We now investigate the theoretical foundations of proof by contradiction. In each case you should also think about how a direct or contrapositive proof. In mathematics, a proof by infinite descent, also known as fermats method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. Its a principle that is reminiscent of the philosophy of a certain fictional detective. What are the \objects of number theory analogous to the above description. Proof by contradiction top this technique is based on assuming the existence of elements in the domain that satisfy the hypothesis and not the conclusion, which then leads logically to a contradiction. Proof by contradiction this is an example of proof by contradiction. In particular, you can certainly find lots of books written by perfectly good mathematicians in which the proof is described as a proof by contradiction. Proving something by contradiction can be a very nice method when it works, and there are many proofs in mathematics made easier or, indeed, possible by it. We start from the implication from the right to the left. Well need this method in chapter 20, when we cover the topic of uncountability. For example, here are some problems in number theory that remain unsolved. 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. Use the method of proof by contradiction to prove the following statements.
Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. If we have a rational number x and an irrational number y such that their difference x y is rational. The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. Hardy described proof by contradiction as one of a mathematicians finest weapons, saying it is a far finer gambit than any. Consequently 1 is not an upper bound for n if it were, since 1 proofs require some kind of binary choice between possibilities. From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. The number of friends of a person x is an integer k with 0 k n. A friendly introduction to number theory is an introductory undergraduate text designed to entice nonmath majors into learning some mathematics, while at the same time teaching them how to think mathematically.
We start with the language of propositional logic, where the rules for proofs are very straightforward. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Proofs are all about logic, but there are different types of logic. Specifically, were going to break down three different methods for proving stuff mathematically. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory. Long story short, deductive proofs are all about using a general theory. 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. Common mistakes in proofs show that if x is real number, then x2 is positive. The infinite primes and museum guard proofs, explained. Proof theory is not an esoteric technical subject that was invented to support a formalist doctrine in the philosophy of mathematics. Apr 03, 2018 former mcdonalds worker does a number theory proof duration.
Contents preface ix introduction x i fundamentals 1. However, contradiction proofs tend to be less convincing and harder to write than. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by nonmajors with the exception in the last three chapters where a background in analysis, measure theory and abstract algebra is required. Theorem if x and y are rational numbers, then xy is a rational number. Perhaps the most obvious thing to say is that there is an epistemological relationship between the two, such that. In this exercise, we prove that the square root of 3 is an irrational number. So, there is an odd number that is equal to an even 5.
We know that the sum of two integers cant produce a noninteger result, therefore a contradiction arises, as the proof demonstrates that two integers can produce a noninteger value. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. There is an indication of a considerable simpli cation of my forcing treatment, by restricting the construction of. An introduction to proof by contradiction, a powerful method of mathematical proof. Proof by contradiction albert r meyer contradiction.
Assume, by way of contradiction, that there are a nite number of primes p 1. The e orts to prove a conjecture, may sometimes require a deeper understanding of the theory in. Eulers theorem is a generalization of fermats little theorem and the proof presented here has the same form as the proof presented above for fermats little theorem. Since a rational number where a and b are integers and b. This shows that p divides m2, which again shows that p divides m. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Sometimes its hard to prove the whole theorem at once, so you split the proof into several. It is a particular kind of the more general form of argument known as reductio ad absurdum. Division given two integers, say a and b, the quotient ba may or may not be an integer e. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. A good one sentence answer is that number theory is the study of the integers, i. Im having trouble understanding proofs by contradiction. In the proof, youre allowed to assume x, and then show that y is true, using x. Sometimes the negation of a statement is easier to disprove leads to a contradiction than the original statement is to prove.
Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. Prove by contradiction that the difference of any rational number and any irrational number is irrational. It concentrates on the proof theory of classical logic. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures.
Id like to know what were assuming and how to start. Proof reasoning by contradiction, assume n is bounded from above. Another importance of a mathematical proof is the insight that it may o er. A direct proof, or even a proof of the contrapositive, may seem more satisfying.
Proof by contradiction proof by contradiction also known as indirect proof or the technique or method of reductio ad absurdum is just one of the few proof techniques that are used to prove mathematical propositions or theorems. The statement \a implies b can be written symbolically. If we were formally proving by contradiction that sally had paid her ticket, we would assume that she did not pay her ticket and deduce that therefore she should have got a nasty letter from the council. If an assertion implies something false, then the assertion itself must be false. Often proof by contradiction has the form proposition p q. Let set sbe the set of numbers which cannot be written as a product of primes. The book starts with the basics of set theory, logic and truth tables, and counting.
Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. I just need to be pointed in the right direction, because ive been looking at this for ages and its not clicking. It covers the basic background material that an imo student should be familiar with. If there is a person y whose number of friends is n. Dear jamie, supreme integral with feynmans trick duration. The approach of proof by contradiction is simple yet its consequence and result are remarkable.
We use a proof by contradiction and assume that the square root of 3 is actually a rational number, and we set. Chapter 17 proof by contradiction this chapter covers proofby contradiction. The first welfare theorem any walrasian equilibrium gives a paretoe cient allocation. 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.
This handbook covers the central areas of proof theory, especially the mathematical aspects of proof theory, but largely omits the philosophical aspects of proof theory. Assume snot empty, so it has a smallest element nby wop. In this case, we implicitly assume all of number and set theory to tackle the problem. Properties of integers proving existential statements universal statements and basic techniques of direct proof common mistakes getting proofs started disproof by counterexample proof by contradiction. Being able to write down a valid proof may indicate that you have a thorough understanding of the problem. In proof by contradiction, you assume your statement is not true, and then. In proof by contradiction, we show that a claim p is true by showing that its negation p. Chapter 17 proof by contradiction university of illinois. Some beautiful theorems with beautiful proofs dan quint. The proof of independence of ch by forcing is added. Proof by contradiction often works well in proving statements of the form. Then, by the principle of induction, the propositional form px is true.
Another useful dose of maths for everyone by dr sarada. There are a number of major changes in the fourth edition. Some material on proof by contradiction has been moved forward to chapter 8. This conditional statement being false means there exist numbers a and b for which a,b. To prove that p is true, assume that p is not true. Then, the book moves on to standard proof techniques. More recently, in the 20th century, the intuitionists went further and came to regard proof by contradiction as an invalid method of reasoning. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. This is a contradiction, because m and n have no common divisors. Based on the assumption that p is not true, conclude something impossible.
Many of the statements we prove have the form p q which, when negated, has the form p. Proofs are given when appropriate, or when they illustrate some insight or important idea. Browse other questions tagged elementary number theory proof writing or ask your own question. Chapter 6 proof by contradiction mcgill university. Proof by contradiction a proof by contradiction is a proof that works as follows. These notes serve as course notes for an undergraduate course in number the ory. For any set a, the empty set is an element of the power set of a. Just as the proof above for 2, this shows that p divides n which means that p2 divides n2. The difference of any rational number and any irrational number is irrational.
The point of departure for a proof by contradiction is the assumption that the statement to be proved is false. Suppose for the sake of contradiction that it is not true that 2 is irrational. Im running things by memory and not by understanding what a contradiction is. This is apowerful prooftechnique that can be extremely useful in the right circumstances. These perfect numbers have seen a great deal of mathematical studyindeed, many of the basic theorems of number theory stem from the investigation of the. This is really a special case of proof by contrapositive where your \if is all of mathematics, and your. So again, there is a decimal number that is equal to an number the conclusion is false even number your turn using a proof by contradiction approach, investigate the following statement. R and r has the least upper bound property, then n has a least upper bound.
Preface inwriting this book i have been motivated by the desire to create a highqualitytextbookthatcostsalmostnothing. The reason is that the proof setup involves assuming x, px, which as we know from section 2. This textbook is designed to help students acquire this essential skill, by developing a working knowledge of. 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. The problems are culled from various sources, many from actual contests. 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. Anderson and greg welty what is the relationship between the laws of logic and the existence of god. In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Number theory concerns the former case, and discovers criteria upon which one can. This is really a special case of proof by contrapositive where your \if is all of mathematics, and your \then is the statement you are trying to prove. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. There is a new chapter on mathematical induction chapter 26. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad.
To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. This alternative proof can be generalised to show that is irrational when is not a square number. This fact is then used in place of primitive roots as a tool to. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. I hadnt realized that i hadnt written it in the notes. The sum of two even numbers is not always even that would mean that there are two even numbers out there in the world somewhere thatll give us an odd number when we add them. The empty set is a subset of a, hence it is an element of the power set of a. Mar 26, 2018 when i first started studying proofs, i found it somehow unsatisfying that a proof by contradiction doesnt end by sort of unrolling the contradiction backward, along the lines of this is a contradiction, so the statement before it must be false, which means the statement before that must be false and so on, until wed worked our way back to the original assumption. For every positive rational number x, there is a positive rational number y for which y proof, proof by contradiction, proof by induction and proof by contrapositive.
Proof techniques stanford university computer science. Proof by contradiction prove that a statement p is true using proof by contradiction. The proof by contradiction is grounded in the fact that any proposition must. The basic goal of this semester is to prove ve or so elegant results in micro theory. When i first started studying proofs, i found it somehow unsatisfying that a proof by contradiction doesnt end by sort of unrolling the contradiction backward, along the lines of this is a contradiction, so the statement before it must be false, which means the statement before that must be false and so on, until wed worked our way back to the original assumption. It is used in the proof that a polynomial of degree d has at most d roots modulo p. Still, there seems to be no way to avoid proof by contradiction. The proof by contradiction method makes use of the equivalence p p f 0 where f 0 is any contradiction. These numbers cant be equal, so this is a contradiction. Browse other questions tagged elementary number theory proof verification or ask your own question.