Often proof by contradiction has the form proposition p q. 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. 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. Brouwe r claimed that proof by contradiction was sometimes invalid. Hence,theassumption is false and the statement is true. However, contradiction proofs tend to be less convincing and harder to write than direct proofs or proofs by contrapositive. Beginning around 1920, a prominent dutch mathematician by the name of l. W e now introduce a third method of proof, called proof by contradiction. Proof by contradiction relies on the simple fact that if the given theorem p is true, then.
The idea is to assume the hypothesis, then assume the conclusion is false. If an assertion implies something false, then the assertion itself must be false. Each theorem is followed by the otes, which are the thoughts on the topic, intended to give a deeper idea of the statement. We start by identify and giving names to the building blocks which make up an argument. Proofs by contradiction template for proving a statement by contradiction. Mathematical proofmethods of proofproof by contradiction.
The proof by deduction section also includes a few practice questions, with solutions in a separate file. A proof by contradiction is a proof that works as follows. Complete proofs using proof by defines the rational number. The proof of this corollary illustrates an important technique called proof by contradiction. Annotations tools when marking corrections to your proofs, the majority of tools you will use will be found in the annotations section. Proof by contradiction is typically used to prove claims that a certain type of object cannot exist.
Alternatively, you can do a proof by contradiction. Common types of proofs proof by contradiction assume the statement to be proved is false show that it implies an absurd or contradictory conclusion hence the initial statement must be true application of modus tollens. Assume to the contrary that there is a solution x, y where x and y are positive integers. The truth values for two statements p and q are given in figure 1. Reductio ad absurdum, which euclid loved so much, is one of a mathematicians finest weapons. Jul 12, 2019 a proof by contradiction is often used to prove a conditional statement \p \to q\ when a direct proof has not been found and it is relatively easy to form the negation of the proposition. In this document we will try to explain the importance of proofs in mathematics, and to give a you an idea what are mathematical proofs. State you have reached a contradiction and what the contradiction entails. Chapter 17 proof by contradiction university of illinois. In fact proofs by contradiction are more general than indirect proofs. On the other hand, proof by contradiction relies on the simple fact that if the given theorem p is true, the. Proofs by contradiction are useful for showing that something is impossible and for proving the converse of already proven results.
This topic has a huge history of philosophic conflict. Notes on proof by contrapositive and proof by contradiction. 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. Proof by contradiction albert r meyer contradiction. The proof by contradiction method makes use of the equivalence. Many of the statements we prove have the form p q which, when negated, has the form p.
Since the sum of two even numbers 2 a and 2 b must always be an integer thats divisible by 2, this contradicts the supposition that the sum of two even numbers is not always even. If pis a conjunction of other hypotheses and we know one. Proving conditional statements by contradiction 107 since x. Id like to know what were assuming and how to start. In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can be derived from that of contradiction. State what the negation of the original statement is. So were going to be talking about proofs of lots of things that were trying to understand. The basic concept is that proof by contrapositive relies on the fact that p.
The literature refers to both methods as indirect methods of proof. P is true, and often that is enough to produce a contradiction. In the proof, youre allowed to assume x, and then show that y is true, using x. By grammar, i mean that there are certain commonsense principles of logic, or proof techniques, which you can. Aleads to absurdity, and proofs by negation, which proofs a statement ais false by showing that aleads to absurdity. You must include all three of these steps in your proofs. Hardy pictured below, he describes proof by contradiction as one of a mathematicians finest weapons.
If you have done both of these actions and you still cannot annotate your proofs then please let your contact person know and they will resupply the pdfs. Thats what proofs are about in mathematics and in computer science. In an proof by contradiction we prove an statement s which may or may not be an implication by assuming s and deriving a contradiction. The reason is that the proof setup involves assuming. 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. 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. It includes disproof by counterexample, proof by deduction, proof by exhaustion and proof by contradiction, with examples for each. 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 direct proof. Proof methods mathematical and statistical sciences. Proof by contradiction is often the most natural way to prove the converse of an already proved theorem. Chapter 6 proof by contradiction mcgill university. If stuck, you can watch the videos which should explain the argument step by step.
Find the vertex of the parabola and go to the left and the right by, say, 1. Proof by contradiction is closely related to proof by contrapositive, and the two are sometimes confused, though they are distinct methods. If these assumptions always lead to a contradiction. Turner october 22, 2010 1 introduction proofs are perhaps the very heart of 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. Indeed, remarkable results such as the fundamental theorem of arithmetic can. Negation 3 we have seen that p and q are statements, where p has truth value t and q has truth value f. So this is a valuable technique which you should use sparingly. These words have very precise meanings in mathematics which can di. Each theorem is followed by the \notes, which are the thoughts on the topic, intended to give a deeper idea of the statement. Im running things by memory and not by understanding what a contradiction is. Do not edit the pdf files even if you have the means to do so. Proof by contradiction often works well in proving statements of the form.
What is the logical negation of the statement that fis a decreasing function. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. When we derive this contradiction it means that one of our assumptions was untenable. The advantage of a proof by contradiction is that we have an additional assumption with which to work since we assume not only \p\ but also \\urcorner q\.
A proof by contradiction is often used to prove a conditional statement \p \to q\ when a direct proof has not been found and it is relatively easy to form the negation of the proposition. Suppose for the sake of contradiction that it is not true that 2 is irrational. After experimenting, collecting data, creating a hypothesis, and checking that hypothesis. Mar 30, 2018 a powerpoint covering the proof section of the new alevel both years. Statements, proofs, and contradiction proof by contradiction 1. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. A contradiction is any statement of the form q and not q. Now we just need a nice, formal statement using our mad lib fillintheblank from the reading.
A powerpoint covering the proof section of the new alevel both years. In mathematics, we use induction to prove mathematical statements involving integers. The converse of the pythagorean theorem the pythagorean theorem tells us that in a right triangle, there is a simple relation between the two leg lengths a and b and the hypotenuse length, c, of a right triangle. Prove by contradiction that v 2 is not a rational number, i.
An introduction to proof by contradiction, a powerful method of mathematical proof. Page 2 viewing your documents viewing two page spreads you may wish to view your document in a two page format to see the layout as it would be in print. The negation of the claim then says that an object of this sort does exist. Then present some argument that leads to a contradiction. 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. Presumably we have either assumed or already proved p to be true so that nding a contradiction implies that. Unlike the other sciences, mathematics adds a nal step to the familiar scienti c method. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. 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. In logic and mathematics proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that assuming the proposition to be false leads to a contradiction.
And in particular, were going to look at a proof technique now called proof by contradiction, which is probably so familiar that you never noticed you were using it. Proof by contradiction this is an example of proof by contradiction. Contradiction proofs this proof method is based on the law of the excluded middle. In practice, you assume that the statement you are trying to prove is false and then show that this leads to a contradiction any contradiction. This will give you some reference to check if your proofs are correct. Proofs by contradiction this is also called reductio ad absurdum. Reviewed by david miller, professor, west virginia university on 41819.
Prove this by contradiction, and use the mean value theorem. The direct method is not very convenient when we need to prove a negation of some statement. The main distinction is that a proof by contrapositive applies only to statements that can be written in the form i. Assume, for the sake of contradiction, that the statement is false. It should give you data to plug into the mean value theorem. Basic proof techniques washington university in st. Show that this is not an if and only if statement by giving a counterexample to the converse. We have reached a contradiction, so our assumption was false. Essentially, if you can show that a statement can not be false, then it must be 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. The possible truth values of a statement are often given in a table, called a truth table. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. We sometimes prove a theorem by a series of lemmas.
Its a principle that is reminiscent of the philosophy of a certain fictional detective. It is a particular kind of the more general form of argument known as reductio. You will nd that some proofs are missing the steps and the purple notes will hopefully guide you to complete the proof yourself. In practice, you assume that the statement you are trying to prove is false and then show that this leads to. 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 to prove the original theorem statement using a direct proof. The metaphor of a toolbox only takes you so far in mathematics. Proofs by contradiction can be somewhat more complicated than direct proofs, because the contradiction you will use to prove the result is not always apparent from the proof statement itself. Its doubtful if you really understand something if you can explain why its true. Proofs by contradiction can be somewhat more complicated than direct proofs, because the contradiction you will use to prove the result is. The vocabulary includes logical words such as or, if, etc.