Proof by contrapositive pdf free

It is often equivalent to proof by contrapositive, though it is subtly different see the examples. A primer for logic and proof appalachian state university. Logic and proof converse, inverse, and contrapositive. We look at another formal mathematical proof method for use in number theory. Please let me know if you use book of proof in your classes and i will update the adoptions list. From rstorder logic we know that the implication p q is equivalent to. If that is not possible, then try proof by contrapositive. It can be modified to only focus on converse, inverse, contrapositive or just biconditionals. The primary reason is that an entire method of proof, called proof by contraposition, is based upon. 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.

For instance, if it rains, then they cancel school. Proof by contraposition is a method of proof which is not a method all its. Proof by contradiction a proof by contradiction is a proof that works as follows. These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas. Moveable gray and white boxes reveal key information.

I figured the best way to prove that is the contrapositive, i. To prove a statement of the form \if a, then b, do the following. Pdf a statement, the contrapositive and the inverse. Set theory proof with contrapositive mathematics stack. The book is available on my web page for free, and the paperback versions.

Basic proof techniques washington university in st. Contrapositive proof example proposition suppose n 2z. Chapter 6 proof by contradiction mcgill university. So this is a valuable technique which you should use sparingly. Based on the assumption that p is not true, conclude something impossible. In that proof we needed to show that a statement p. 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. Study at advanced higher maths level will provide excellent preparation for your studies when at university. Notes on proof by contrapositive and proof by contradiction. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause.

Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements. Some universities may require you to gain a continue reading. The basic concept is that proof by contrapositive relies on the fact that p. Compare your proof with the proof by contrapositive in task 3. So, to prove if p, then q by the method of contrapositive means to prove if. Since q2 is an integer and p2 2q2, we have that p2 is even.

Calgary remix, i can proceed as follows line 1 is the premise. Book of proof is available for free download at the authors website. By saying that the two propositions are equivalent we mean that. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive. Similarly, a statements converse and its inverse are always either both true or both false. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. On my journey to improve my mathematical rigour i have covered direct proofs and proof by contradiction. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. That is, the arbitrary x is part of the antecedent. The proof began with the assumption that p was false, that is that.

Proofs and mathematical reasoning university of birmingham. Since proof by contrapositive involves negating certain logical statements, one has to be careful. We simplify the proof by only considering inputfree programs which. To prove that p is true, assume that p is not true.

Whenever a conditional statement is true, its contrapositive is also true and vice versa. 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. Sometimes the contrapositive of a formula is easier to prove than the original formula. A very common mistake is to use proof by contradiction when a proof by contrapositive would be much more clear. By the closure property, we know b is an integer, so we see that 3jn2. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is 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. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. On the other hand, proof by contradiction relies on the simple fact. 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. This lesson targets the converse, inverse, contrapositive of a conditional and then introduces biconditionals in the study of logic.

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. We discuss the difference between contrapositive and. The contrapositive of the statement has its antecedent and consequent inverted and flipped. For all real numbers, if eqr2 eq is irrational, then r is irrational. Shorser the contrapositive of the statement \a b i. 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. Contents preface vii introduction viii i fundamentals 1.

In this post i will cover the third method for proving theorems. In other words, we want to be able to read a conditional statement if p then q and immediately \see the contrapositive. Converses, contrapositives and proof by the contrapositive. In line 3, in order to ultimately arrive at a contradiction, i. The statement \a implies b can be written symbolically.

Contrapositive definition of contrapositive by the free. Discrete mathematics proof by contraposition youtube. It is very important to realize that these two implications are not logically equivalent. If you like what you see, feel free to subscribe and follow me for updates.

Using a fitchstyle natural deduction proof editor and checker associated with forall x. The second proposition is called the contrapositive of the rst proposition. The proves the contrapositive of the original proposition. 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. Then, the book moves on to standard proof techniques. For instance, suppose we want to prove if mathamath, then mathbmath.