theory plays a crucial role in computing, for example, in the design of randomized algo. Sets are fundamental discrete structures that form the basis of more complex discrete. The two, however, as a course name, describe the same thing. Recursively defined functions and sets, structural induction.
Discrete math is not the name of a branch of mathematics, like number theory, algebra, calculus, etc. 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. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines. Here one set of axioms is given, and comments on variations follow. Number theory athe math behind the rsa crypto system.
The material is o ered as the secondyear course comp 2804 discrete structures ii. An important feature of a set is that its elements are \distinct or \uniquely identi able. A graph drawn in a plane in such a way that if the vertex set of graph can be partitioned into two non empty disjoint subset x and y in such a way that each edge of g has one end in x and one end in y. A set is typically expressed by curly braces, fgenclosing its elements. In this method it is easy to judge if a relation is reflexive, symmetric or transitive just by looking at the matrix. The set is a welldefined collection of definite objects of perception or thought and the georg cantor is the father of set theory. We now mention some set operations that enable us in generating new sets from existing ones.
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. Theres something like 7 or 8 other types of relations. The present definition of a set may sound very vague. These are notes on discrete mathematics for computer scientists. This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. For the notion of structure in mathematical logic, see structure mathematical logic. A set is an unordered collection of objects definition 2 the objects in a set are called the elements, or members, of the set. Applied discrete structures al doerr university of massachusetts lowell ken levasseur university of massachusetts lowell may 12, 2019. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart.
Formal definition there are some differences in exactly what axioms are used to define a ring. Cs381 discrete structures discrete mathematics web course material last update august 2, 2009 note. Please help to improve this article by introducing more precise. R tle a x b means r is a set of ordered pairs of the form a,b. Choose the most appropriate definition of plane graph. Reasonable efforts have been made to test interactive exercises and tools that have been developed here. The set of natural numbers or nonnegative integers, denoted by n, is defined. A binary relation r on a set x defines a directed graph.
Inductively defined sets to define a set s inductively, we need to give 3 things. In contrast to continuous sets and continuous functions, the concept of a local minimum cannot be defined unless a suitable metric or distance between the solutions is defined. We look at some relations that are of interest in mathematics. Our definition is informal because we do not define what a collection is. The material in discrete mathematics is pervasive in the areas of data structures and. Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable.
The term set is intuitively understood by most people to mean a collection of objects that are called elements of the set. A course in discrete structures cornell university. Cartesian product ab not equal to ba cartesian product denoted by is a binary operator which is usually applied between sets. Discrete structures are somewhat like constructs for discrete mathematics, but also deals with the topic matter. Discrete structures homework 1 solutions cs2800 sets and functions 1. A set may also be thought of as grouping together of single objects into. Often, the objects in a set have similar properties. A set is an unordered collection of distinct objects. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. 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.
We can list each element or member of a set inside curly brackets like this. Set definition a set is an unordered collection of different elements. Applied discrete structures class notes, section 1. For one thing, discrete probability theory illustrates how a lot of fairly dry material from chapter 4 is used. A key feature of modern computer science is the ability to specify the functionality of computer programs in terms of mathematical expressions. A set may also be thought of as grouping together of single objects into a whole. A set is one of the most fundamental object in mathematics.
Sets notes introduction i notes introduction ii notes unl cse. Discrete structures lecture notes stanford university. Cantor took the idea of set to a revolutionary level, unveiling its true power. Discrete mathematics thirdedition seymour lipschutz, ph. A brief history of sets a set is an unordered collection of objects, and as such a set is determined by the objects it contains. In computer science, directed graphs are used to represent knowledge e. Discrete mathematics functions a function assigns to each element of a set, exactly one element of a related set. By inventing a notion of size of set he was able compare different forms of in nity and, almost incidentally, to shortcut several traditional mathematical arguments. In this zeroone is used to represent the relationship that exists between two sets.
In this chapter, we will cover the different aspects of set theory. The ellipsis can also be used to define an infinite set, as in the following. Discrete structure solved mcqs computer science solved mcqs. A set l on which a partial ordering relation reflexive, antisymmetric, and transitive r is defined is called a partially. Symmetry and asymmetry of solutions in discrete variable structural optimization 633 fig.
We conclude with several exercises from our text that are similar to those that are assigned. Discrete mathematics representing relations geeksforgeeks. Discrete mathematicsset theory wikibooks, open books. Many different systems of axioms have been proposed. A set can be defined as a unordered collection of entities that are related because they obey a certain rule. Discrete mathematics set theory cardinality how to compute the cardinality of a set.
Come up with a definition for f 1 extends f 2 where f 1 and f 2 are partial functions from a to b, to capture the informal idea that f 1 does everything f 2 does, and maybe more. Nov 25, 2016 set operations in discrete mathematics 1. Richard mayr university of edinburgh, uk discrete mathematics. From this simple beginning, an increasingly complex and useful. This is much simpler than nyc, where we oldtimers cant find the bmt any more. 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. The aim of this part of the discrete mathematics course is to introduce fundamental concepts and techniques in set theory in preparation for its many applications in computer science.
The union of two sets a and b is the set that contains all elements in a, b, or both. 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. A x b means r is a set of ordered pairs of the form a,b where a. Now, about the applications of set relations in speci. Set theory and types of set in discrete mathematics.
An assertion involving predicates is satisfiable if there is a universe and an interpretation. If ais a set and ais an element of it, we write a2a. It deals with continuous functions, differential and integral calculus. Give one or more rules telling how to construct a new element from an existing element in s. 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. Therecursive stepgives the rules for forming new elements in the set from those already known to be in the set. A set is an unordered collection of different elements. Finally, i became convinced that a short introduction to discrete probability was needed. Before the 19th century it was uncommon to think of sets as completed objects in their own right.
588 1143 995 23 574 117 1431 808 220 1353 1378 899 1377 1360 1188 1474 194 222 515 11 1425 742 1082 350 1400 975 1399 366 1328 472 724 1108 25 844 1215 1450 917 500 815 553 1340 418 863 1442