The negation of any tautology is a contradiction, and the negation of contradiction is a tautology. Discrete mathematics download ebook pdf, epub, tuebl, mobi. Another useful dose of maths for everyone by dr sarada. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous. These problem may be used to supplement those in the course textbook. First assume p is true, and then show that for some proposition r, r is true and r is true that is, we show p r r is true 11. Contradiction definition is act or an instance of contradicting.
Learn the definition of this method and observe how it is applied to proving a statements truth value. Discrete mathematics sets, russells paradox, and halting. However, contradiction proofs tend to be less convincing and harder to write than. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. Problems on discrete mathematics1 ltex at january 11, 2007. Discrete mathematics mathematical proof techniques. Proof by contradiction also known as indirect proof or the method of reductio ad absurdum is a common proof technique that is based on a very simple principle. Many of the statements we prove have the form p q which, when negated, has the form p. He was solely responsible in ensuring that sets had a home in mathematics. The statement \a implies b can be written symbolically. Contradiction definition of contradiction by merriamwebster. I much research on consistent versions of set theory zermelos zfc, russells type theory etc. In order to show that n, pn holds, it suffices to establish the following two properties. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational.
This lecture now we have learnt the basics in logic. Chapter 17 proof by contradiction university of illinois. A real number x is rational if x ab for some a, b are integers. Mathematical proofmethods of proofproof by contradiction. He developed two types of trans nite numbers, namely, trans nite ordinals and trans nite. 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. Discrete mathematics sets, russells paradox, and halting problem 2026 undecidability of halting problem i undecidability of halting problem proved by alan turing in 1936 i proof is quite similar to russells paradox instructor. Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move. For many students, the method of proof by contradiction is a tremendous gift and a trojan horse, both of which follow from how strong the method is. 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. Definition 1 an integer n is a perfect square if n k2 for some integer k. Ok, so i understand i need to prove that the contradiction is wrong for the proof to be true. Basic proof techniques washington university in st.
Besides reading the book, students are strongly encouraged to do all the. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. No matter what the individual parts are, the result is a true statement. Function terminology examples i what is the range of this function. Definition steps example prove by contradiction is an irrational number is an irrational number propositio n proof 10. This is apowerful prooftechnique that can be extremely useful in the right circumstances. A tautology in math and logic is a compound statement premise and conclusion that always produces truth. A proof by contradiction establishes that p is true by assuming that p is false and arriving at a contradiction, which is any proposition of the form r. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs.
One of several different ways to prove a statement in mathematics is proof by contradiction. Its a principle that is reminiscent of the philosophy of a certain fictional detective. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. The proof by contradiction method makes use of the equivalence p p f 0 where f 0. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. Here are some good examples of proof by contradiction. Sep 25, 2017 i this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction and proof by contrapositive. To prove a theorem, assume that the theorem does not hold. I since we have a contradiction, one can prove any nonsense using naive set theory. An open introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. Oct 15, 2014 we discuss the idea of proof by contradiction and work through a small example to prove that there is no smallest positive rational number. Click download or read online button to get discrete mathematics book now. We arrive at a contradiction when we are able to demonstrate that a statement is both simultaneously true and false, showing that our assumptions are inconsistent.
On this quizworksheet, youre going to be subjected to questions that will cover topics like the application of proof by contradiction, as well as assumptions, and how to determine the necessity. We are going to apply the logical rules in proving mathematical theorems. A compound propositioncan be created from other propositions using logical connectives. Rosen this is probably the most popular textbook for discrete math. The two integers will, by the closure property of addition, produce another member of the set of integers. Greek philosopher, aristotle, was the pioneer of logical reasoning. The aim of this book is not to cover discrete mathematics in.
Induction examples the principle of mathematical induction. Induction is a defining difference between discrete and continuous mathematics. In that proof we needed to show that a statement p. Rosen discrete mathematics and its applications, by kenneth h. Chapter 6 proof by contradiction mcgill university.
In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can be derived from that of contradiction. Wherever i mention chapters page numbers, i am referring to the 7th edition. Direct proof contrapositive proof by contradiction proof by cases 3. Four basic proof techniques used in mathematics youtube. If it were rational, it would be expressible as a fraction ab in lowest terms, where a and b are integers, at least one of which is odd.
Discrete mathematics propositional logic tutorialspoint. To discourage guessing, the problem will be graded as follows. The statement \a implies b can be written symbolically as \a b. There are some issues with this example, both historical and pedagogical. This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. This lecture covers proof by contradiction and proof by contrapositive. Interesting examples 30 what if the lower right corner is also removed.
The opposite of a tautology is a contradiction or a fallacy, which is always false. Discrete mathematics propositional logic the rules of mathematical logic specify methods of reasoning mathematical statements. Hauskrecht course administration homework 1 first homework assignment is out today will be posted on the course web page due next. Examples of objectswith discrete values are integers, graphs, or statements in logic. Discrete here is used as the opposite of continuous. In discrete math, most implications involving unknown quantities are, implicitly, universal statements.
Chapter 17 proof by contradiction this chapter covers proofby contradiction. Discrete mathematics available for download and read online in other formats. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. We take a look at an indirect proof technique, proof by contradiction. Well need this method in chapter 20, when we cover the topic of uncountability. The proof by contradiction method makes use of the equivalence p p f 0 where f 0 is any contradiction one way to show that the latter is as follows. This site is like a library, use search box in the widget to get ebook that you want. Therefore our supposition is wrong, and the proof by contradiction is complete. It deals with continuous functions, differential and integral calculus. The mathematics in these applications is collectively called discrete mathematics.
Discrete mathematics sets, russells paradox, and halting problem instructor. Theorem for every, if and is prime then is odd proof we will prove by contradiction the original statement is. Also, x is irrational if it is not rational, that is if x. Covers set theory, logic, counting, permutations and combinations, functions, relations, number theory, proofs, and formal grammar. Proof by contradiction this is an example of proof by contradiction. Richard mayr university of edinburgh, uk discrete mathematics. Discrete mathematics proof by contradiction youtube. Discrete structures lecture notes stanford university. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. The aim of this book is not to cover discrete mathematics in depth it should be clear. Let pbe the statement maria learns discrete mathematics. Division algorithm when an integer is divided by a positive integer, there is aquotientand. This contradiction means the statement cannot be proven false. A compound proposition that is neither a tautology or a contradiction is called a.
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