Basic proof techniques washington university in st. The reason is that the proof setup involves assuming. On the other hand, proof by contradiction relies on the simple fact that if the given theorem p is true, the. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. The truth values for two statements p and q are given in figure 1. Suppose for the sake of contradiction that it is not true that 2 is irrational. 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. Hence,theassumption is false and the statement is true. The basic concept is that proof by contrapositive relies on the fact that p. A powerpoint covering the proof section of the new alevel both years. 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. Proofs by contradiction template for proving a statement by contradiction. In mathematics, we use induction to prove mathematical statements involving integers. 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.
State you have reached a contradiction and what the contradiction entails. Mar 30, 2018 a powerpoint covering the proof section of the new alevel both years. In practice, you assume that the statement you are trying to prove is false and then show that this leads to. In the proof, youre allowed to assume x, and then show that y is true, using x. Proofs by contradiction are useful for showing that something is impossible and for proving the converse of already proven results. 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. An introduction to proof by contradiction, a powerful method of mathematical proof. Reductio ad absurdum, which euclid loved so much, is one of a mathematicians finest weapons. Indeed, remarkable results such as the fundamental theorem of arithmetic can. Thats what proofs are about in mathematics and in computer science. Proofs by contradiction can be somewhat more complicated than direct proofs, because the contradiction you will use to prove the result is. Annotations tools when marking corrections to your proofs, the majority of tools you will use will be found in the annotations section. Assume to the contrary that there is a solution x, y where x and y are positive integers. Negation 3 we have seen that p and q are statements, where p has truth value t and q has truth value f.
W e now introduce a third method of proof, called proof by contradiction. 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. The possible truth values of a statement are often given in a table, called a truth table. It includes disproof by counterexample, proof by deduction, proof by exhaustion and proof by contradiction, with examples for each. By grammar, i mean that there are certain commonsense principles of logic, or proof techniques, which you can. Prove this by contradiction, and use the mean value theorem.
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. Chapter 17 proof by contradiction university of illinois. Chapter 6 proof by contradiction mcgill university. Proof by contradiction this is an example of proof by contradiction.
Many of the statements we prove have the form p q which, when negated, has the form p. Proof by contradiction relies on the simple fact that if the given theorem p is true, then. Im running things by memory and not by understanding what a contradiction is. The direct method is not very convenient when we need to prove a negation of some statement. Brouwe r claimed that proof by contradiction was sometimes invalid. 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. 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. The proof by contradiction method makes use of the equivalence. 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. P is true, and often that is enough to produce a contradiction. Contradiction proofs this proof method is based on the law of the excluded middle. Aleads to absurdity, and proofs by negation, which proofs a statement ais false by showing that aleads to absurdity. So were going to be talking about proofs of lots of things that were trying to understand.
Mathematical proofmethods of proofproof by contradiction. If stuck, you can watch the videos which should explain the argument step by step. Its doubtful if you really understand something if you can explain why its true. If these assumptions always lead to a contradiction. Id like to know what were assuming and how to start. Beginning around 1920, a prominent dutch mathematician by the name of l. A proof by contradiction is a proof that works as follows.
However, contradiction proofs tend to be less convincing and harder to write than direct proofs or proofs by contrapositive. These words have very precise meanings in mathematics which can di. The metaphor of a toolbox only takes you so far in mathematics. A contradiction is any statement of the form q and not q. Turner october 22, 2010 1 introduction proofs are perhaps the very heart of mathematics. 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. Find the vertex of the parabola and go to the left and the right by, say, 1. 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.
Proof by contradiction is typically used to prove claims that a certain type of object cannot exist. The proof of this corollary illustrates an important technique called proof by contradiction. We have reached a contradiction, so our assumption was false. When we derive this contradiction it means that one of our assumptions was untenable.
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. Proof by contradiction albert r meyer contradiction. 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. After experimenting, collecting data, creating a hypothesis, and checking that hypothesis. 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. Each theorem is followed by the otes, which are the thoughts on the topic, intended to give a deeper idea of the statement.
State what the negation of the original statement is. 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\. Assume, for the sake of contradiction, that the statement is false. Unlike the other sciences, mathematics adds a nal step to the familiar scienti c method. Do not edit the pdf files even if you have the means to do so. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note.
You must include all three of these steps in your proofs. In fact, the apt reader might have already noticed that both the constructive method and contrapositive method can be derived from that of contradiction. Essentially, if you can show that a statement can not be false, then it must be true. The negation of the claim then says that an object of this sort does exist. Statements, proofs, and contradiction proof by contradiction 1. 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. Notes on proof by contrapositive and proof by contradiction. The proof by deduction section also includes a few practice questions, with solutions in a separate file. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. What is the logical negation of the statement that fis a decreasing function. It is a particular kind of the more general form of argument known as reductio.
Presumably we have either assumed or already proved p to be true so that nding a contradiction implies that. We start by identify and giving names to the building blocks which make up an argument. Proofs by contradiction this is also called reductio ad absurdum. Reviewed by david miller, professor, west virginia university on 41819.
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 often works well in proving statements of the form. Prove by contradiction that v 2 is not a rational number, i. We sometimes prove a theorem by a series of lemmas. The idea is to assume the hypothesis, then assume the conclusion is false. Complete proofs using proof by defines the rational number. 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. 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. 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 closely related to proof by contrapositive, and the two are sometimes confused, though they are distinct methods. You will nd that some proofs are missing the steps and the purple notes will hopefully guide you to complete the proof yourself. It should give you data to plug into the mean value theorem. Proving conditional statements by contradiction 107 since x.
Each theorem is followed by the \notes, which are the thoughts on the topic, intended to give a deeper idea of the statement. 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. Then present some argument that leads to a contradiction. The main distinction is that a proof by contrapositive applies only to statements that can be written in the form i. Alternatively, you can do a proof by contradiction. Hardy pictured below, he describes proof by contradiction as one of a mathematicians finest weapons. If an assertion implies something false, then the assertion itself must be false. This topic has a huge history of philosophic conflict. In fact proofs by contradiction are more general than indirect proofs. If pis a conjunction of other hypotheses and we know one.
So this is a valuable technique which you should use sparingly. Often proof by contradiction has the form proposition p q. Now we just need a nice, formal statement using our mad lib fillintheblank from the reading. Its a principle that is reminiscent of the philosophy of a certain fictional detective. 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. Show that this is not an if and only if statement by giving a counterexample to the converse. 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.
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