Proof methods and strategy in discrete mathematics pdf

The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Professor sachkovs aim is to focus attention on results that illustrate the methods described. Discrete mathematics and its applications, seventh edition. Youll find discrete mathematics notes questions as you will go further through the pdf file. Existence proof examples show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways. Methods of proving a proof of the proposition of the form x px is called an existence proof sometimes, we can find an element s, called a. Besides reading the book, students are strongly encouraged to do all the. Methods of proof direct, contradiction, conditional, contraposition. Discrete mathematics proof methodsintroduction to sets 752. Four basic proof techniques used in mathematics youtube. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students.

Before we explore and study logic, let us start by spending some time motivating this topic. To know the fundamental ide as of mathematical logic, concepts of set theory and boolean algebra uniti. Nonetheless, this skill is well within the reach of the willing student. The methodology of mathematics has been spectacularly successful, and it has spawned many other elds. 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. Transparencies to accompany rosen, discrete mathematics and its. Hence, x 2 4k 2 22k 2 which is even since it is an integer which is divisible by 2. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. Throughout the book the application of mathematical reasoning is emphasized to solve problems while the authors guide the student in thinking about, reading. We are going to apply the logical rules in proving mathematical theorems. The backbone of any good proof is a complete, watertight argument. A vacuous proof of an implication happens when the hypothesis of the implication. To impart the mathematical concepts and numerical methods required to computer science. Discrete mathematics proof methods and strategy 1842 direct proof to prove p.

These methods provide the overall approach and strategy of proofs. Remember, when you write mathematics, you should keep your readers perspective in mind. Since the mathematical methods for achieving this depend a great deal on the type of. Ideas of mathematical logic, concepts of set theory and boolean algebra. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall. Vacuous proof trivial proof direct proof indirect proof proof by contradiction proof by cases. This lecture now we have learnt the basics in logic. Mathematical proofs can themselves be represented formally as discrete structures. View notes l5proofmethodsv2f15 from eecs 203 at university of michigan. This is the simplest and easiest method of proof available to us. If every pair of people in a group has met, well call the group a club. In the twentieth century, computer programming and applied statistics developed from o shoots of mathematics into disciplines of their own. Discrete math basic proof methods overview of basic proof methods 1.

Mathematicians reduce problems to the manipulation of symbols using a set of rules. Catalog description math 248 methods of proof in mathematics 4 units prerequisite. Direct proofs the product of two odd numbers is odd. We are not logicians, but programming methodologists, and this text reflects that perspective. Based on work supported in part by the nsf under grant cda9214957 and darpa. One innovation uses several major threads to help weave core topics into a cohesive whole. Mathematical induction strong induction and well ordering the. Understanding these methods is a key component of learning how to read and construct mathematical proofs.

Keywords teaching mathematics, equational logic, discrete mathematics. Ma8351 discrete mathematics syllabus notes question paper question banks with answers anna university. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. Exhaustively enumerate di erent possibilities, and prove the theorem for each case in many proofs, one needs to combine several di erent strategies. Discrete mathematics lecture 5 fall 2015 um eecs 203 l5. Proof methods and strategy mathematics stack exchange. These active and wellknown authors have come together to create a fresh, innovative, and timely approach to discrete math. Proof techniques, proof by contradiction, mathematical induction. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics.

This is the heavy weaponry of proofs, and almost anything will yield to it. We have merged all the notes in a single pdf as discrete mathematics notes pdf. New to discrete mathematics disjunctive normal form. If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach. Lets agree that given any two people, either they have met or not. Basic proof techniques washington university in st.

Methods of proofs recall we discussed the following methods of proofs. Constructive the proof is given by finding an element such that is true. A trusted guide to discrete mathematics with proof. 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. Ma8351 discrete mathematics syllabus notes question banks. He was solely responsible in ensuring that sets had a home in mathematics. Here the author presents some complex problems of discrete mathematics in a simple and unified form using an original, general combinatorial scheme. I have made money from other survey sites but made double or triple with for the same time and effort. In a constructive proof one attempts to demonstrate p q directly. Deductive proof proof by deduction inductive proof proof by induction proof by contradiction. Discussion the rst two methods of proof, the \trivial proof and the \vacuous proof are certainly the easiest when they work.

Now in a newly revised edition discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. A proof of a proposition of the form is called an existence proof. Discrete mathematics proof by contradiction duration. Show that there are irrational numbers r and s such that rs is rational. We are then allowed to use the truth of the assumption in 1, 2, or 3 in the proof. 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. The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. Discrete mathematics and its applications, seventh edition answers to chapter 1 section 1. A distinctive aspect of the book is the large number of asymptotic formulae derived. 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 notes are made using simple language which would be very helpful and thats the reason why we call it discrete mathematics easy notes. Proof by enumeration proofs and symbolic logic s is the the theorem to be proven, expressed as a proposition or predicate deductive proof. What are strategies for writing proofs in discrete. The aim of this book is not to cover discrete mathematics in depth it should be clear from the description above that such a task would be illde. A binary relation from a to b is a subset of a cartesian product a x b. For whichever method you are trying, choose a strategy. Form a truth table in which the premises are columns, and the conclusion is the last col umn.