Proof by contrapositive pdf

We can also assume x is odd, else we are in case 1. 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. We like to demonstrate that pn is true for all n 2n. 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. Proof by contradiction relies on the simple fact that if the given theorem p is true, then. Study at advanced higher maths level will provide excellent preparation for your studies when at university. The primary reason is that an entire method of proof, called proof by contraposition, is based upon. Four basic proof techniques used in mathematics youtube. The proof began with the assumption that p was false, that is that. The basic concept is that proof by contrapositive relies on the.

By the closure property, we know b is an integer, so we see that 3jn2. The proves the contrapositive of the original proposition. The statement \a implies b can be written symbolically. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. 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. 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. Logic and proof converse, inverse, and contrapositive. We prove the implication p q indirectly by proving the contrapositive. Programs are notoriously buggy, and theres a growing community of researchers and practitioners trying to. Basic proof techniques washington university in st.

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. Similarly, a statements converse and its inverse are always either both true or both false. Since the negation of conclusion implies the negation of hypothesis. Proof by contradiction can be applied to a much broader class of statements than proof by contraposition, which only works for implications. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. Contrapositive proof example proposition suppose n 2z. The creative commons license allows you to freely use or share the books pdf, in full or in part. Prove that if x2 xy is odd, then x is odd and y is even. Contrapositivesupposenj12,sothereisaninteger c forwhich 12. Proof by contrapositive o remember that the converse is not always true.

In order for 4 to divide n2 3, n cannot not be an integer. In the contrapositive proof we are turning it around and. 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. To prove a statement of the form \if a, then b, do the following.

Since proof by contrapositive involves negating certain logical statements, one has to be careful. By saying that the two propositions are equivalent we mean that. Then, the book moves on to standard proof techniques. 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.

Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. This part of the paper explores the differences and similarities that exist between proof by contraposition and proof by contradiction. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause. 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. The purpose of this technique is often to get a starting situation to work with. The notion of proof by contradiction is often confused with the indirect method for proving an implication we discussed earlier. Chapter 17 proof by contradiction university of illinois. The literature refers to both methods as indirect methods of proof.

Dec 24, 2014 we look at an indirect proof technique, proof by contraposition. In fact proofs by contradiction are more general than indirect. Contrapositivesupposen j 12,sothereisanintegerc forwhich 12. So this is a valuable technique which you should use sparingly.

O the contrapositive is similar to the converse, but is always true. Use the method of contrapositive proof to prove the following. These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas. 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. Proof by contradiction versus proof by contraposition. Shorser the contrapositive of the statement \a b i. Copious examples of proofs 19 rewrite it in each of the three forms and prove each. In this method of proof, there is no contradiction to be found. The basic concept is that proof by contrapositive relies on the fact that p. Some problems have structures that make it more obvious to try induction or contradiction. In this post i will cover the third method for proving theorems.

The contrapositive of the statement has its antecedent and consequent inverted and flipped. Using a fitchstyle natural deduction proof editor and checker associated with forall x. Mat231 transition to higher math proof by contradiction fall 2014 3 12. Discrete mathematics proof by contraposition youtube. Then x2 xy 2k2 2ky 22k2 ky is even since 2k2 ky 2z. Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements. Discrete mathematics proof by contraposition thetrevtutor. Contrapositive supposenj12,sothereisaninteger c forwhich 12. Rather our aim is to show, usually through a direct argument, that the contrapositive statement is true. Proof by contraposition often times, in order to show p. However, today we want try another approach that works well here and in other important cases where a contrapositive proof may not. There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. The contrapositive method is a direct proof of q p.

But there are proofs of implications by contradiction that cannot be directly rephrased into proofs by contraposition. This proof method is used when, in order to prove that px. Please let me know if you use book of proof in your classes and i will update the adoptions list. 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. In line 2, i assume q and so start a subproof which is indented according to fitch notation. 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. 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. 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 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. Calgary remix, i can proceed as follows line 1 is the premise. For any set a, the empty set is an element of the power set of a. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. The contrapositive proof goes the opposite way of the direct proof. Mathematical proofmethods of proofproof by contrapositive.

Using proof by contradiction vs proof of the contrapositive. Notes on proof by contrapositive and proof by contradiction. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. We are showing that if q is false then p must be false. The second proposition is called the contrapositive of the rst proposition. For all integers mand n, if the product of mand nis even, then mis even or nis even. Chapter 6 proof by contradiction mcgill university.

We start with the language of propositional logic, where the rules for proofs are very straightforward. The method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive. Based on the assumption that p is not true, conclude something impossible. If x is even, then x 2k for some k 2z, so x2y2 2y 2k2y2 2y 4k2y2 2y 22k2y2 2y. Proof by induction principle of mathematical induction. If p leads to a contradiction, then p cant be true, and therefore p must be true. On my journey to improve my mathematical rigour i have covered direct proofs and proof by contradiction. If not b then not a o if it is raining, the sidewalk is wet. I so a divides any multiple of b i thus a divides bc, c an integer.

To prove that p is true, assume that p is not true. In that proof we needed to show that a statement p. Whenever a conditional statement is true, its contrapositive is also true and vice versa. This is the simplest and easiest method of proof available to us. 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.

We prove the proposition p by contradiction by proving an implication of the form. A proof by contradiction is a proof that works as follows. Converses, contrapositives and proof by the contrapositive. Mathematicalwriting 107 proposition suppose a, b2z andn n.

Our goal is to get to the point where we can do the contrapositive mentally. Proofs using contrapositive and contradiction methods. To show that pn holds for all natural numbers n, it su ces to establish the following. In a constructive proof one attempts to demonstrate p q directly. This textbook is designed to help students acquire this essential skill, by developing a working knowledge of. In other words, we want to be able to read a conditional statement if p then q and immediately \see the contrapositive. That is, the arbitrary x is part of the antecedent.

From rstorder logic we know that the implication p q is equivalent to. 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. The empty set is a subset of a, hence it is an element of the power set of a. Proof by contradiction a proof by contradiction is a proof that works as follows. 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.

Since q2 is an integer and p2 2q2, we have that p2 is even. The three forms are 1 direct if n2 is even, n is even. To that end, assume x and y are not both odd, meaning that either x is even or y is even. It is very important to realize that these two implications are not logically equivalent.