The book starts with the basics of set theory, logic and truth tables, and counting. There are several proof technique and such introductory courses perhaps under discrete mathematics and similar with lecture notes on the net. One of the best known examples of proof by contradiction is the provof that 2 is irrational. A direct proof, but starting with the contrapositive. This is because it is easier to transforminformationabout xintoinformationabout7. Show that not p or q is logically equivalent to notp and notq.
A proof by contrapositive of p q is a direct proof of its contrapositive. The notion of proof by contradiction is often confused with the indirect method for proving. This is logically equivalent to the statement \if i go to class, then it is not raining today. From rstorder logic we know that the implication p q is equivalent to.
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. 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. A proof by contraposition contrapositive is a direct proof of the contrapositive of a statement. Based on the assumption that p is not true, conclude something impossible.
Discrete mathematics proof by contraposition youtube. So if we want to prove the rst statement, it su ces to prove the second statement which is called the contrapositive. It is important to be extremely pedantic when interpreting a contraposition. The second proposition is called the contrapositive of the rst proposition. Sometimes youll be asked to prove a claim thats not in a good form for a direct proof. It is very important to realize that these two implications are not logically equivalent. However, today we want try another approach that works well here and in other important cases where a contrapositive proof may not. Many of the statements we prove have the form p q which, when negated, has the form p. From rstorder logic we know that the implication p q is equivalent. There is a connection between contrapositive proofs and proof by contradiction.
Alternatively, you can do a proof by contradiction. Mat231 transition to higher math proof by contradiction fall 2014 3 12. In 1891, georg cantor assumed that there is a set t that contains every infinitelengt. 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. The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false. Though the proofs are of equal length, you may feel that the contrapositive proof. To prove that p is true, assume that p is not true. Contrapositive examples the infinite series module. But the real point of this example is to show that in general, you cant check a claim about an in. The proves the contrapositive of the original proposition. 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. 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. O sometimes the direct route is just too difficult to deduce.
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 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. 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. I tried to do a proof by contradiction like you did, but it was not accepted. Mathematicalwriting 107 proposition suppose a, b2z andn n. Dec 24, 2014 we look at an indirect proof technique, proof by contraposition.
In fact, the contrapositive is true because the original statement is true. Shorser the contrapositive of the statement \a b i. The basic concept is that proof by contrapositive relies on the. 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. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. In a direct proof you start with the hypothesis of a statement. There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true.
For our next example, consider the following proposition concerninganintegerx. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. Further examples mccpdobson3111 example provebyinductionthat11n. Direct proof p q is proved by showing that if p is true then q follows indirect proof show the contrapositive q p. The method of contradiction is an example of an indirect proof. Likedirectproof,thetechniqueofcontrapositiveproofis. To prove that p q is false, find an example of where p is true. Discussion the rst two methods of proof, the \trivial proof and the \vacuous proof are certainly the easiest when they work. Converses, contrapositives and proof by the contrapositive.
If this is mysterious, we have some methods the reader can use to understand it more fully. A direct proof, or even a proof of the contrapositive, may seem more satisfying. This proof, and consequently knowledge of the existence of. The statement \a implies b can be written symbolically. We obtain the desired conclusion in both cases, so the original statement is true. But, of course, there is room for disagreement about what. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. The statement \a implies b can be written symbolically as \a b. For all integers mand n, if the product of mand nis even, then mis even or nis even. The basic concept is that proof by contrapositive relies on the fact that p. Look at richard hammacks book of proof for a detailed discussion of proof forms.
Proof by contrapositive o this proof technique makes a lot of proofs so much easier. That is, the arbitrary x is part of the antecedent. The three forms are 1 direct if n2 is even, n is even. Contrapositivesupposen j 12,sothereisanintegerc forwhich 12. 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. A proof by contradiction induction cornell university.
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. If p leads to a contradiction, then p cant be true, and therefore p must be true. Hauskrecht methods of proving theorems basic methods to prove the theorems. In a direct proof we assume the hypothesis together with axioms and other theorems previously proved and we derive the conclusion from them. So this is a valuable technique which you should use sparingly. Logical equivalence guarantees that this is a valid proof method. This video is part of a discrete math course taught at the university of cinc.
The contrapositive is certainly true because the entire province of bc is a part of canada. Chapter 17 proof by contradiction this chapter covers proofby contradiction. An indirect proof or proof by contrapositive consists of proving the contrapositive of the desired implication, i. If not b then not a o if it is raining, the sidewalk is wet. O the contrapositive is similar to the converse, but is always true. Still, there seems to be no way to avoid proof by contradiction. Contrapositivesupposenj12,sothereisaninteger c forwhich 12. In a direct proof of this conditional, we start with the assumption that 21n is. In the proof, youre allowed to assume x, and then show that y is true, using x. Chapter 9 contrapositiveproof wenowexamineanalternativetodirectproofcalledcontrapositive proof. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. We use informal proofs to illustrate different methods of proving theorems cs 441 discrete mathematics for cs m.
In most of the following examples the underlying \theorem may be a fact that is well known to you. The second proposition is called the contrapositive of the first proposition. Study at advanced higher maths level will provide excellent preparation for your studies when at university. Then, the book moves on to standard proof techniques. The purpose in presenting them, however, is not to surprise you with new mathematical facts, but.
For our next example, consider the following proposition concerning an integer x. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs. The contrapositive method is a direct proof of q p. 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. So the contrapositive of this statement must also be true. Thanks virmaior, the thing is with fitch, proving a contradiction is not that simple, putting two contradictory premises following each other doesnt suffice. In fact, we have already implicitly used this strategy when we used truth tables to show that certain propositions were true or valid.
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. We look at an indirect proof technique, proof by contraposition. Chapter 6 proof by contradiction mcgill university. For instance, suppose we want to prove if mathamath, then mathbmath. By saying that the two propositions are equivalent we mean that. Notes on proof by contrapositive and proof by contradiction. Chapter 17 proof by contradiction university of illinois. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p.
To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. Contrapositive proof example proposition suppose n 2z. 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. Proof by contrapositive o remember that the converse is not always true. 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. Whenever a conditional statement is true, its contrapositive is also true and vice versa. Logic and proof converse, inverse, and contrapositive. The basic concept is that proof by contrapositive relies on. Similarly, a statements converse and its inverse are always either both true or both false. Copious examples of proofs 19 rewrite it in each of the three forms and prove each.
We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. It seems that the question was designed not to accept proofs by contradiction. On my journey to improve my mathematical rigour i have covered direct proofs and proof by contradiction. As an example, here is a proof by contradiction of. Proof by contrapositive california state university, fresno. We started with direct proofs, and then we moved on to. Proof by contradiction a proof by contradiction is a proof that works as follows. 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. 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. By the closure property, we know b is an integer, so we see that 3jn2. What are some interesting examples of proof by contrapositive.
44 938 371 767 782 986 649 890 175 1383 1058 43 1127 596 272 681 122 7 1273 185 267 987 1450 46 665 1252 31 372 1490 1232