Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Basic set theory a set is a many that allows itself to be thought of as a one. Discrete mathematics rules of inference and mathematical. The objects in a set are called the elements, or members, of the set. Most of the questions on the final will be small variations on problems you have already seen on previous exams or problem sets. Discrete mathematicsset theorypage 2 wikibooks, open. A formal proof of the conclusion c based on the set of. Aims this course will develop the intuition for discrete mathematics reasoning involving numbers and sets. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. Topics covered include logic and the nature of mathematical proof, set theory, relations and functions, matrices and systems of linear equations, algebraic.
A thats what we must end up with do change just one part of the expression at a time, using just one of the set laws at a time dont miss steps out, and change two things at once. Sequences and series are among the most important applications of these ideas. Set theory is the branch of mathematics that studies sets, which are collections of objects, such as blue, white, red or the infinite set of all prime numbers. Download pdf basic discrete mathematics logic set theory.
S px means that there exists at least one element x of s for which px is true. In fact, the desire to advance the modern set theory had been the desire of mathematicians who wanted to pursue ultimate rigorousness in mathematics. Beginning with a a lucid introduction to logic and set theory, the book goes on to cover. Duplicates dont contribute anythi ng new to a set, so remove them. This is a book about discrete mathematics which also discusses mathematical reasoning and logic. Set theory \a set is a many that allows itself to be thought of as a one. Write your own proofs in set theory and discrete mathematics. The order of the elements in a set doesnt contribute.
Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. Elements of discrete mathematics virginia commonwealth university. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Methods of proofs, including mathematical induction. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Thanks for contributing an answer to mathematics stack exchange. Click download or read online button to get basic discrete mathematics book now. This discrete mathematics is way different from the typical maths i have been doing since young. The beginning of set theory as a branch of mathematics is. Read discrete mathematics online, read in mobile or kindle. The strong tradition, universality and neutrality of set theory make it rm common ground on which to provide uni cation between seemingly disparate areas and notations of computer science. Issues about data structures used to represent sets and the computational cost of set operations. But even more, set theory is the milieu in which mathematics takes place today. S px means that px is true for every x in the set s.
Proofs are to mathematics what spelling is to poetry. Lecture notes on discrete mathematics july 30, 2019. Discrete mathematics combines a balance of theory and applications with mathematical rigor and an accessible writing style. Cis 275 introduction to discrete mathematics catalog description basic set theory and symbolic logic. Although the results of securing our mathematical foundations turn. Click download or read online button to basic discrete mathematics logic set theory and probability book pdf for free now. Topics covered include logic and the nature of mathematical proof, set theory, relations and functions, matrices and systems of linear equations, algebraic structures, boolean algebras, and a thorough treatise on graph. They are not guaranteed to be comprehensive of the material covered in the course. Discrete mathematics is designed to serve as a textbook for a singlesemester undergraduate course on the discrete and combinatorial mathematics. Discrete mathematicsset theory wikibooks, open books. Set theory is one of the most rigorous study in mathematics.
Discrete mathematics is an essential part of the foundations of theoretical computer science, statistics, probability theory, and algebra. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. If i were to sub in the definition, it would lead me to nowhere where i can use whatever law there is in set theory. These notes provide a very brief background in discrete mathematics. An introduction to the discrete paradigm in mathematics and computer science. Course objectives to develop logical thinking and its application to computer science to emphasize the importance.
It contains chapters on set theory, logic, algebra matrix algebra and boolean algebra, and graph theory with applications. Written with clarity, humor and relevant realworld examples, basic discrete mathematics is a wonderful introduction to discrete mathematical reasoning. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Translate natural language statements to and from formal propositional logic. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. The set theory material introduces the basic notions of set, element, union, intersection. Discrete structures lecture notes stanford university.
Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. Discrete mathematics for computer science department of computer. This alone assures the subject of a place prominent in human culture. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. Basic discrete mathematics logic set theory and probability download basic discrete mathematics logic set theory and probability ebook pdf or read online books in pdf, epub, and mobi format.
This is a textbook on discrete mathematics for undergraduate students in computer science and mathematics. A proof is a valid argument that establishes the truth of a theorem as the conclusion. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Basic discrete mathematics download ebook pdf, epub. Learn and master all of discrete math logic, set theory, combinatorics, graph theory, etc 4. The book is invaluable for a discrete and finite mathematics course at the freshman undergraduate level or for selfstudy since there are full solutions to the exercises in an appendix. Problems on discrete mathematics1 ltex at january 11, 2007. Demorgans law and the distributive law of union over intersection duration. A set is typically determined by its distinct elements, or members, by. Learning how to write mathematical proofs takes time and hard work. Topics include logic, set theory, number theory, induction, recursion, counting techniques, and graph theory.
If s is a set and px is a statement about x, then the notation. Set theory basic building block for types of objects in discrete mathematics. This course is a prerequisite for all theory courses as well as discrete mathematics ii, algorithms i, security part ib and part ii, arti. Set theory is likely to be around long after most presentday programming languages have faded from memory. The beginning of combinatorics as we know it today. The originators of the basic concepts of discrete mathematics, the mathematics of finite structures, were the hindus, who knew the formulae for the number of permutations of a set of n elements, and for the number of subsets of cardinality k in a set of n elements already in the sixth century. Methods of proving common mistakes in proofs strategies. Many examples of formal and informal proofs are given. The theory of graphs is a major branch of discrete mathematics. Discrete mathematics and combinatorics download ebook.
For those of you new to abstract mathematics elementary does not mean simple though much of the material. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Download discrete mathematics ebook free in pdf and epub format. The final exam will be thursday, 7 december, 9amnoon in the normal classroom the final will cover everything in the course, with an emphasis on the most important concepts that have appeared in at least two places. This books gives an introduction to discrete mathematics for beginning undergraduates. Its a basis for mathematicspretty much all mathematics can be formalised in set theory. Few full derivations of complex mathematical theorems from set theory. Partially ordered sets and sets with other relations have applications in several areas in discrete mathematics, countable sets including finite sets are the main focus. This chapter will be devoted to understanding set theory, relations, functions. From this simple beginning, an increasingly complex and useful. Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial.
Although the results of securing our mathematical foundations turn out to be rather negative and we. These problem may be used to supplement those in the course textbook. I use this book for a discrete math textbook when i am really teaching an introduction to proof and not discrete math. Set theory is an important language and tool for reasoning. A set is an unordered collection of distinct objects. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematical proofs. The text is divided into two parts, the first of which constitutes the core of a onesemester course covering proofs, predicate calculus, set theory, elementary number theory, relations, and functions, and the second of which applies this material to a more advanced study of selected topics in pure mathematics, applied mathematics, and computer. The book began as a set of notes for the discrete mathematics course at the. Browse other questions tagged discretemathematics elementarysettheory or. This site is like a library, use search box in the widget to get ebook that you want. Discrete mathematics i computer science tripos, part 1a paper 1 natural sciences tripos, part 1a.