In the contrapositive proof we are turning it around and. Chapter 17 proof by contradiction university of illinois. I am looking more for a set of guidelines or intuitions or heuristics that might suggest that trying to use the contrapositive to prove the mathematical statement might be a good idea. The contrapositive proof goes the opposite way of the direct proof. Calgary remix, i can proceed as follows line 1 is the premise. In a direct proof of this conditional, we start with the assumption that 21n is. Reviewed by david miller, professor, west virginia university on 41819. His writing is not always easy to follow, as the following frag. Hauskrecht methods of proving theorems basic methods to prove the theorems. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. In line 3, in order to ultimately arrive at a contradiction, i. In this paper, we shall examine the argumentations which.
O proof by deduction o proof by contrapositive o proof by contradiction. A proof provides a means for guaranteeing such claims. A proof is a sequence of logical statements, one implying another, which gives. 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. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause. Mathematical proofmethods of proofproof by contrapositive. 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. The proof by contraposition method makes use of the equivalence p q q p to show that the conditional statement p q is true, we first assume q is true, and use axioms, definitions, proved theorems, with rules of inference, to show p is also true 8. Contrapositive examples the infinite series module. The equivalence between a statement and its contrapositive is so obvious for an expert that, usually, he does not need any explanation.
An indirect proof or proof by contrapositive consists of proving the contrapositive of the desired implication, i. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. On my journey to improve my mathematical rigour i have covered direct proofs and proof by contradiction. Some problems have structures that make it more obvious to try induction or 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. But, of course, there is room for disagreement about what. Proofs if the statement to be proved is not in the form p q but perhaps simply a statement p, then there is no contrapositive. Rather our aim is to show, usually through a direct argument, that the contrapositive. We discuss the difference between contrapositive and direct proofs, explain why contrapositive. We started with direct proofs, and then we moved on to. 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.
There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. We are showing that if q is false then p must be false. However, in a proof by contradiction, we assume that p is true and q is false and arrive at some sort of illogical statement such as 12. Set theory proof with contrapositive stack exchange. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. Free pdf version part i fundamentals chapter 1 sets a ll of mathematics can. Proof by contraposition proof by contraposition rests on the fact that an implication q and its p contrapositive p q not implies not q p are two logically equivalent statements. 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.
Discrete mathematics proof by contraposition youtube. The method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive. Pdf a statement, the contrapositive and the inverse. Logically, theorem i is exactly the same as this theorem, which is called the contrapositive of theorem i. Proofs and mathematical reasoning university of birmingham. By the closure property, we know b is an integer, so we see that 3jn2. In a direct proof we assume the hypothesis together with axioms and other theorems previously proved and we derive the conclusion from them. This is because it is easier to transforminformationabout xintoinformationabout7. A proof by contrapositive of p q is a direct proof of its contrapositive. For example, the contrapositive of the theorem if it rains then there are clouds in the sky. Simon singh a proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true.
We will show that if n is an integer and n is even then is even. Using a fitchstyle natural deduction proof editor and checker associated with forall x. Notice that we are not trying to show that p infers q as that would be the same as to say q infers p. The contrapositive is certainly true because the entire province of bc is a part of canada.
This follows because the product of two even numbers is even. In fact, the contrapositive is true because the original statement is true. 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. Here are some examples of conjunction, disjunction and negation.
Contrapositivesupposen j 12,sothereisanintegerc forwhich 12. Jun 17, 2016 proof by contradiction examples part 5 david metzler. Still, there seems to be no way to avoid proof by contradiction. The underlying reasoning is that since a conditional statement is logically equivalent to its contrapositive, if the contrapositive is true, then the statement must also be true. Contrapositive proof example proposition suppose n 2z. How do you know when to use proof by contradiction or contrapositive. Preface inwriting this book i have been motivated by the desire to create a highqualitytextbookthatcostsalmostnothing.
Proving an existential statement without a domain using the method of constructive proof. Shorser the contrapositive of the statement \a b i. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs. Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements. Notes on proof by contrapositive and proof by contradiction. Chapter 5 contrapositive proof chapter 6 proof by contradiction chapter 7. Chapter 9 contrapositiveproof wenowexamineanalternativetodirectproofcalledcontrapositive proof. The basic concept is that proof by contrapositive relies on the. We look at another formal mathematical proof method for use in number theory. This video is part of a discrete math course taught at the university of cinc. I figured the best way to prove that is the contrapositive, i got this far. In this method of proof, there is no contradiction to be found. The contrapositive of p q is the formula q p, and it is equivalent.
It is very important to realize that these two implications are not logically equivalent. Use proof by contradiction or contrapositive proof, as appropriate, for each of the following. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. If p leads to a contradiction, then p cant be true, and therefore p must be 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. Logical equivalence guarantees that this is a valid proof method. The negation of a statement simply involves the insertion of the word not at the proper part of the statement. 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 of the proposition of the form x px is called an existence proof sometimes, we can find an element s, called a witness, such that ps is true this type of existence proof is constructive sometimes, we may have nonconstructive existence proof, where we do not find the witness 20. 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. Proof by contrapositive o remember that the converse is not always true. Study at advanced higher maths level will provide excellent preparation for your studies when at university. O the contrapositive is similar to the converse, but is always true. Cs 70 discrete mathematics and probability theory fall.
I then go into some discussion of the distinction between the contrapositive of a statement and its converse. Based on the assumption that p is not true, conclude something impossible. From rstorder logic we know that the implication p q is equivalent. The statement \a implies b can be written symbolically. Suppose a 2z, if a2 is not divisible by 4, then a is odd. Pdf book of proof third edition david long academia. 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.
Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. A direct proof, or even a proof of the contrapositive, may seem more satisfying. Likedirectproof,thetechniqueofcontrapositiveproofis. Oct 11, 2014 writing a proof, trivial and vacuous proofs, direct proofs, proof by contrapositive, proof by cases, proof evaluations slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Proof by contrapositive california state university, fresno. The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false. Sometimes proving the contrapositive of a statement is easier than proving the. In line 2, i assume q and so start a subproof which is indented according to fitch notation. So this is a valuable technique which you should use sparingly. We look at an indirect proof technique, proof by contraposition. Proofs using contrapositive and contradiction methods. Then, the book moves on to standard proof techniques. Converses, contrapositives and proof by the contrapositive. Discussion the rst two methods of proof, the \trivial proof and the \vacuous proof are certainly the easiest when they work.
A proof by contraposition contrapositive is a direct proof of the contrapositive of a statement. What are the converse, contrapositive, and inverse. The statement \a implies b can be written symbolically as \a b. The proofwriting workbook william paterson university. We use informal proofs to illustrate different methods of proving theorems cs 441 discrete mathematics for cs m. We all like to assert things, and few of us like to say things that turn out to be false. Cs 70 discrete mathematics and probability theory fall 2009 satish rao, david tse note 2 proofs intuitively, the concept of proof should already be familiar. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. The contrapositive method is a direct proof of q p. To prove that p is true, assume that p is not true. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. In this post i will cover the third method for proving theorems. Chapter 6 proof by contradiction mcgill university.
The proves the contrapositive of the original proposition. That is, the arbitrary x is part of the antecedent. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. I so if a2 is not divisible by 4 is true, then a is odd is also true. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. Basic form of a proof of an existential statement using the method of constructive proof. Here is another approach to writing a proof of theorem i. Proof by contradiction a proof by contradiction is a proof that works as follows. This document models those four di erent approaches by proving the same proposition four times over using each fundamental method. The basic concept is that proof by contrapositive relies on the fact that p.
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