The basic concept is that proof by contrapositive relies on the. This video is part of a discrete math course taught at the university of cinc. The basic concept is that proof by contrapositive relies on the fact that p. If p leads to a contradiction, then p cant be true, and therefore p must be true. I remember in first year my professor showed that p. In fact, the contrapositive is true because the original statement is true. Chapter 5 contrapositive proof e now examine an alternative to direct proof called contrapositive proof. In other words, we want to be able to read a conditional statement if p then q and immediately \see the contrapositive. In line 3, in order to ultimately arrive at a contradiction, i. The equivalence between a statement and its contrapositive is so obvious for an expert that, usually, he does not need any explanation. To prove a statement of the form \if a, then b, do the following. Then, the book moves on to standard proof techniques. Notice that we did not alert the reader that we are doing a proof of the contrapositive.
The primary reason is that an entire method of proof, called proof by contraposition, is based upon. A proof by contraposition contrapositive is a direct proof of the contrapositive of a statement. Logic and proof converse, inverse, and contrapositive. O the contrapositive is similar to the converse, but is always true. By saying that the two propositions are equivalent we mean that. Dec 24, 2014 we look at an indirect proof technique, proof by contraposition.
The second proposition is called the contrapositive of the rst proposition. Using a fitchstyle natural deduction proof editor and checker associated with forall x. Set theory proof with contrapositive stack exchange. 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. Undergraduate students understanding of the contraposition. Contrapositive proof example proposition suppose n 2z. Proof by contradiction a proof by contradiction is a proof that works as follows. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. Proof by induction o there is a very systematic way to prove this. 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.
There is a connection between contrapositive proofs and proof by contradiction. The statement \a implies b can be written symbolically as \a b. 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. Box842014 virginiacommonwealthuniversity richmond,virginia,23284 bookofproof. Based on the assumption that p is not true, conclude something impossible. Pdf a statement, the contrapositive and the inverse. Through stepbystep worked solutions to exam questions available in the online study pack we cover everything you need to know about proof by contrapositive to pass your final exam as well as students studying advanced higher mathematics, the resources will benefit young adults studying alevel mathematics and undergraduates who need a little extra help. But the initial assumption does reveal that we are doing a proof by contrapositive. Lets now prove the equivalence of a compound statement that will be useful when studying methods of proof. On the other hand, proof by contradiction relies on the simple fact.
In mathematic logic, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive. So, to prove if p, then q by the method of contrapositive means to prove if. Whenever a conditional statement is true, its contrapositive is also true and vice versa. Our goal is to get to the point where we can do the contrapositive mentally. Pdf the equivalence between a statement and its contrapositive is so obvious for an expert that, usually, he does not need any. We are showing that if q is false then p must be false. More precisely, proofs by contrapositive are a proper subset of proofs by contradiction every proof by contraposition can be reformulated as a proof by contradiction, as you correctly noticed. It is very important to realize that these two implications are not logically equivalent. For instance, suppose we want to prove if mathamath, then mathbmath. 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. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Contrapositive examples the infinite series module. P is true, then deduce new statements until you have deduced some statement c and its negation.
The contrapositive of the statement has its antecedent and consequent inverted and flipped. The easiest proof i know of using the method of contraposition and possibly the nicest example of this technique is the proof of the lemma we stated in section 1. Calgary remix, i can proceed as follows line 1 is the premise. Chapters 4 through 6 are concerned with three main techniques used for proving theorems. O proof by deduction o proof by contrapositive o proof by contradiction. The book starts with the basics of set theory, logic and truth tables, and counting.
Thanks virmaior, the thing is with fitch, proving a contradiction is not that simple, putting two contradictory premises following each other doesnt suffice. It seems that the question was designed not to accept proofs by contradiction. 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 proves the contrapositive of the original proposition. That is, the arbitrary x is part of the antecedent. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note.
When an element possesses one of two or more properties, it can be useful to divide the proof into parts called cases, i. It can be modified to only focus on converse, inverse, contrapositive or just biconditionals. This site is like a library, use search box in the widget to get ebook that you want. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause. This lesson targets the converse, inverse, contrapositive of a conditional and then introduces biconditionals in the study of logic. Proof by contrapositive california state university, fresno.
The contrapositive method is a direct proof of q p. Notes on proof by contrapositive and proof by contradiction. Please let me know if you use book of proof in your classes and i will update the adoptions list. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. The logical implication if p then q has the truth table p q pmath\rightarrowmathq f f t f t t t f f t t t in other words, if the premise p is false, or if both the premise and. Since proof by contrapositive involves negating certain logical statements, one has to be careful. If not b then not a o if it is raining, the sidewalk is wet. The contrapositive is certainly true because the entire province of bc is a part of canada. Click download or read online button to get a transition to mathematics with proofs book now. So this is a valuable technique which you should use sparingly. Moveable gray and white boxes reveal key information.
Contents preface vii introduction viii i fundamentals 1. Propositional statements direct proof splash at berkeley. To prove that p is true, assume that p is not true. Metamath a computer language for mathematical proofs. Like direct proof, the technique of contrapositive proof is used to prove conditional statements of the form if p, then q. In this method of proof, there is no contradiction to be found. We sometimes prove a theorem by a series of lemmas.
Similarly, a statements converse and its inverse are always either both true or both false. Jul 18, 2011 i ran into some difficulties trying to show prove the contrapositive law cpl. 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. I tried to do a proof by contradiction like you did, but it was not accepted. Since q2 is an integer and p2 2q2, we have that p2 is even. For instance, if it rains, then they cancel school. Assume q and then use the rules of inference, axioms, definitions. Book of proof is available for free download at the authors website.
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. Obviously i cant make use of the cpl within its own proof. This uses the fact that for every two statements p and q, the implication p q and its contrapositive are logically equivalent. Rather our aim is to show, usually through a direct argument, that the contrapositive statement is true. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs. A transition to mathematics with proofs download ebook. The contrapositive proof goes the opposite way of the direct proof.
I so a divides any multiple of b i thus a divides bc, c an integer. 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 method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive. In the contrapositive proof we are turning it around and. Click download or read online button to get 100 mathematical proof book now. 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. Proofs and mathematical reasoning university of birmingham. It is often equivalent to proof by contrapositive, though it is subtly different see the examples. 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. Discrete mathematics proof by contraposition youtube. Preface inwriting this book i have been motivated by the desire to create a highqualitytextbookthatcostsalmostnothing. Proof by contrapositive o remember that the converse is not always true.
P by showing that the truth tables for both statements were the same for all possible truth values of p and q. Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. The creative commons license allows you to freely use or share the books pdf, in full or in part, provided you acknowledge it as the authors work. We look at an indirect proof technique, proof by contraposition. By the closure property, we know b is an integer, so we see that 3jn2. What is an easy statement to prove by direct, contrapositive. Shorser the contrapositive of the statement \a b i. The two approaches are not equivalent, even if they are quite similar and related.
There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. 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. I attempted to show 1 independently before realizing that simply saying obviously, since a is a subset of b might be the best way to say it, though not the most formalrigorous. Converses, contrapositives and proof by the contrapositive. Proofs using contrapositive and contradiction methods.