Proof by induction principle of mathematical induction. For all integers mand n, if the product of mand nis even, then mis even or nis even. 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. This is the simplest and easiest method of proof available to us. On my journey to improve my mathematical rigour i have covered direct proofs and 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. The statement \a implies b can be written symbolically. Then x2 xy 2k2 2ky 22k2 ky is even since 2k2 ky 2z. By saying that the two propositions are equivalent we mean that. Discrete mathematics proof by contraposition youtube. To that end, assume x and y are not both odd, meaning that either x is even or y is even. Proof by contradiction versus proof by contraposition. We prove the implication p q indirectly by proving the contrapositive. The method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive.
The primary reason is that an entire method of proof, called proof by contraposition, is based upon. Chapter 17 proof by contradiction university of illinois. In fact proofs by contradiction are more general than indirect. Mathematicalwriting 107 proposition suppose a, b2z andn n. Rather our aim is to show, usually through a direct argument, that the contrapositive statement is true. From rstorder logic we know that the implication p q is equivalent to. 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.
In line 2, i assume q and so start a subproof which is indented according to fitch notation. The basic concept is that proof by contrapositive relies on the. So this is a valuable technique which you should use sparingly. So, to prove if p, then q by the method of contrapositive means to prove if. That is, the arbitrary x is part of the antecedent. Notice that we are not trying to show that p infers q as that would be the same as to say q infers p. In a constructive proof one attempts to demonstrate p q directly. 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.
There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. Basic proof techniques washington university in st. This part of the paper explores the differences and similarities that exist between proof by contraposition and proof by contradiction. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. The literature refers to both methods as indirect methods of proof. The notion of proof by contradiction is often confused with the indirect method for proving an implication we discussed earlier. Mathematical proofmethods of proofproof by contrapositive. By the closure property, we know b is an integer, so we see that 3jn2. In the contrapositive proof we are turning it around and. It is very important to realize that these two implications are not logically equivalent. For any set a, the empty set is an element of the power set of a.
In this post i will cover the third method for proving theorems. Logic and proof converse, inverse, and contrapositive. 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. Study at advanced higher maths level will provide excellent preparation for your studies when at university. If p leads to a contradiction, then p cant be true, and therefore p must be true. To prove that p is true, assume that p is not true. Our goal is to get to the point where we can do the contrapositive mentally. Whenever a conditional statement is true, its contrapositive is also true and vice versa.
However, today we want try another approach that works well here and in other important cases where a contrapositive proof may not. The second proposition is called the contrapositive of the rst proposition. Some problems have structures that make it more obvious to try induction or contradiction. 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. Shorser the contrapositive of the statement \a b i. Use the method of contrapositive proof to prove the following. We prove the proposition p by contradiction by proving an implication of the form.
Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Calgary remix, i can proceed as follows line 1 is the premise. In this method of proof, there is no contradiction to be found. Contrapositive proof example proposition suppose n 2z. Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements. Proof by contradiction a proof by contradiction is a proof that works as follows. We start with the language of propositional logic, where the rules for proofs are very straightforward. Converses, contrapositives and proof by the contrapositive. Using proof by contradiction vs proof of the contrapositive. In that proof we needed to show that a statement p. Discrete mathematics proof by contraposition thetrevtutor.
Based on the assumption that p is not true, conclude something impossible. But there are proofs of implications by contradiction that cannot be directly rephrased into proofs by contraposition. Contrapositive supposenj12,sothereisaninteger c forwhich 12. 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 purpose of this technique is often to get a starting situation to work with.
Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. Since q2 is an integer and p2 2q2, we have that p2 is even. Proof by contraposition often times, in order to show p. We are showing that if q is false then p must be false. A proof by contradiction is a proof that works as follows. In mathematic logic, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive. Using a fitchstyle natural deduction proof editor and checker associated with forall x. To show that pn holds for all natural numbers n, it su ces to establish the following. These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas. 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. Copious examples of proofs 19 rewrite it in each of the three forms and prove each. Contrapositivesupposen j 12,sothereisanintegerc forwhich 12. The empty set is a subset of a, hence it is an element of the power set of a.
Please let me know if you use book of proof in your classes and i will update the adoptions list. Mat231 transition to higher math proof by contradiction fall 2014 3 12. Proof by contrapositive o remember that the converse is not always true. In other words, we want to be able to read a conditional statement if p then q and immediately \see the contrapositive.
Programs are notoriously buggy, and theres a growing community of researchers and practitioners trying to. For a computer scientist, some of the most important things to prove are the correctness of programs and systemswhether a program or system does what its supposed to. Notes on proof by contrapositive and proof by contradiction. If x is even, then x 2k for some k 2z, so x2y2 2y 2k2y2 2y 4k2y2 2y 22k2y2 2y.
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. To prove a statement of the form \if a, then b, do the following. Prove that if x2 xy is odd, then x is odd and y is even. The contrapositive of the statement has its antecedent and consequent inverted and flipped. 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 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. The creative commons license allows you to freely use or share the books pdf, in full or in part. The proof began with the assumption that p was false, that is that. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause. This proof method is used when, in order to prove that px. This textbook is designed to help students acquire this essential skill, by developing a working knowledge of. Proof by contradiction can be applied to a much broader class of statements than proof by contraposition, which only works for implications.
Contrapositivesupposenj12,sothereisaninteger c forwhich 12. The proves the contrapositive of the original proposition. Four basic proof techniques used in mathematics youtube. O the contrapositive is similar to the converse, but is always true. The three forms are 1 direct if n2 is even, n is even. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. Similarly, a statements converse and its inverse are always either both true or both false. The basic concept is that proof by contrapositive relies on the fact that 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.
Some universities may require you to gain a continue reading. Then, the book moves on to standard proof techniques. Proofs using contrapositive and contradiction methods. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. Dec 24, 2014 we look at an indirect proof technique, proof by contraposition. Since the negation of conclusion implies the negation of hypothesis. If not b then not a o if it is raining, the sidewalk is wet. Sep 25, 2017 i this video i prove the statement the sum of two consecutive numbers is odd using direct proof, proof by contradiction, proof by induction and proof by contrapositive. The contrapositive method is a direct proof of q p. The contrapositive proof goes the opposite way of the direct proof. We like to demonstrate that pn is true for all n 2n.
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