View notes l5proofmethodsv2f15 from eecs 203 at university of michigan. Transparencies to accompany rosen, discrete mathematics and its. A binary relation from a to b is a subset of a cartesian product a x b. Form a truth table in which the premises are columns, and the conclusion is the last col umn. Lets agree that given any two people, either they have met or not. Exhaustively enumerate di erent possibilities, and prove the theorem for each case in many proofs, one needs to combine several di erent strategies. If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach. Ideas of mathematical logic, concepts of set theory and boolean algebra. Discrete mathematics with proof, second edition continues to facilitate an uptodate understanding of this important topic, exposing readers to. Proof by enumeration proofs and symbolic logic s is the the theorem to be proven, expressed as a proposition or predicate deductive proof. Youll find discrete mathematics notes questions as you will go further through the pdf file.
Throughout the book the application of mathematical reasoning is emphasized to solve problems while the authors guide the student in thinking about, reading. Before we explore and study logic, let us start by spending some time motivating this topic. Hence, x 2 4k 2 22k 2 which is even since it is an integer which is divisible by 2. 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. The backbone of any good proof is a complete, watertight argument. What are strategies for writing proofs in discrete. We are then allowed to use the truth of the assumption in 1, 2, or 3 in the proof.
Discrete mathematics proof methodsintroduction to sets 752. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. To know the fundamental ide as of mathematical logic, concepts of set theory and boolean algebra uniti. Discrete mathematics proof by contradiction duration. Nonetheless, this skill is well within the reach of the willing student. In the twentieth century, computer programming and applied statistics developed from o shoots of mathematics into disciplines of their own. 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. New to discrete mathematics disjunctive normal form.
The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. 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. Ma8351 discrete mathematics syllabus notes question banks. 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. A vacuous proof of an implication happens when the hypothesis of the implication. Show that there are irrational numbers r and s such that rs is rational.
A distinctive aspect of the book is the large number of asymptotic formulae derived. This is the heavy weaponry of proofs, and almost anything will yield to it. Constructive the proof is given by finding an element such that is true. Discrete mathematics lecture 5 fall 2015 um eecs 203 l5. Mathematical induction strong induction and well ordering the. The methodology of mathematics has been spectacularly successful, and it has spawned many other elds. If every pair of people in a group has met, well call the group a club. Since the mathematical methods for achieving this depend a great deal on the type of. 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.
In a constructive proof one attempts to demonstrate p q directly. Methods of proofs recall we discussed the following methods of proofs. Rather, we discuss a number of selected results and methods, mostly from the areas of. 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. Remember, when you write mathematics, you should keep your readers perspective in mind. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. We have merged all the notes in a single pdf as discrete mathematics notes pdf. Discussion the rst two methods of proof, the \trivial proof and the \vacuous proof are certainly the easiest when they work.
R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. For whichever method you are trying, choose a strategy. Discrete mathematics proof methods and strategy 1842 direct proof to prove p. This lecture now we have learnt the basics in logic.
Deductive proof proof by deduction inductive proof proof by induction proof by contradiction. The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false. 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. Discrete mathematics and its applications, seventh edition. Here the author presents some complex problems of discrete mathematics in a simple and unified form using an original, general combinatorial scheme. Direct proofs the product of two odd numbers is odd. A trusted guide to discrete mathematics with proof. Catalog description math 248 methods of proof in mathematics 4 units prerequisite. Discrete mathematics and its applications, 7th edition, by k. A proof of a proposition of the form is called an existence proof. Ma8351 discrete mathematics syllabus notes question paper question banks with answers anna university. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs.
This is the simplest and easiest method of proof available to us. 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. The ability to reason using the principles of logic is key to seek the truth which is our goal in mathematics. Basic proof techniques washington university in st.
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. We are going to apply the logical rules in proving mathematical theorems. He developed two types of trans nite numbers, namely, trans nite ordinals and trans nite. Proof techniques, proof by contradiction, mathematical induction.
He was solely responsible in ensuring that sets had a home in mathematics. Below are some effective methods of proof that might aim you in the right direction. Professor sachkovs aim is to focus attention on results that illustrate the methods described. Methods of proof direct, contradiction, conditional, contraposition. Discrete mathematics with proof, 2nd edition wiley. Besides reading the book, students are strongly encouraged to do all the. To impart the mathematical concepts and numerical methods required to computer science. This notes are made using simple language which would be very helpful and thats the reason why we call it discrete mathematics easy notes. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall. These methods provide the overall approach and strategy of proofs.
Discrete mathematics and its applications, seventh edition answers to chapter 1 section 1. Four basic proof techniques used in mathematics youtube. Vacuous proof trivial proof direct proof indirect proof proof by contradiction proof by cases. Discrete math basic proof methods overview of basic proof methods 1.