Proof by contradiction is closely related to proof by contrapositive, and the two are sometimes confused, though they are distinct methods. The main distinction is that a proof by contrapositive applies only to statements that can be written in the form i. Contradiction there can only be a satisfying assignment if you use the other truth value for x. 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. The sum of two positive numbers is always positive. After that, many authors have focused their attention on the difficulties involved in the contradiction argument employed in the proof, e. In that proof we needed to show that a statement p. To prove a statement by contradiction, you show that the negation of the statement is impossible, or leads to a 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. To prove that p is true, assume that p is not true. Proof by contradiction a proof by contradiction is a proof that works as follows.
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 nonexample one is not the only process that can lead to a proof by contradiction. 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. It will actually take two lectures to get all the way through this. Reviewed by david miller, professor, west virginia university on 41819. Alternatively, you can do a proof by contradiction. The opposite of a tautology is a contradiction, a formula which is always false.
In proof by contradiction, we show that a claim p is true by showing that its negation p. Assume for the sake of contradiction that there was a largest real number. It is a logical law that if a then b is always equivalent to if not b then not a this is called the contrapositive, and is the basis to proof by contrapositive, so a only if b is equivalent to if a then b as well when proving an if and only if proof directly, you must make sure that the equivalence you are proving holds in all steps of the proof. Weve got our proposition, which means our supposition is the opposite. A postive integer n is evenly divisible by 11 if, and only if, the difference of the sums of the digits in the even and odd positions in n is divisible by 11. The proof began with the assumption that p was false, that is that. Many of the statements we prove have the form p q which, when negated, has the form p. The infinite primes and museum guard proofs, explained. Further researches are needed to point out some others.
Chapter 6 proof by contradiction mcgill university. Since this is an if, and only if theorem, we must prove two implications. If, and only if california state university, fresno. A proof of the heineborel theorem university of utah. There are only two steps to a direct proof the second step is, of course, the tricky part. 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. Contradiction proofs this proof method is based on the law of the excluded middle. O assume the conclusion is false o find a contradiction. Simplify the formula by replacing x by this truth value and repeat the process. This topic has a huge history of philosophic conflict.
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. There are no natural number solutions to the equation x2 y2 1. For the sake of contradiction, suppose there are only finitely many. Chapter 17 proof by contradiction university of illinois.
Brouwe r claimed that proof by contradiction was sometimes invalid. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. If our supposition in a proof by contradiction was there exists some integer n such that the product of n and its reciprocal does not. Its a principle that is reminiscent of the philosophy of a certain fictional detective. Contents preface vii introduction viii i fundamentals 1. Propositional logic propositional resolution propositional theorem proving unification today were going to talk about resolution, which is a proof strategy.
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. B is false, and use this to deduce and obvious contradiction of from c is true and c. A proof of the heineborel theorem theorem heineborel theorem. Here are some good examples of proof by 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. In either case, it should be apparent that 3 cannot. Proof by contradiction on if and only if statements. Beginning around 1920, a prominent dutch mathematician by the name of l. Notes on proof by contrapositive and proof by contradiction. 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. On the other hand, proof by contradiction relies on the simple fact that if the given theorem p is true, the. To prove a theorem of the form a if and only if b, you first prove if a then b, then you prove if b then a, and thats enough to complete the proof.
A contradiction is any statement of the form q and not q. Mar 26, 2018 the proof well give dates back to euclid, and our version of his proof uses one of the oldest tricks in the book and the book. If we know q is true then p q is true regardless of the truth value of p. For instance, suppose we want to prove if mathamath, then mathbmath. The proof well give dates back to euclid, and our version of his proof uses one of the oldest tricks in the book and the book. In classical logic, particularly in propositional and firstorder logic, a proposition is a contradiction if and only if. In such a proof, we assume the opposite of what we really want to prove, and then reason from there until we reach a statement that is clearly impossible. In other words, a contradiction is false for every assignment of truth values to its simple components. We now only have to get past the recognition problem. 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. 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\. So this is a valuable technique which you should use sparingly.
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. Assume to the contrary there is a rational number pq, in reduced form, with p not equal to zero, that satisfies the equation. From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. Copious examples of proofs many examples follow theorem 2. First, it is well known that proving by contradiction is a complex activity for the students of various scholastic levels. Based on the assumption that p is not true, conclude something impossible. Truthtables,tautologies,andlogicalequivalences mathematicians normally use a twovalued logic. Proof by contradiction this is an example of proof by contradiction. To prove that the statement if a, then b is true by means of direct proof, begin. 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. So to prove a statement of the form a if and only if b, you really have to do.
A subset s of r is compact if and only if s is closed and bounded. If we must prove a is not evenly divisible by 3 if a 21 is evenly divisble by 3. A postive integer n is evenly divisible by 9 if, and only if, the sum of the digits of n is divisble by 9. There are some issues with this example, both historical and pedagogical. Proof by contradiction relies on the simple fact that if the given theorem p is true, then. O that is the basic structure of a proof by contradiction. Basic proof techniques washington university in st. Logic and proof examples proof by contradiction examples.
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. This is the simplest and easiest method of proof available to us. Then an even number is equal to an odd number, which is a contradiction. If pis a conjunction of other hypotheses and we know one. This is not the only way to perform an indirect proof there is another technique called proof by contrapositive.
The sum of two positive numbers is not always positive. Thanks for contributing an answer to mathematics stack exchange. 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. 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. Proof by contradiction o well, you were all here during the last two minutes.
To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. An introduction to proof by contradiction, a powerful method of mathematical proof. A proof by contradiction in this case has the logical form p p r. Next proof by mathemaitcal induction back to proof by contrapositive. Proof by contradiction is always a viable approach. For a set of premises and a proposition, it is true that. First, well look at it in the propositional case, then in the firstorder case. Say were trying to prove by contradiction that if n 2 is an odd number, then n is also odd for all integers n. There exist two positive numbers a and b that sum to a negative number. In practice, you assume that the statement you are trying to prove is false and then show that this leads to. Essentially, if you can show that a statement can not be false, then it must be true. Proof by contradiction with rational and irrational numbers.