This is an important technique for proving mathematical results. Proof by contradiction in logic and mathematics is a proof that determines the truth of a statement by assuming the proposition is false, then working to show its falsity until the result of that assumption is a contradiction. 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. Proving this directly via constructive proof would probably be very difficult if not impossible. We shall show that you cannot draw a regular hexagon on a square lattice. This booklet consists of problem sets for a typical undergraduate discrete mathematics course aimed at computer science students. This might be my all time favorite proof by contradiction. A proof by contradiction establishes that p is true by assuming that p is false and arriving. If it is a miracle, any sort of evidence will answer, but if. We obtain the desired conclusion in both cases, so the original statement is true. Proof by contradiction a proof by contradiction is a proof that works as follows. Problems on discrete mathematics1 ltex at january 11, 2007. In each case you should also think about how a direct or contrapositive proof would work. This proof, and consequently knowledge of the existence of.
On the analysis of indirect proofs example 1 let x be an integer. As an example of this, consider the following proposition and its proof. Whats a simple example of a proof by contradiction. Discrete structures mathematical proof techniques 1638 another example i recall. Proof by contradiction process negate the conclusion. Well need this method in chapter 20, when we cover the topic of uncountability. To prove that p is true, assume that p is not true.
It is an indirect proof technique that works like this. I have made money from other survey sites but made double or triple with for the same time and effort. As the name suggests, this is a book on algorithms, which is a topic that relies heavily on discrete math. O proof by deduction o proof by contrapositive o proof by contradiction o proof by induction. Proof by contradiction versus proof by contraposition this part of the paper explores the differences and similarities that exist between proof by contraposition and proof by contradiction. Attempts to do so have led to the strange world of constructive mathematics. In the introduction example, the goal was to prove that there is no largest number, so the proof begins with the premise that there is. Mathematical proof methods of proof proof by contradiction.
Form a truth table in which the premises are columns, and the conclusion is the last col umn. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. The statement \a implies b can be written symbolically as \a b. This lecture now we have learnt the basics in logic. 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. This is the template for dam discrete and argumentative mathematics. Suppose that i wanted to prove that the cube root of. The method of contradiction is an example of an indirect proof. Direct proof and counterexample 1 in this chapter, we introduce the notion of proof in mathematics. Actually, we will see a proof of this for v 2 shortly. Direct proof an argument in which we prove a proposition in its originallystated form is called a direct proof. In that proof we needed to show that a statement p. Template for proofs in discrete and argumentative mathematics.
Hopefully this short introduction will shed some light on what the subject is about and what you can expect as you move. Now we just need a nice, formal statement using our mad lib fillintheblank from the reading. Proof by contradiction often involves clever application of proven knowledge to arrive at a contradiction. This lesson defines both direct and indirect proofs and, in turn, points out the differences between them. Induction examples the principle of mathematical induction. It is not a discrete math book, but a few of the topics on this page are covered briefly in clrs some in the appendix, some in regular chapters. Proof by contradiction albert r meyer contradiction. There are some issues with this example, both historical and pedagogical.
Discrete mathematics for computer scienceproof wikiversity. Chapter 17 proof by contradiction university of illinois. Discrete mathematics inductive proofs saad mneimneh 1 a weird proof contemplate the following. Then, the book moves on to standard proof techniques.
In discrete math, most implications involving unknown quantities are, implicitly, universal statements. Induction is a defining difference between discrete and continuous mathematics. The following simple but wonderful proof is at least as old as euclids book the elements. If so, i could write up a nice, detailed solution, but i wonder if the main issue is in understanding how proofs by contradiction work and are logically valid. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. Well also look at some examples of both types of proofs in both abstract and realworld. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs.
A logical argument establishing the truth of the theorem given the truth of the axioms. We will learn more about the integers in chapter 4. Give a direct proof of the theorem if is an odd integer, then 2 is odd. Proof by contradiction is one of the most important proof methods. Hauskrecht proofs with quantifiers universally quantified statements. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematical proofs. If we add or subtract an odd number from an even number, we get an odd number, so 3n. All major mathematical results you have considered. These problem may be used to supplement those in the course textbook. Still, there seems to be no way to avoid proof by contradiction.
Proof by contradiction is a very interesting form of proof, in which we make an assumption usually were not allowed to make assumptions when doing proofs and use the assumption to arrive at a contradiction. A real number x is rational if x ab for some a, b are integers. Believe it or not, this type of proof is used fairly frequently by mathematicians. If you try to do this, you will find that if you make your hexagon very large, then you can get somewhat close to. The literature refers to both methods as indirect methods of proof. A counterexample is an example that disproves a universal for all statement. Jul 06, 2017 a proof by contradiction is a method of proving a statement by assuming the hypothesis to be true and conclusion to be false, and then deriving a contradiction. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. In this next example, the variables are real numbers. Therefore our supposition is wrong, and the proof by contradiction is complete.
Once we have a contradiction, we know that the assumption cant be true. Many of the statements we prove have the form p q which, when negated, has the form p. Discrete mathematics proof by contradiction youtube. In the proof, youre allowed to assume x, and then show that y is true, using x. You must include all three of these steps in your proofs. This confused state of affairs extends well beyond italy. Show the statement holds fo r one example but we do not have the witness example. A direct proof, or even a proof of the contrapositive, may seem more satisfying. Anyrational numbercan be written in the form p q where p and q are integers and have no common. 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. In the example proof of \\sqrt2 \s irrationality, what is the key underlying assumption.
When you have an idea or when someone tells you something, test the idea by trying examples. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. Proof by contradiction this is an example of proof by contradiction. Often proof by contradiction has the form proposition p q. A proof is a sequence of logical deductions, based on accepted assumptions and previously proven statements and verifying that a statement is true. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. This lecture covers proof by contradiction and proof by contrapositive. Discrete mathematics mathematical proof techniques.
Based on the assumption that p is not true, conclude something impossible. Preface this book is designed for a one semester course in discrete mathematics for sophomore or junior level students. What constitutes a proof may vary, depending on the field. And now were going to call explicit attention to it, and think about it. Since the sum of two even numbers 2a and 2b must always be an integer thats divisible by 2, this contradicts the supposition that the sum of two even numbers is not always even.
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. Here are some good examples of proof by contradiction. To rock a proof by contradiction, follow these steps. State what the negation of the original statement is.
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. Dec 22, 2014 71 videos play all discrete math 1 thetrevtutor discrete math 5. Epps discrete mathematics with applications 2011 is. Discrete mathematics i fall 2011 7 university of hawaii proof by contradiction example. Proof by contradiction assume that for some values of n, pn is. However, there is a general method, the principle of mathematical induction. Certainly we cannot draw that conclusion from just the few above examples. Use the method of proof by contradiction to prove the following statements. The book starts with the basics of set theory, logic and truth tables, and counting. Prove the property holds for all examples can be tricky proof by cases to divides the proof. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. To illustrate, below is an example of a proof by contraposition. However, contradiction proofs tend to be less convincing and harder to write than.
Begin with the premise that whatever you are attempting to prove, the opposite is true. In mathematics, a formal proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. Alternatively, you can do a proof by contradiction. Mar 03, 2011 proof techniques, proof by contradiction, mathematical induction. Deductive proof example if fx is even, then fx is not onetoone. If an assertion implies something false, then the assertion itself must be false. Non examples and proof by contradiction samuele antonini department of mathematics university of pisa, italy researches in mathematics education about proof by contradiction revealed some difficulties of the students but also that this kind of argumentation comes spontaneously in certain situations. Chapter 17 proof by contradiction this chapter covers proofby contradiction.
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. This is apowerful prooftechnique that can be extremely useful in the right circumstances. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Mathematical proofmethods of proofproof by contradiction. Informal proofs cs 441 discrete mathematics for cs m. Another useful dose of maths for everyone by dr sarada. The proof began with the assumption that p was false, that is that. Proof by contradiction forms the bedrock of all kinds of theorems we take for granted, like the fact that intersecting lines cross at only one point, or that the square root of 2 is an irrational number. Also, x is irrational if it is not rational, that is if x. In the united states, many textbooks fail to clearly distinguish between these two types of proof. Clrs is the primary suggested textbook for my algorithms course. Typically relies on the proof by contradiction negate the existentially quantified statement and show that it implies a contradiction. Direct proof contrapositive proof by contradiction proof. 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.
In this video well talk about the structure of proof by contradiction, and then do a small example where we prove a theorem that no integer is both even and odd. Definition 1 an integer n is a perfect square if n k2 for some integer k. Proof techniques, proof by contradiction, mathematical induction. Chapter 6 proof by contradiction mcgill university. Definition steps example prove by contradiction is an irrational number is an irrational number propositio n proof 10. A good example of this is by proving that is irrational. Obtaining counterexamples is a very important part of mathematics, because doing mathematics requires that you develop a critical attitude toward claims. We take a look at an indirect proof technique, proof by contradiction. Some forms of direct proof are discussed in sections 1.