Chapter 9 contrapositiveproof wenowexamineanalternativetodirectproofcalledcontrapositive proof. Contradiction and contraposition carleton university. The proof by contraposition method makes use of the equivalence p q q p to show that the conditional statement p q is true, we first assume q is true, and use. Since it is a rare occasion when we are able to get by with one of these two methods of proof, we turn to some we are more likely to need. Therefore, we have proven the result by contraposition. Though the proofs are of equal length, you may feel that the contrapositive proof. Calgary remix, i can proceed as follows line 1 is the premise. Proof by contradiction relies on the simple fact that if the given theorem p is true, then.
Discrete mathematics proof by contraposition youtube. Prove the following statement by contraposition if a product of two positive real numbers is greater than 100, then at. 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. The method of contradiction is an example of an indirect proof. The purpose in presenting them, however, is not to surprise you with new mathematical facts, but. For our next example, consider the following proposition concerning an integer x. Chapter 6 proof by contradiction mcgill university. Study at advanced higher maths level will provide excellent preparation for your studies when at university. You are probably familiar with at least a little canadian geography, and know that british columbia bc is a geographical area within canada. In another sense this method is indirect because a proof by contraposition can usually be recast as a proof by contradiction fairly easily. There is an indication of a considerable simpli cation of my forcing treatment, by. Proof by contrapositive california state university, fresno. Likedirectproof,thetechniqueofcontrapositiveproofis. The difference is that in a proof by contradiction, we have the extra assumption \pa\ to work with and in certain situations, it provides us with a more convenient way of arriving at a.
Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. Homework statement a prove that if n is an integer and n 2 is a multiple of 3, then n is a multiple of 3. A proof by contraposition contrapositive is a direct proof of the contrapositive of a statement. In line 3, in order to ultimately arrive at a contradiction, i. Using a fitchstyle natural deduction proof editor and checker associated with forall x. Mathematical writing the following list comes from our text. Mat231 transition to higher math contrapositive proof fall 2014.
But there are proofs of implications by contradiction that cannot be directly rephrased into proofs by contraposition. Direct proofs proof by contraposition math berkeley. This is apowerful prooftechnique that can be extremely useful in the right circumstances. Many of the statements we prove have the form p q which, when negated, has the form p. A proof by contradiction induction cornell university. In one sense this proof technique isnt really all that indirect. Proof by contradiction often works well in proving statements of the form. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. Learn the definition of this method and observe how it is applied to proving a statements truth value. Proof by contraposition can be an e ective approach when a traditional direct proof is tricky, or it can be a di erent way to think about the substance of a problem.
The proof of independence of ch by forcing is added. Well need this method in chapter 20, when we cover the topic of uncountability. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. We look at an indirect proof technique, proof by contraposition.
I hadnt realized that i hadnt written it in the notes. Proofs using contrapositive and contradiction methods. In a proof by contraposition both the starting point and the aim are clear. A proof by contraposition can be converted into a proof by contradiction by adding the assumption that the premise is true and then showing that there is a contradiction. Notes on proof by contrapositive and proof by contradiction. Logic and proof converse, inverse, and contrapositive. Prove, using contradiction, that at least one student must have received at least k marks in the exam. 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 this case, one might wonder why we have a proof by contradiction instead of a proof by contraposition since we end up with \\neg pa\ anyway. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. Contrapositive examples the infinite series module.
The sum of two even numbers is not always even that would mean that there are two even numbers out there in the world somewhere thatll give us an odd number when we add them. The proves the contrapositive of the original proposition, and so completes the proof. Proof by contradiction can be applied to a much broader class of statements than proof by contraposition, which only works for implications. Shorser the contrapositive of the statement \a b i.
In most of the following examples the underlying \theorem may be a fact that is well known to you. Whenever a conditional statement is true, its contrapositive is also true and vice versa. For instance, suppose we want to prove if mathamath, then mathbmath. It is important to be extremely pedantic when interpreting a contraposition. From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. Proof by contradiction versus proof by contraposition this part of the paper explores the differences and similarities that exist between proof by contraposition and proof by contradiction. Given integers a and b and an n 2n, we say that a and b are congruent modulo n if nja b.
Moreover, i have found that these logical exercises make me a much more e ective debugger of. Some universities may require you to gain a continue reading. One of the most famous proofs of all time ironically, one that is incorrectly cited as an example of proof by contradiction1 is a proof by contraposition. In a direct proof you start with the hypothesis of a statement. Sometimes youll be asked to prove a claim thats not in a good form for a direct proof. To prove that p is true by contrapositive, we prove the following statement. The reason is that the proof setup involves assuming x,px, which as we know from section 2. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. It is important to be extremely pedantic when interpreting a. We obtain the desired conclusion in both cases, so the original statement is true. 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.
In 1891, georg cantor assumed that there is a set t that contains every infinitelengt. 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. Suppose a 2z, if a2 is not divisible by 4, then a is odd. The two approaches are not equivalent, even if they are quite similar and related. To prove that p q is false, find an example of where p is true. Proof by contraposition examples kent state university.
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. Existence proof examples show that there is a positive integer that can be written as the sum of cubes of positive integers. 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. One way to get started is to do a few more examples. This is because it is easier to transforminformationabout xintoinformationabout7. To illustrate, below is an example of a proof by contraposition. Often proof by contradiction has the form proposition p q. Proofs by contraposition and contradiction physics forums. To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. The literature refers to both methods as indirect methods of proof. The primary reason is that an entire method of proof, called proof by contraposition, is based upon exactly this.
However, contradiction proofs tend to be less convincing and harder to write than. The statement \a implies b can be written symbolically. 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. 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. For our next example, consider the following proposition concerninganintegerx.
One of several different ways to prove a statement in mathematics is proof by contradiction. This video is part of a discrete math course taught at the university of cinc. Examples of congruence 7 3 mod 2 since 7 3 4 is a multiple of 2. P is true, and often that is enough to produce a contradiction. Squaring both sides, we get 2 a2b2 thus, a2 2b2, so a2 is even. Chapter 17 proof by contradiction this chapter covers proofby contradiction. Think of this as \a is equivalent to b pssst, as long as we are using modulo n. Proof by contradiction this is an example of proof by contradiction.
Moreover, i have found that these logical exercises make me a much more e ective debugger of code that involves conditional statements. In this post i will cover the third method for proving theorems. This proof method is applied when the negation of the theorem statement is easier to be shown to lead to an absurd not true situation than proving the original theorem statement using a direct proof. In this one, a direct proof would be awkward and quite di cult, so contrapositive is the way to go. Simon singh a proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true.
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