Proof methods and strategy in discrete mathematics pdf

This notes are made using simple language which would be very helpful and thats the reason why we call it discrete mathematics easy notes. Deductive proof proof by deduction inductive proof proof by induction proof by contradiction. Ma8351 discrete mathematics syllabus notes question banks. 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. I have made money from other survey sites but made double or triple with for the same time and effort. These methods provide the overall approach and strategy of proofs. Ideas of mathematical logic, concepts of set theory and boolean algebra. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. What are strategies for writing proofs in discrete.

He developed two types of trans nite numbers, namely, trans nite ordinals and trans nite. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. Discussion the rst two methods of proof, the \trivial proof and the \vacuous proof are certainly the easiest when they work. Proof techniques, proof by contradiction, mathematical induction. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. The methodology of mathematics has been spectacularly successful, and it has spawned many other elds. Discrete mathematics proof methodsintroduction to sets 752. Exhaustively enumerate di erent possibilities, and prove the theorem for each case in many proofs, one needs to combine several di erent strategies.

Rather, we discuss a number of selected results and methods, mostly from the areas of. This is the simplest and easiest method of proof available to us. Direct proofs the product of two odd numbers is odd. Discrete mathematics with proof, second edition continues to facilitate an uptodate understanding of this important topic, exposing readers to. Methods of proof direct, contradiction, conditional, contraposition. Mathematicians reduce problems to the manipulation of symbols using a set of rules. We are not logicians, but programming methodologists, and this text reflects that perspective. View notes l5proofmethodsv2f15 from eecs 203 at university of michigan. The backbone of any good proof is a complete, watertight argument. Since the mathematical methods for achieving this depend a great deal on the type of. We have merged all the notes in a single pdf as discrete mathematics notes pdf. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall. To impart the mathematical concepts and numerical methods required to computer science. In a constructive proof one attempts to demonstrate p q directly.

The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. Throughout the book the application of mathematical reasoning is emphasized to solve problems while the authors guide the student in thinking about, reading. Discrete mathematics lecture 5 fall 2015 um eecs 203 l5. Discrete mathematics with proof, 2nd edition wiley. Understanding these methods is a key component of learning how to read and construct mathematical proofs. 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. Youll find discrete mathematics notes questions as you will go further through the pdf file. Below are some effective methods of proof that might aim you in the right direction. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students.

Keywords teaching mathematics, equational logic, discrete mathematics. Here the author presents some complex problems of discrete mathematics in a simple and unified form using an original, general combinatorial scheme. Methods of proofs recall we discussed the following methods of proofs. For whichever method you are trying, choose a strategy. In the twentieth century, computer programming and applied statistics developed from o shoots of mathematics into disciplines of their own. Constructive the proof is given by finding an element such that is true. Hence, x 2 4k 2 22k 2 which is even since it is an integer which is divisible by 2. This lecture now we have learnt the basics in logic. Lets agree that given any two people, either they have met or not. This is the heavy weaponry of proofs, and almost anything will yield to it. 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.

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. 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. An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach. A proof of a proposition of the form is called an existence proof. Besides reading the book, students are strongly encouraged to do all the. A binary relation from a to b is a subset of a cartesian product a x b. We are then allowed to use the truth of the assumption in 1, 2, or 3 in the proof. Before we explore and study logic, let us start by spending some time motivating this topic.

Transparencies to accompany rosen, discrete mathematics and its. Ma8351 discrete mathematics syllabus notes question paper question banks with answers anna university. Discrete mathematics and its applications, seventh edition answers to chapter 1 section 1. Mathematical proofs can themselves be represented formally as discrete structures. 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. Catalog description math 248 methods of proof in mathematics 4 units prerequisite. Mathematical induction strong induction and well ordering the. 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. Discrete mathematics proof by contradiction duration. He was solely responsible in ensuring that sets had a home in mathematics.

Vacuous proof trivial proof direct proof indirect proof proof by contradiction proof by cases. To know the fundamental ide as of mathematical logic, concepts of set theory and boolean algebra uniti. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. Discrete math basic proof methods overview of basic proof methods 1. If every pair of people in a group has met, well call the group a club. Professor sachkovs aim is to focus attention on results that illustrate the methods described. Nonetheless, this skill is well within the reach of the willing student. Based on work supported in part by the nsf under grant cda9214957 and darpa. A vacuous proof of an implication happens when the hypothesis of the implication. One innovation uses several major threads to help weave core topics into a cohesive whole.

The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false. Remember, when you write mathematics, you should keep your readers perspective in mind. A distinctive aspect of the book is the large number of asymptotic formulae derived. Basic proof techniques washington university in st. Show that there are irrational numbers r and s such that rs is rational. Discrete mathematics and its applications, seventh edition. New to discrete mathematics disjunctive normal form.