This update brings some reorganization of topics and new examples and exercises. Discrete here is used as the opposite of continuous. 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. This course will roughly cover the following topics and speci c applications in computer science. From a night class at fordham university, nyc, fall, 2008. I read through the power point given by my teacher. Today we introduce set theory, elements, and how to build sets. Sets, logic, proofs, probability, graph theory, etc discrete math 9. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. Discrete math cheat sheetstudy sheetstudy guide in pdf. Discrete mathematics and its applications by kenneth h.
Introduction to formal methods for reasoning and to mathematical tech niquesbasictocomputerscience. Mathematics is a discipline in which working the problems is essential to the understanding of the material contained in this book. Predicate logic richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics. Comprehensive presentations can be found in mitzenmacher and upfal 3, ross 4, 5, and grimmett and stirzaker 2. The cartesian product of s and t, denoted by s x t, is the set of all ordered pairs s,t, where s. Below is a collection of problems that relate to topics in discrete mathematics. Cheat sheetstudy sheet for a discrete math class that covers sequences, recursive formulas, summation, logic, sets, power sets, functions, combinatorics, arrays and matrices. Topicsincludepropositionallogic, discrete mathematics, and linear algebra. Students are strongly encouraged to keep up with the exercises and the sequel of concepts as they are going along, for mathematics builds on itself. He was solely responsible in ensuring that sets had a home in mathematics. Submitted by prerana jain, on august 11, 2018 set theory. Discrete mathematicsset theoryexercises wikibooks, open.
But even more, set theory is the milieu in which mathematics takes place today. The objects that make up a set also known as the elements or members of a set can be anything. Discrete mathematics sets, russells paradox, and halting. He had defined a set as a collection of definite and distinguishable objects selected by the mean. The set sthat contains no element is called the empty set or the null set and is denoted by fgor a set that has only one element is called a singleton set. A binary relation from a to b is a subset of a cartesian product a x b. Well now look at various ways in which sets may be related to one another. A set may also be thought of as grouping together of single objects into a whole. In this article, we will learn about the introduction of sets and the different types of set which is used in discrete mathematics. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene.
Because the fundamentals of set theory are known to all mathematicians, basic problems in the subject seem elementary. 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. However, the rigorous treatment of sets happened only in the 19th century due to the german mathematician georg cantor. Here are three simple statements about sets and functions. I am attempting to work on some proofs for my math assignment, but ill be honest in that i am really struggling to understand them. Discrete mathematics for computer science some notes jean gallier abstract.
Sets, relations and functions, sequences, sums, cardinality of sets richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics. This video is an updated version of the original video released over two years ago. 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. Jul 11, 2017 today we introduce set theory, elements, and how to build sets. Discrete mathematics relations whenever sets are being discussed, the relationship between the elements of the sets is the next thing that comes up. A set can be written explicitly by listing its elements using set bracket. Sets are conventionally denoted with capital letters. Besides reading the book, students are strongly encouraged to do all the. A set is an unordered collection of different elements. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines. R tle a x b means r is a set of ordered pairs of the form a,b. 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. But its only when we apply sets in different situations do they become the powerful building block of mathematics that they are. A beautiful exposition of discrete probability can be found in chapter 8 of concrete mathematics, by graham, knuth, and patashnik 1.
If necessary, redraw the diagram to eliminate any empty regions. The aim of this book is not to cover discrete mathematics in depth it should be clear. The set is a welldefined collection of definite objects of perception or thought and the georg cantor is the father of set theory. Discrete mathematics boolean algebra with introduction, sets theory, types of sets, set operations, algebra of sets, multisets, induction, relations, functions and algorithms etc. These problem may be used to supplement those in the course textbook. Discrete mathematicsdiscrete mathematics and itsand its applicationsapplications seventh editionseventh edition chapter 2chapter 2 setssets lecture slides by adil aslamlecture slides by adil aslam lecture slides by adil aslam email address is. Discrete mathematics for computer science some notes. These are notes on discrete mathematics for computer scientists. Among the fields covered by discrete mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic. Set theory and types of set in discrete mathematics. Are all sets that i just randomly banged on my keyboard to produce. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator.
Discrete mathematics sets, russells paradox, and halting problem 2026 undecidability of halting problem i undecidability of halting problem proved by alan turing in 1936 i proof is quite similar to russells paradox instructor. Hauskrecht basic discrete structures discrete math study of the discrete structures used to represent discrete objects. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Discrete mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. 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. A spiral workbook for discrete mathematics covers the standard topics in a sophomorelevel course in discrete mathematics.
Number theory athe math behind the rsa crypto system. The mathematics in these applications is collectively called discrete mathematics. Discrete mathematics pdf notes dm lecture notes pdf. This alone assures the subject of a place prominent in human culture. Then xis the set of even integers between 0 and 12. Ross 4 contains an enormous amount of examples and is very easy to read. Two sets and are said to be equal if and only if they have exactly the same elements. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. Discrete mathematicsset theory wikibooks, open books for. Problems on discrete mathematics1 ltex at january 11, 2007. In this chapter, we will cover the different aspects of set theory. Now as a word of warning, sets, by themselves, seem pretty pointless.