Pdf the equivalence between a statement and its contrapositive is so obvious for an expert that, usually, he does. Converses, contrapositives and proof by the contrapositive. Since statement a follows from b, and b follows from a, these two statements are equivalent. 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. Shorser the contrapositive of the statement \a b i. The basic concept is that proof by contrapositive relies on the. Set theory proof with contrapositive mathematics stack. Notes on proof by contrapositive and proof by contradiction. 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.
Since proof by contrapositive involves negating certain logical statements, one has to be careful. The statement \a implies b can be written symbolically. Prove that every amount of postage of 12 cents or more can be formed using just 4cent and 5cent stamps. Mathematical proofs a transition to advanced mathematics. Sep 10, 2012 this video describes proof by contraposition, a method of proving a conditional statement by constructing a direct proof of its contrapositive.
So, to prove if p, then q by the method of contrapositive means to prove if. Pdf a statement, the contrapositive and the inverse. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. Sep 25, 2014 this video demonstrates a proof by contrapositive. Already, 60% of 2018 has been sunspot free, and the minimum wont happen until late 2019 at the earliest, and possibly not until 2020. In the case of negation of a statement the logic involved in it get completely reversed. The contrapositive of the statement has its antecedent and consequent inverted and flipped. Before we prove this, lets look at some examples and see if we can see a pattern.
Jul, 2011 once i finished up the post on contrapositive proofs i spend the better part of an hour feeling i wasnt quite finished with the topic. What is the difference between contrapositive and negation. In mathematic logic, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive. Contraposition the law of traditional logic that holds that if from a certain proposition a there follows the proposition b, then from the negation of the proposition b follows the negation of the proposition a or in the language of conditional propositions. That is, write the rst sentence or two of the proof that is, the assumptions, and the last sentence of the proof that is, the conclusion. Presentation slides in pdf and latex formats have been created to accompany every chapter. What is an easy statement to prove by direct, contrapositive. 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.
By the closure property, we know b is an integer, so we see that 3jn2. If ab 0 and bc contrapositive of the given statement. But, from the parity property, we know that an integer is not odd if, and only if, it is even. Mathematicalwriting 107 proposition suppose a, b2z andn n. Contents preface vii introduction viii i fundamentals 1. 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. Try not to think about this particular point too much.
Likedirectproof,thetechniqueofcontrapositiveproofis. 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. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Proof by contrapositive direct proof if a, then b note a. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. Setting up proof by contrapositive katherine stange, spring 2018 for each theorem, set up the form of a proof by contrapositive. Chapter 17 proof by contradiction university of illinois. If a number is not even, then it is not divisible by 2. The book is available on my web page for free, and the paperback. The method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive. I figured the best way to prove that is the contrapositive, i got this far. A proof is a method of establishing the truthfulness of an implication. The proves the contrapositive of the original proposition. The three forms are 1 direct if n2 is even, n is even.
Contrapositivesupposen j 12,sothereisanintegerc forwhich 12. If p leads to a contradiction, then p cant be true, and therefore p must be true. So this is a valuable technique which you should use sparingly. To start viewing messages, select the forum that you want to visit from the selection below. Contrapositive examples the infinite series module. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. This video describes proof by contraposition, a method of proving a conditional statement by constructing a direct proof of its contrapositive. It is available for download to qualified instructors from the. 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. So, you had seen popular proof techniques such as direct proof and contrapositive proof. It focuses on catherine, a young woman who has spent years caring for her.
Contrapositive supposenj12,sothereisaninteger c forwhich 12. This video is part of a discrete math course taught at the university of cinc. It is often equivalent to proof by contrapositive, though it is subtly different see the examples. One of the most acclaimed plays of the 19992000 season, proof is a work that explores the unknowability of love as much as it does the mysteries of science. Proofs and mathematical reasoning university of birmingham. Heres a site with a boggling number of scans of old radio, tv, amateur, and swl publications, books and magazines both, all of them apparently legally downloadable for free. Proof contrapositive contrapositive statement if n is not. In fact, the contrapositive is true because the original statement is true. That is, for all integers n, if n is not odd, then n 2 is not odd. The contrapositive is certainly true because the entire province of bc is a part of canada.
Contrapositive proof example proposition suppose n 2z. It is free math help boards we are an online community that gives free mathematics help any time of the day about any problem, no matter what the level. Start studying geometry converse, inverse, and contrapositive. For all integers n, if n is even, then n 2 is even. One of the most acclaimed plays of the 19992000 season, proof is a work that explores the unknowability of love as much as it does the. To prove a statement of the form \if a, then b, do the following. Apr 01, 2016 there is a big difference between the contrapositive and negation of a mathematical statement. For instance, the proposition all cats are mammals can be restated as the conditional if something is a cat, then it is a mammal. The statement \a implies b can be written symbolically as \a b.
For example, a total postage of 12 cents uses three 4cent stamps. The book can be downloaded from my web page in pdf format for free, and the print version costs considerably less. Once i finished up the post on contrapositive proofs i spend the better part of an hour feeling i wasnt quite finished with the topic. Mathematical induction is a special way of proving things, most commonly used to established the given statement for all natural numbers. In this post, i would like to touch on another technique in proving which is proof by induction. Mathematical proofmethods of proofproof by contrapositive.
Contrapositive definition of contrapositive by the free. 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. Set theory proof with contrapositive mathematics stack exchange. Therefore, the contrapositive \if a,then b is also true. The basic concept is that proof by contrapositive relies on the fact that p. Proof contrapositive contrapositive statement if n is not even then 1 n 6 1 from cse 20 at university of california, san diego. If no box contains 4 or more balls then it is not true that 40 balls were distributed among 10 boxes. In general, if one of the two statements are true, then the other must be true a proof is beyond what is required for the ism. The contrapositive method is a direct proof of q p.
Copious examples of proofs 19 rewrite it in each of the three forms and prove each. In logic, the contrapositive of a conditional statement is formed by negating both terms and. I figured the best way to prove that is the contrapositive, i. Contrapositive example with sets the infinite series module. There is a big difference between the contrapositive and negation of a mathematical statement.
The contrapositive is logically equivalent to the original statement, so if we prove that the contrpositive is true then the original statement must also be true. A contradiction is a statement which is always false. In other words, the conclusion if a, then b is inferred by constructing a proof of the claim if not. I will assume that x is odd and y is even without loss of generality, since x and y are commutative. Chapter 9 contrapositiveproof wenowexamineanalternativetodirectproofcalledcontrapositive proof. Negating the two propositions, the statement we want to prove has the form. A proof is a valid argument that establishes the truth of a theorem as the conclusion statements in a proof can include the axioms. On the analysis of indirect proofs example 1 let x be an integer. Lets start by looking at a few small values for k and see if both sides of the equation give the same results. Proof is the winner of the 2001 pulitzer prize for drama. 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.
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