By the closure property, we know b is an integer, so we see that 3jn2. However, today we want try another approach that works well here and in other important cases where a contrapositive proof may not. To prove that p q is false, find an example of where p is true. Alternatively, you can do a proof by contradiction. 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. But, of course, there is room for disagreement about what. Contrapositive examples the infinite series module. As an example, here is a proof by contradiction of. But the real point of this example is to show that in general, you cant check a claim about an in. Chapter 9 contrapositiveproof wenowexamineanalternativetodirectproofcalledcontrapositive proof. The basic concept is that proof by contrapositive relies on the. We look at an indirect proof technique, proof by contraposition.
Contrapositive proof example proposition suppose n 2z. A proof by contraposition contrapositive is a direct proof of the contrapositive of a statement. Show that not p or q is logically equivalent to notp and notq. One of the most famous proofs of all time ironically, one that is incorrectly cited as an example of proof by contradiction1 is a proof by contraposition. It is important to be extremely pedantic when interpreting a contraposition. Though the proofs are of equal length, you may feel that the contrapositive proof. In a direct proof of this conditional, we start with the assumption that 21n is. Proof by contrapositive o remember that the converse is not always true. From rstorder logic we know that the implication p q is equivalent. The basic concept is that proof by contrapositive relies on the fact that p. This proof, and consequently knowledge of the existence of. It seems that the question was designed not to accept proofs by contradiction. Shorser the contrapositive of the statement \a b i. In most of the following examples the underlying \theorem may be a fact that is well known to you.
Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. To prove that p is true, assume that p is not 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. Simon singh a proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true. In 1891, georg cantor assumed that there is a set t that contains every infinitelengt. Notes on proof by contrapositive and proof by contradiction.
Notes on proof by contrapositive and proof by contradiction hyunyoung lee august 29, 2019 in this short lecture note, i explain the di erence between proof by contrapositive and proof by contradiction. Hauskrecht methods of proving theorems basic methods to prove the theorems. The three forms are 1 direct if n2 is even, n is even. Discrete mathematics proof by contraposition youtube.
Copious examples of proofs 19 rewrite it in each of the three forms and prove each. An indirect proof or proof by contrapositive consists of proving the contrapositive of the desired implication, i. Further examples mccpdobson3111 example provebyinductionthat11n. In fact, the contrapositive is true because the original statement is true. Contrapositivesupposenj12,sothereisaninteger c forwhich 12. We started with direct proofs, and then we moved on to. Notes on proof by contrapositive and proof by contradiction hyunyoung lee august 31, 2018 in this short lecture note, i will explain the di erence between proof by contrapositive and proof by contradiction, which seem to cause easily some confusions. Contrapositivesupposen j 12,sothereisanintegerc forwhich 12. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. Similarly, a statements converse and its inverse are always either both true or both false. Chapter 6 proof by contradiction mcgill university.
The second proposition is called the contrapositive of the first proposition. Proof by contraposition can be an e ective approach when a traditional direct proof is tricky, or it can be a di erent way to think about the substance of a problem. This video is part of a discrete math course taught at the university of cinc. For our next example, consider the following proposition concerninganintegerx. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. O sometimes the direct route is just too difficult to deduce. In a direct proof we assume the hypothesis together with axioms and other theorems previously proved and we derive the conclusion from them. The statement \a implies b can be written symbolically as \a b. The proves the contrapositive of the original proposition. Logic and proof converse, inverse, and contrapositive.
So the contrapositive of this statement must also be true. A direct proof, or even a proof of the contrapositive, may seem more satisfying. That is, the arbitrary x is part of the antecedent. Unlike the method of trivial proof, this method may be applied to any implication. Jul 06, 2017 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.
Still, there seems to be no way to avoid proof by contradiction. A direct proof, but starting with the contrapositive. In this post i will cover the third method for proving theorems. While weve seen that its possible for a statement to be true while its converse is false, it turns out that the contrapositive is better behaved. Proof by contrapositive california state university, fresno. What are some interesting examples of proof by contrapositive. So this is a valuable technique which you should use sparingly. Many of the statements we prove have the form p q which, when negated, has the form p. A statement and its contrapositive are logically equivalent, in the sense that if the statement is true, then its contrapositive is true and vice versa.
Whenever a conditional statement is true, its contrapositive is also true and vice versa. So if we want to prove the rst statement, it su ces to prove the second statement which is called the contrapositive. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. The purpose in presenting them, however, is not to surprise you with new mathematical facts, but. By saying that the two propositions are equivalent we mean that.
Logical equivalence guarantees that this is a valid proof method. From rstorder logic we know that the implication p q is equivalent to. However, indirect methods such as proof by contradiction can also be used with contraposition, as, for example, in the proof of the irrationality of the square root of 2. On my journey to improve my mathematical rigour i have covered direct proofs and proof by contradiction. In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a. Discussion the rst two methods of proof, the \trivial proof and the \vacuous proof are certainly the easiest when they work. We use informal proofs to illustrate different methods of proving theorems cs 441 discrete mathematics for cs m. The basic concept is that proof by contrapositive relies on. Look at richard hammacks book of proof for a detailed discussion of proof forms. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. Thanks virmaior, the thing is with fitch, proving a contradiction is not that simple, putting two contradictory premises following each other doesnt suffice.
A proof by contradiction induction cornell university. Mat231 transition to higher math proof by contradiction fall 2014 3 12. For instance, suppose we want to prove if mathamath, then mathbmath. A proof by contrapositive of p q is a direct proof of its contrapositive. The statement \a implies b can be written symbolically. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. Chapter 17 proof by contradiction university of illinois. Converses, contrapositives and proof by the contrapositive. It is very important to realize that these two implications are not logically equivalent. The method of contradiction is an example of an indirect proof. Jul 10, 2017 one of the most famous proofs of all time ironically, one that is incorrectly cited as an example of proof by contradiction1 is a proof by contraposition. There are several proof technique and such introductory courses perhaps under discrete mathematics and similar with lecture notes on the net. I tried to do a proof by contradiction like you did, but it was not accepted. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever.
In fact, we have already implicitly used this strategy when we used truth tables to show that certain propositions were true or valid. One of the best known examples of proof by contradiction is the provof that 2 is irrational. The contrapositive is certainly true because the entire province of bc is a part of canada. O the contrapositive is similar to the converse, but is always true. Reading up on different methods for proving things i realized that a contrapositive proof is a really clever thing to used and often a better way to prove things than a proof by contradiction.
Likedirectproof,thetechniqueofcontrapositiveproofis. In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. Existence proof examples show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways. The connection is that any proof by contrapositive can trivially be turned into a proof by contradiction, and most proofs by contradiction can either be turned into a contrapositive proof or else has a simpler hidden direct proof. Then, the book moves on to standard proof techniques. In the proof, youre allowed to assume x, and then show that y is true, using x.
For all integers mand n, if the product of mand nis even, then mis even or nis even. This is logically equivalent to the statement \if i go to class, then it is not raining today. Proof by contrapositive o this proof technique makes a lot of proofs so much easier. We obtain the desired conclusion in both cases, so the original statement is true. The book starts with the basics of set theory, logic and truth tables, and counting. There is a connection between contrapositive proofs and proof by contradiction. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p.
In a direct proof you start with the hypothesis of a statement. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. The contrapositive method is a direct proof of q p. Mathematicalwriting 107 proposition suppose a, b2z andn n. For example, the assertion if it is my car, then it is red is equivalent to if that car is not red, then it is not mine. Dec 24, 2014 we look at an indirect proof technique, proof by contraposition. Study at advanced higher maths level will provide excellent preparation for your studies when at university. Chapter 17 proof by contradiction this chapter covers proofby contradiction. The notion of proof by contradiction is often confused with the indirect method for proving.
If p leads to a contradiction, then p cant be true, and therefore p must be true. If this is mysterious, we have some methods the reader can use to understand it more fully. Direct proof p q is proved by showing that if p is true then q follows indirect proof show the contrapositive q p. The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false. The second proposition is called the contrapositive of the rst proposition. Conclusion that proof of the contrapositive proves the original statement example if a sum of two real numbers is less than 50, then at least one of the numbers is less than 25. Proof by contradiction a proof by contradiction is a proof that works as follows. This is because it is easier to transforminformationabout xintoinformationabout7. For our next example, consider the following proposition concerning an integer x. Sometimes youll be asked to prove a claim thats not in a good form for a direct proof. Based on the assumption that p is not true, conclude something impossible.