Weve got our proposition, which means our supposition is the opposite. This proof method is applied when the negation of the theorem statement is easier to be shown to lead to an absurd not true situation than proving the original theorem statement using a. Proof by contradiction example let us assume that the original statement is false. Contrapositive, contradiction, induction proofs, proof by. Brouwe r claimed that proof by contradiction was sometimes invalid. It is a particular kind of the more general form of.
Jun 17, 2016 a longer example, focusing on a case where we mistakenly prove the converse instead of the original statement, due to a common misunderstanding of the method of proof by contradiction. The story of goliath the giant 1 samuel 17 is a bible classic with a clear message for young and old. Pdf book of proof third edition david long academia. Finding a contradiction means that your assumption is false and therefore the statement is true. There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result.
Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. Assume to the contrary there is a rational number pq, in reduced form, with p not equal to zero, that satisfies the equation. An analysis of these studies points out that the proof by contradiction, from cognitive and didactical points of view, seems to have the form of a paradox. Chapter 17 proof by contradiction university of illinois. An introduction to proof by contradiction, a powerful method of mathematical proof. Alternatively, you can do a proof by contradiction. Proofs cis0082 logic and foundations of mathematics. To prove p, it su ces to assume ot p and derive a contradiction. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. In praise of proofs by contradiction that arent scientific. Therefore, there is a such that, is prime and is even, all at the same time. Notes on proof by contrapositive and proof by contradiction. First of a series of videos showing examples of proof by contradiction, for an upcoming number theory course. Its a principle that is reminiscent of the philosophy of a certain fictional detective.
The metaphor of a toolbox only takes you so far in mathematics. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. We take the negation of the given statement and suppose it to be true. You must include all three of these steps in your proofs. So this is a valuable technique which you should use sparingly. In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can. Then argue that a mathematical contradiction occurs. To prove ot p, it su ces to assume p and derive a contradiction. God however wanted to punish israel for their sins most. To prove p, it su ces to assume \not p and derive a contradiction. 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. This textbook is designed to help students acquire this essential skill, by developing a working knowledge of. In that proof we needed to show that a statement p.
Proof by contradiction a proof by contradiction is a proof that works as follows. For any set a, the empty set is an element of the power set of a. The proof began with the assumption that p was false, that is that. Proof by contradiction is informally used to refer to twodi erent rules of inference. More examples of proof by contradiction math and multimedia. Negating the two propositions, the statement we want to prove has the form. Proofs in mathematics and computer science require a precisely stated proposition to be. This document models those four di erent approaches by proving the same proposition four times over using each fundamental method. We suppose such an integer does exist, and will reach a contradiction. Proof by contradiction relies on the simple fact that if the given theorem p is true, then. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Proof by contradiction california state university, fresno. Chapter 6 proof by contradiction mcgill university.
The empty set is a subset of a, hence it is an element of the power set of a. Below are several more examples of this proof strategy. We start with the language of propositional logic, where the rules for proofs are very straightforward. In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can be derived from that of contradiction. To prove that p is true, assume that p is not true. Inequality proof by contradiction mathematics stack exchange. The sum of two positive numbers is always positive. A proof provides a means for guaranteeing such claims. Jun 16, 2016 first of a series of videos showing examples of proof by contradiction, for an upcoming number theory course. Aug 28, 2014 proof by contradiction is one of the major proof techniques in mathematics.
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. State you have reached a contradiction and what the contradiction entails. Proof by contradiction this is an example of proof by contradiction. The proofwriting workbook william paterson university. First, it is well known that proving by contradiction is a complex activity. From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. To prove the statement a implies b, a proof by contradiction assumes that both a and not b are true, and. 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. Proof by contradiction is also possible with young children.
To prove that the statement if a, then b is true by means of direct proof, begin by assuming a is true and use this information to deduce that b is true. Unfortunately, not all proposed proofs of a statement in mathematics are actually correct, and so some e ort needs to be put into veri cation of such a proposed proof. Based on the assumption that p is not true, conclude something impossible. A longer example, focusing on a case where we mistakenly prove the converse instead of the original statement, due to a common misunderstanding of the method of proof by contradiction. State what the negation of the original statement is.
Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. On the other hand, proof by contradiction relies on the simple fact that if the given theorem p is true, the. Reviewed by david miller, professor, west virginia university on 41819. Our first example of a proof by contradiction dates back more than 2000. To prove a statement by contradiction, you show that the negation of the statement is impossible, or leads to a contradiction. The weird and wonderful world of constructive mathematics. It is a particular kind of the more general form of argument known as reductio ad absurdum. The use of this fact forms the basis of the technique of proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of. 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. Basic proof techniques washington university in st. Contradictions about david david and saul when did saul meet david. Giant discovery in israel uncovers proof of goliaths rule. However, equally important and less studied, is the role of goliath and the philistines as the physical and ideological enemies of david and the messianic dynasty. Use proof by contradiction or contrapositive proof, as appropriate, for each of the following.
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. Hardy described proof by contradiction as one of a mathematicians finest weapons, saying it is a far finer gambit than any. Cs 70 discrete mathematics and probability theory fall 2009. The sum of two positive numbers is not always positive. Mathematical proofmethods of proofproof by contradiction. Proof theory was created early in the 20th century by david hilbert to prove. Hardy pictured below, he describes proof by contradiction as one of a mathematicians finest weapons. Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. This topic has a huge history of philosophic conflict. Many of the statements we prove have the form p q which, when negated, has the form p. Fall 2009 satish rao, david tse note 2 proofs intuitively, the concept of proof should already be familiar.
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. Beginning around 1920, a prominent dutch mathematician by the name of l. On the analysis of indirect proofs example 1 let x be an integer. The basic concept is that proof by contrapositive relies on the fact that p. Mathematical writing by donald knuth, tracy larrabee, and paul m. There exist two positive numbers a and b that sum to a negative number. The proof above actually commits another sin against mathematical exposition, namely the unnecessary use of proof by contradiction. 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. We all like to assert things, and few of us like to say things that turn out to be false. To prove \not p, it su ces to assume p and derive a contradiction.
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