Preface this is a book about discrete mathematics which also discusses mathematical reasoning and logic. Pdf invitation to discrete mathematics semantic scholar. Discrete mathematics for computer science some notes jean gallier abstract. Cis 275 introduction to discrete mathematics catalog description basic set theory and symbolic logic. Four basic proof techniques used in mathematics youtube. Browse other questions tagged discrete mathematics elementaryset. Besides reading the book, students are strongly encouraged to do all the. This course is a prerequisite for all theory courses as well as discrete mathematics ii, algorithms i, security part ib and part ii, arti. Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Guide to proofs on discrete structures stanford university. 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. However, the rigorous treatment of sets happened only in the 19th century due to the german math ematician georg cantor. The steps of the proofs are not expressed in any formal language as e.

Mathematical logic and proofs mathematics libretexts. Direct proof and counterexample 1 in this chapter, we introduce the notion of proof in mathematics. These are notes on discrete mathematics for computer scientists. In writing and speaking mathematics, a delicate balance is maintained between being formal and not getting bogged down in minutia.

I have since printed out the pdf and it has made learning mathematics significantly easier. A logical argument establishing the truth of the theorem given the truth of the axioms. Combinatorics how to count things, how to make combinatorial objects that have desired properties. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. Aims 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. For us, these topics will be primarily a means to practice proofbased mathematics. Mat230 discrete math mathematical induction fall 2019 12 20. Examples 1 and 2 demonstrate proofs for this simpler case. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students.

Propositional logic richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics. 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. Discrete math, proving a set is a subset of another. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. Discrete mathematics for computer science some notes. Discrete mathematics virginia commonwealth university. A lemma is a helping theorem or a result which is needed to prove a theorem. Mathematics is really about proving general statements via arguments, usually called proofs. Uniqueness proof, discrete math help mathematics stack. Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move. Discrete mathematics an open introduction by oscar levin most of us read math textbooks, but i. This part illustrates the method through a variety of examples.

Once these are in place, and are accepted and understood, then one can begin proving theorems. Methods of proofs, including mathematical induction. Boolean expressions and functions discrete mathematics. Nov 28, 2018 a pdf file that combines the proof of theorem 8. Mathematical induction is used to prove that each statement in a list of statements is true. This section will teach you the format of writing a proof, and walk you through some example proofs. Discrete mathematics with proof, second edition continues to facilitate an uptodate understanding of this important topic, exposing readers to a wide range of modern and technological applications.

The proof of the first two parts is left as an exercise. An axiom is a statement that is assumed to be true, or in the case of a. Demorgans law and the distributive law of union over intersection duration. Discrete structures lecture notes stanford university. Discrete mathematics recurrence relation in this chapter, we will discuss how recursive techniques can derive sequences and be used for solving counting problems. Examples of objectswith discrete values are integers, graphs, or statements in logic. Mathematical induction, is a technique for proving results or establishing statements for natural numbers. Discrete mathematics, second edition in progress january, 2020 springer. Number theory and cryptography richard mayr university of edinburgh, uk richard mayr university of edinburgh, uk discrete mathematics.

Think of digital watches versus analog watches ones where the. A function f from a to b is called onto, or surjective, if and only if for every element b. We now present three simple examples to illustrate this. This video is an updated version of the original video released over two years ago.

Discrete mathematics mathematical proof techniques 32 example, cont i i i i instructor. A guide to proofwriting to accompany discrete mathematics and. 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. Methods of proving a direct proof of a conditional statement p q first assumes that p is true, and uses axioms. Therefore, alice is either a math major or a csi major. This is the standard way of proving subset relations. Discrete mathematics for computer scienceproof wikiversity. Introduction to number theory i number theory is the branch of mathematics that deals with integers and their properties i number theory has a number of applications in computer science, esp. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs. Discrete mathematics deals with objects that come in discrete bundles, e. 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 continuous functions, differential and integral calculus. 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.

I this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction. Automated proof checking in introductory discrete mathematics classes. A set a is said to be a proper subset of b if and only if a b and a b. Introduction to proofs to understand written mathematics, one must understand what makes up a correct mathematical argument, that is, a proof. There are plenty of examples of statements which are hard to prove directly, but whose contrapositive can easily be proved directly. He developed two types of trans nite numbers, namely, trans nite ordinals and trans nite. Lecture notes on discrete mathematics july 30, 2019. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. We start with some given conditions, the premises of our argument, and from these we find a consequence of interest, our conclusion. The difference of a and b, denoted by a b, is the set containing those elements that are in a but not in b. Instructors who have adopted the text may request code to solve knapsack problems. Form a truth table in which the premises are columns, and the conclusion is the last column. Think of digital watches versus analog watches ones where the second hand loops around continuously without stopping.

For the student, my purpose was to present material in a precise, readable manner, with the concepts and techniques of discrete mathematics clearly presented and demonstrated. Discrete mathematics direct proofs examples youtube. A theory is consistent if it cant prove both p and notp for any p. Discrete mathematics inductive proofs saad mneimneh 1 a weird proof contemplate the following. At this point, we have seen a few examples of mathematical proofs. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs.

You should also read chapters 2 and 3 of the textbook, and look at the exercises at the end of these chapters. Some other forms of argument fallacies can lead from true statements to an incorrect conclusion. In contrast, continuous mathematics deals with objects that vary continuously, e. In the previous examples weve seen, weve tried to prove that a.

Introduction bijection and cardinality school of mathematics. Discrete mathematics cardinality 179 how to count elements in a set how many elements are in a set. Aims this course will develop the intuition for discrete mathematics reasoning involving numbers and sets. A lively and entertaining style is combined with rigorous mathematics, and the many illustrations. Intro rules of inference proof methods rules of inference for propositional logic which rule of inference is used in each argument below. A course in discrete structures cornell university.

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. Induction is a powerful tool that can be used to prove many other sorts of statements. The argument may use other previously established statements, such as theorems. Can we say that this infinite set is larger than that infinite set. Outline 1 divisibility and modular arithmetic 2 primes and greatest common divisors 3 solving congruences. In this video we tackle a divisbility proof and then prove that all integers are the difference of two squares. Automated proof checking in introductory discrete mathematics. Discrete mathematics pdf notes dm lecture notes pdf. Discrete mathematics mathematical proof techniques.

This tool is the principle of mathematical induction. Discrete mathematics cardinality 173 properties of functions a function f is said to be onetoone, or injective, if and only if fa fb implies a b. This is a course on discrete mathematics as used in computer science. Argument in mathematics, an argument is a sequence of propositions called premises followed by a proposition called conclusion a valid argument is one that, if all its premises. The problem is, as you no doubt know from arguing with friends, not all arguments are good arguments. I have made money from other survey sites but made double or triple with for the same time and effort. A set a is said to be a proper subset of b if and only. Similar and parallel discussions can be found in the life contingencies book of jordan 1967 and the actuarial mathematics book of bowers et al. An integer a is called a perfect square if there exists an. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous. In order to prove this two steps are needed, prove existence and. Mar 03, 2011 proof techniques, proof by contradiction, mathematical induction. Problems on discrete mathematics1 ltex at january 11, 2007.

Discrete mathematics and its applications seventh edition. Proof techniques, proof by contradiction, mathematical induction. Sep 25, 2017 new to proving mathematical statements and theorem. We denote that a is a proper subset of b with the notation a b. Not just for proving the validity of algebraic equations. Mat230 discrete math mathematical induction fall 2019 5 20 using mathematical induction is a bit like setting up cascading dominos.

The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Discrete mathematics mathematical proof techniques 1432 proof by cases i in some cases, it is very di cult to prove a theorem by applying the same argument in all cases. We sometimes prove a theorem by a series of lemmas corollary. Mathematical induction is a mathematical technique which is used to prove a statement. Discrete mathematics functions 46 onto functions i a function f from a to b is calledontoi for every element y 2 b, there is an element x 2 a such that fx y. Discrete mathematics functions 46 onto functions i a function f from a to b is calledontoi for every element. Does it make sense at all to ask about the number of elements in an infinite set. We sometimes prove a theorem by a series of lemmas. Preface inwriting this book i have been motivated by the desire to create a highqualitytextbookthatcostsalmostnothing.

These problem may be used to supplement those in the course textbook. There are a few standard methods for proving an implication, and a couple of points that apply to all proofs. Invitation to discrete mathematics is at once an introduction and a thoroughly comprehensive textbook for courses in combinatorics and graph theory. Discrete mathematics recurrence relation tutorialspoint. Outline mathematical argument rules of inference 2. Discrete mathematics direct proofs direct proof dartmouth cs. My goal was to show the relevance and practicality of discrete mathematics to.

Today we introduce set theory, elements, and how to build sets. Course objectives to develop logical thinking and its application to computer science to emphasize the importance. He was solely responsible in ensuring that sets had a home in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion is true under the assumption that the premisses are true. Discrete mathematics introduction to number theory 219 divisibility. Come up with your own examples of a true statement, a false statement. One takes the axiom to be given, and to be so obvious and plausible that no proof is required. Inference rules and proof methods school of electrical. The difference of a and b is also called the complement of b with respect to a.

529 866 1568 107 835 1193 1500 1003 920 1106 1305 96 860 1074 1290 331 786 1278 84 274 997 1003 1090 836 416 787 1265 374 750 267 1265 1481