contrapositive proof example Example. 1. Contrapositive. 2. The video shows how these are related. The proof tools of contrapositive and contradiction are introduced. Anything is congruent That's what obtuse means. ) II. 1. This means there is an integer k so that n = 2k. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. (Otherwise, if each set were a subset of the other, the sets would be equal, contrary to our hypothesis. Proof: Assume that x is even. 3. Proof by Contrapositive example Contrapositivehypothesis x is rational definition x= p/qfor integers pand q. If nº is odd, then n is odd. If the Cubs do not win tomorrows game, then they will not win the pennant. Example 1: Prove the following statement by contraposition: If a product of two positive real numbers is greater than 100, then at least one of the number is greater than 10. Follow Contrapositive help understanding these specific examples from Graph Theory. is even, then -is even. Suppose n is even. In other words, one is true if and only if the other is true. If you are struggling trying to prove something directly, one option would be to try a contrapositive proof. For every pair of real numbers 4 and y , if x +5 is irrational, then 4 is irrational or 5 is irrational. If nº is odd, then n is odd. Sentences (i) and (ii) are both statements (only the ﬁrst of which is true). If n and m are negative, then nm is a positive integer. We have shown that ¬(n is odd) → ¬(3n + 2 is odd), thus its contrapositive (3n + 2 is odd) → (n is odd) is A conditional statement is an if-then statement. End of proof: Therefore \(n\) can be written as the sum of consecutive integers. By the closure property, we know b is an integer, so we see that 3jn2. Therefore \(7x+9 = 2b+1\), where b is the integer \(7a+4\). To prove . Example 4: Prove the following statement by contradiction: For all integers n, if n 2 is odd, then n is odd. For a quadratic polynomial f x ax bx c() 2 to have roots its (c) Proof by contrapositive Proof by contrapositive, also known as proof by contraposition is based on the logical equivalence between a conditional sentence and its contrapositive. [We take the negation of the given statement and suppose it to be true. To prove that this implication holds, let us first construct a truth table for the proposition P Q. Try to find a direct proof, and then give up and do the contrapositive: assume , and prove that . If that is not possible, then try proof by contrapositive. Since B is of the form 2k + 1withk = 2m2 ≠ 2m + 1, we conclude that B is odd. 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. –Will show that 3x is rational and therefore not irrational. </p> If you are able to use modus tollens, then the result would follow quickly. Prove: (p →q) ∧(q →s) ⇒(p →s) 1. So, n = 2k for some integer k. A direct proof is difficult, so a proof by contrapositive is preferable. (Contrapositive) Let integer n be given. Assume q is false (¬q is true. Proof by Contrapositive Example • Prove: If n2 is an even integer, then is even. 3. This is because there are some birds, like the penguin, that don’t fly at all. Let the contrapositive of Statement A be Statement B , B: If an element, y, is not in S, then it cannot be in T. The inverse of p !q is ˘p !˘q. Example Prove that if n is an integer such that n 3 + n 2 + n is odd, then n is odd. If and are two integers with opposite parity, then their sum must be odd. 4, namely that for any integer n, if n2is even then nis even. The proof of the infinitude of primes is also presented using both methods. (c) Proof by contrapositive Proof by contrapositive, also known as proof by contraposition is based on the logical equivalence between a conditional sentence and its contrapositive. " This follows logically from our initial The inverse is " If an object is not red, then it does not have color. Example If you are a UNL student you are a cornhusker. End of proof: Therefore \(a^2 + b^2\) is even. For example, the contrapositive of the theorem “if it rains then there are clouds in the sky” is the Example: Prove that if n is an integer and 3n + 2 is odd, then n is odd. 3 Proof by contradiction We end with a description of proof by contradiction. Proof by contrapositive, contradiction, and smallest counterexample. Suppose that and real numbers. Then either x+y≠0 or x-y≠0. 1. the contrapositive was true because the original statement was true the given statement and its contrapositive are equivalent To get a sense of why this would be so, the next example takes a closer look at the contrapositive. Proof: Let x be a real number in the range given, namely x > 1. (Contrapositive) Suppose x is not odd. Here is an example of that using a different proof checker: However, if you have to derive this using introduction and elimination rules, the following might work using a proof provided by the authors of forallx (page 170): This packet will cover "if-then" statements, p and q notation, and conditional statements including contrapositive, inverse, converse, and biconditional. ¬q 1, 2, modus tollens 4. ) 1. ’ Example: proof by contrapositive Prove the following: For every integer n, if -. Follow Contrapositive help understanding these specific examples from Graph Theory. " Proof. 3), but often it is easier and more natural to prove the contrapositive of a sentence. 6. 9x 2U (P(x)). Unwinding Definitions (Getting Started) Constructive Versus Existential Proofs; Counter Examples ; Proof by Exhaustion (Case by Case) What Does "Well Defined" Mean? The Pigeon Hole Principle Examples: Counterexample Prove or disprove: dx+ ye= dxe+ dye. Then 3x−15 = 3(2a)−15 = 6a−15 = 2(?)+1 = 2(3a−8)+1. 3. in some way. Example: Prove by contradiction that if x+y > 5 then either x > 2 or y > 3. Answer: We assume the hypothesis x+y > 5. Then we want to show that x2 6x + 5 is odd. Proof by Contradiction. Cite. The given statement is the contrapositive, not the converse. A conditional statement is logically equivalent to its contrapositive! (This is very useful for proof writing!) The converse of p !q is q !p. : “Let x,y be numbers such that x≠0. We will prove by induction Consider for example: “If ab = 0, then a = 0 or b = 0. Compare proof by contradiction and proof by contrapositive and provide an example of one or the other. Methods of indirect proof: I. Proof. Class notes: Slideshow; Video: Part 1 (describing proof by contrapositive) [16 min] . Logic A proposition derived by negating and permuting the terms of another, equivalent proposition; for example, All not-Y is not-X is the contrapositive Contrapositive - definition of contrapositive by The Free Dictionary A. Thus, domino k fall and remain standing. The negation of "irrational" is simply "not irrational". We would need to find a single example of one of these conditions, any one of which would be a counterexample: A living woman who does not eat food, or; A woman who eats food but who is not alive, or; A nonliving woman who eats food, or; A woman who does not eat food but who is alive That's how you do a proof by contrapositive. 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). Go prove the contrapositive. This also happens in example 3 { the original assertion is true, but the Proof by Contrapositive Proof by Contradiction Special Cases Contrapositive I The contrapositive of the statement \If A, then B" is the statement \If not B, then not A. 4. It makes use of the fact that the statement Proof 7. Thus 3n+ 7 is odd. The converse of a conditional is formed simply by keeping the antecedent and consequent of a conditional in the same place and denying them both. Our proof will attempt to show that this is false. Share. Like to indirect proof examples to identify which type of this. , not odd). Therefore, the contrapositive is if has at least one solution then . Example. Thus your statement of what the contrapositive is is not logically equivalent. For example, the contrapositive of "If it is Sunday, I go to church'' is "If I am not going to church, it is not Sunday. If nº is odd, then n is odd. then the statement "if not B then not A" is also true. So 00:14:41 Prove using proof by contrapositive (Examples #2-4) 00:22:28 What is proof by contradiction? (Examples #5-6) 00:26:44 Show the square root of 2 is irrational using contradiction (Example #7) 00:30:07 Demonstrate by indirect proof (Examples #8-10) 00:33:01 Proof of equivalence (Example #11) 00:35:59 Justify the biconditional statement when written as a contrapositive, but perhaps it is not so obviously false in the original. 1. 3. 2 Contrapositive A contrapositive proof is just a direct proof of the negation. Proof: Suppose x is even. 1. Therefore, instead of proving p ⇒ q, we may prove its contrapositive q ¯ ⇒ p ¯. This latter statement can be proven as follows: suppose that x is not even, then x is odd. For every conditional statement you can write three related statements, the converse, the inverse, and the contrapositive. m 44 4 Reasoning and Proof. The basic concept is that proof by contrapositive relies on the fact that p !q and its contrapositive :q !:p are logically equivalent, thus, if p(x) !q(x) is true for all x then :q(x) !:p(x) is also true for all x, vice versa. This proof method is used when, in or-der to prove that p(x) !q(x) holds for all x, proving that its contrapositive Contraposition is often helpful when an implication has multiple hypotheses, or when the hypothesis specifies multiple objects (perhaps infinitely many). 7). First, translate given statement from informal to formal language: ∀ positive real number r and s, if (r . Therefore, the contrapositive \If A,then B" is also true. However, we can still write down what a counter-example must look like. A proof by contrapositive would look like: Proof: We’ll prove the contrapositive of this statement. Follow steps of Direct Proof to prove not A. Example: Prove that if you pick 22 days from the calendar, at least 4 must fall on A. From rst-order logic we know that the implication P )Q is equivalent to :Q ):P. In Example 2. Do so in such a way as to demonstrate the structure of proof by contrapositive. F. 13 = 2\left ( 6 \right) + 1 13 = 2(6) + 1 or. In this one, a direct proof would be awkward (and quite di cult), so contrapositive is the way to go. ” The contrapositive of the this statement is “If neither a nor b is 0, then ab ≠ 0. Suppose a and b are real numbers. This is false. 3. Doctor Fenton replied: Hi Rahul, I think this is primarily a matter of interpretation. p →q Premise 5. Polak's remarks. If the converse is true, The contrapositive “If the sidewalk is not wet, then it did not rain last night” is a true statement. If we want to prove that A ⇒ B, then we can prove that ¬B ⇒ ¬A. Use the contrapositive the prove that if 49 8 c then c7 2 has no solutions. The only way w is not accepted is if it gets to C. contrapositive :q !:p is true. A. The interval , for example, has no least element. Therefore, the contrapositive is if has at least one solution then . We will attempt to show that 2 is rational. Assume \(n\) is a multiple of 3. Perhaps the most famous example of proof by contradiction is this: 2 is irrational. Direct Proof. If the statement is true, then the contrapositive is also logically true. E. Secondly, we need to know how to write the contrapositive of a statement. > This lesson will present how to determine if a newly written statement is the converse, inverse, or contrapositive of an if-then statement. Contradictive Proof Example Prove the following: No odd integer can be expressed as the sum of three even integers. , that n is even. To illustrate, below is an example of a proof by contraposition. 3. The contrapositive of the above statement is: If x is not even, then x 2 is not even. ) 1. ” !Proof: !Given (contrapositive form): Let !WTS (contrapositive form): … !Conclusion: … 7??? Example! Proof by contrapositive In logic, the contrapositive of a conditional statement is formed by negating both terms and reversing the direction of inference. Prove by contradiction that there is no greatest even integer. To demonstrate (or “prove”) that a statement like this is false, all we need to do is supply an example that shows it is false. Then we have 3n + 2 is odd, and n is even. A contrapositive proof seems more reasonable: assume n is odd and show that n3 +5 is even. First, we formulate the contrapositive: P is so P is 49 8 c . Direct proof. The negation of the statement \There exists x 2R, x2 1 < 0," is the statement \For all x 2R, x2 1 < 0. Contradiction 4. 4. Therefore three is an odd number. Theorem 4. A contradiction occurs {\color {blue}p} \to {\color {red}q} p → q, we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement. Therefore, proving that P → Q is equivalent to proving that ¬ Q →¬ P. Let n 2Z. Proof: Suppose they donÕ t all fall. 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 converse, if n2 is even, then n is even is true by Lemma 2. " Note: As in the example, the contrapositive of any true proposition is also true. 15, which one do you like and why? In general when we prove a theorem of the form \(P \Rightarrow Q\), we do not recommend to start by trying to use proof by contradiction. ) Proof: We will prove the contrapositive; i. Proof Assume that x is even. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining. Then \(7x+9 = 7(2a)+9 = 14a+8+1 = 2(7a+4)+1\). There are two possible cases: , and (we've already assumed ). ] On a further note, it is often the case that the antecedent P can be naturally seen as a negative statement (¬P'). The proof began with the assumption that P was false, that is that ∼P was true, and from this we deduced C∧∼. Since it is an implication, we could use a direct proof: Assume q ¯ is true (hence, assume q is false). Let k > 0 be the lowest numbered domino that remains standing. 3. For example, assuming that the xed set is Z, then the above statement can be written 00:29:17 – Understanding the inverse, contrapositive, and symbol notation; 00:35:33 – Write the statement, converse, inverse, contrapositive, and biconditional statements for each question (Examples #13-14) 00:45:40 – Using geometry postulates to verify statements (Example #15) This logical equivalence is the basis for one of the most important laws of deduction, modus tollens, and for the contrapositive method of proof. For a number to be "not irrational" has 2 cases. Proof: (Direct Proof) Assume that x,y ∈ℝ. Proof: (Contrapositive: If n is even, then 3n + 2 is even) Suppose that the conclusion is false, i. Since 3a−8 ∈ Z, 3x−15 is odd. $\blacksquare$ Let $P$ be the statement "Bob is hungry" and let $Q$ be the statement "Bob will eat lunch". 3. Example Prove: If x+10 is odd, then x is odd Proof (by contrapositive) 1 Math 299 Lecture 21: Proof of the Contrapositive For example, 121 is a perfect square since 121 = 112. This is equivalent contrapositive ¬q → ¬p. As a simple (and arguably artificial) example, compare, for x a real number: 1 (a). Proof: • Assume 3n + 2 is odd and n is even, that is n = 2k, where k an integer. You can solve this problem in the ‘regular way’ if you use some information about prime numbers – but if you do contrapositive, the only thing involved is the distributive property and the definition of even and odd numbers – which contrapositive and proof by contradiction, which seem to cause easily some confusions. By the definition of a rational number, the statement can be made that " If Example. Is l Dillig, CS243: Discrete Structures Mathematical Proof Techniques 13/38 Another This completes the proof. (1) Proof by Contrapositive: In this method, to prove we prove its contrapositive, , instead. The latter implies that n = 2k for some integer k, so that 3n + 2 = 3(2k) + 2 = 2(3k + 1). Suppose x2(y2 2y) is odd. Case 2 : x<0 Case 3: x= 0 3) Proof by using the contrapositive: Fact: A conditional statement and its contrapositive always have the same truth value. Example: Prove that p 2 is irrational. Copyright © Nahid Sultana 2014-2015. e. Part 2: Q )P. Proof: If x= 2c, then x2(y 2 2y) = (2c)2(y 2y) = 4c2(y2 2y). •Theorem: If x is a real number such that 3x irrational then x is irrational. " A conditional statement is an if-then statement. So 3n + 2 is not odd. Proof: To prove that "If A, then B," we will prove the equivalent statement "If (Not B), then (Not A). Suppose ne Z. For example, you could say something like “We will prove the contraposi-tive of this statement, namely, that ” or “By contrapositive; we will instead prove that …” Don't skip this step! It's important for several reasons. The contrapositive is like the contradiction. ) 1. Proof by Contrapositive. This may be the case of a proof by contrapositive which is a "direct proof", possibly accounting for J. “odd” n2 = (2k+1)2 = 4k2 + 4k +1 Therefore, n2 = 2(2k2 + 2k) + 1, which is odd. Purported proofs of the Four Color Theorem continue to stream in. In fact, it is a famous unsolved problem whether there are infinitely many primes that work. But if a/b = √ 2, then a 2 = 2b 2. If it were rational, it would be expressible as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. 1. I Non-mathematical example: \If you are a computer science major, then you are very smart," can be rephrased as 2) Proof by cases – Many proofs divide naturally into cases. Referring to the above example, the contrapositive of \If this gure is a triangle, then it has 104 Proof by Contradiction 6. This works because p!q :q!:p. ÿ When proving a conditional, one can prove the contrapositive statement instead of the original – this is called a contrapositive proof. Example Prove that if n is an integer such that n 3 + n 2 + n is odd, then n is odd. 3. Share. QED 2. Contrapositive proof. Suppose also that nm is not a positive integer. Use the method of proof by contradiction to prove the following statements. (i) One plus one equals two. x = 2k , for some integer k. Proof Variations for Theorem 2 Theorem 2, like most others, can be proven in a number of other ways. € Example 6: Prove that the sum of two odd integers is even. If y= 2c, then x2(y 2 2y) = x2((2c)2 2(2c)) = x(4c2 4c) = 4x2(c2 c). method proof by contrapositive . If you use the contrapositive, you are working with linear independence, which is a set definition with many theorems tied to it, making it much easier to work with. More formally stated. Given a statement p q, the contrapositive is the statement q p, the inverse is p q and the Proof: We assume and prove . 3. Proposition: If x and y are to integers for which x+y is even then x and y have same parity (either both are even or both are odd). A polite signal to any reader of a proof by contradiction is to provide an introductory sentence: One of the best known examples of proof by contradiction is the pro√of that 2 is irrational. e. •Proof : Assume that the statement is false. Proof: The satement will be proven by contraposition. " An object which is blue is not red, and still The converse is " If an object has color, then Contrapositive Proof: Suppose x is even. Compare your proof with the proof by contrapositive in Task 3. " Suppose Not B. 2 In this example, we consider the following sentences. The proof of the irrationality of root two is presented as an example of each. In a proof by contraposition both the starting point and the aim are clear: assume that ¬q is true and show that it logically leads to Australian Senior Mathematics Journal vol . For every conditional statement you can write three related statements, the converse, the inverse, and the contrapositive. The purpose of this technique is often to get a starting situation to work with. If 3jn then n = 3a for some a 2Z. Using the chain of biconditionals: x2 10x+ 25 = 0 ,(x 5)2 = 0 ,x 5 = 0 ,x = 5 we see that :Q ):P is always true. Contrapositive of original statement: If Contrapositive: If Jennifer does not eat food, then Jennifer is not alive. If you can prove that the contrapositive of a statement is true then the original statement must also be true. The video shows how these are related. Direct proof 2. Thus q2 is not divisible by 3. (This method takes advantage of the fact that the contrapositive is logically equivalent to the original implication, which we saw in Section 2. 6. Proof: Suppose a b mod 6. Inverse: If you don't drink Pepsi, then you aren't happy. Start by assuming not B e. 2. 6 Practice with Counter-Examples Please de ne counter-examples for exercises 5{13. 2. If 3x−15 is even, then x is odd. Thus, 3n+2 = 3(2k)+2 = 2(3k+1). 3 Symbolize this statement, taken from the instructions for IRS From 1040, line 10: For each theorem, set up the form of a proof by contrapositive. 1. The set of even integers has no least element. Proof. If x2 6x+ 5 is even, then x is odd. Solution: Use proof by contraposition. Example Theorem: If n is an integer and n2 is even, then n is even. A. If n is odd, then nº is odd. (ii) One plus one equals three. We have shown that if n is an even integer, then n^2 is even. Left hand side, but the circle is synonymous with the theorem proof. Let x;ybe two integers. Any statement that is clearly false is suﬃcient. Solution. (P Q) F. 1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. 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". Therefore, P ,Q. Suppose n = 2k for some k 2Z. e. Example: Write "If the Cubs win the pennant, then they will have won tomorrows game. Counterexamples. { Proof by counterexample: x = 1 2 and y = 1 2 Prove or disprove: \Every positive integer is the sum of two squares of integers" { Proof by counterexample: 3 Prove or disprove: 8x8y(xy x) (over all integers) { Proof by counterexample: x = 1;y = 3;xy = 3; 3 6 1 Greedy Algorithms x^2 = x * x = 0 is an example of something that doesn't have 2 distinct roots. Therefore, proving that P → Q is equivalent to proving that ¬ Q →¬ P. Prove that xand yare odd. Part II: Proof Strategies. ) 1. On the other hand, sentence (iii) is neither true nor false (the truth or falsity depends on the reference The Converse and Contrapositive. T → p. 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). Indirect Proof: Proof by Contraposition Proof by Contraposition: Assume ¬q and show ¬p is true. Find the converse, inverse, and contrapositive. So let’s try the contrapositive. They are compared and contrasted. Now, 5x-7= 5(2 k) -7 = 10 k -7 = 2 (5 k - 4)+1 = 2 l+1, where l=5 k - 4 ∈ ℤ Hence 5x-7 is odd. Thenthereexistsanintegern thatisevenandodd. All major mathematical results you have considered The main (and in this author’s opinion, the only) benefit of a proof by contrapositive is that one can turn such a statement into a constructive one. We already proved this on slides 4 and 5. So you want to know how one deduces X from its contrapositive. Q. 4. 2. 3. 1: Let x ∈ ℤ. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion is true under the assumption that the premisses are true. These two statements are logically equivalent to one another. Explicitly state the contrapositive of what we want to prove. But this is clearly true, since for all , and and are equal. (c) Proof by contrapositive Proof by contrapositive, also known as proof by contraposition is based on the logical equivalence between a conditional sentence and its contrapositive. In this example, the first step in stating the contrapositive of the original instructions to (c) Proof by contrapositive Proof by contrapositive, also known as proof by contraposition is based on the logical equivalence between a conditional sentence and its contrapositive. Share. To prove: For all whole numbers x, if x 2 is even then x is even. Proof of the contrapositive 3. In our example, the contrapositive of "If X is 2 then X is an even number" would read, "If X is NOT an even number Very often, mathematical statements of the form \(\forall x, P(x) \Rightarrow Q(x)\) are established by proving its contrapositive: \(\forall x, eg Q(x) \Rightarrow eg P(x)\). Therefore, we can conclude that Knuth was not a UNL student. 3 Concludethatthestatementtobeprovedistrue. Since one of these integers is even and the other odd, there is no loss of generality to suppose is even and is odd. Chapter 2. 2 Proof by contradiction In proof by contradiction, you assume your statement is not true, and then derive a con-tradiction. Then n = 2x for some x 2Z. 3. This video is part of a Discrete Math course taught at the University of Cinc We look at another formal mathematical proof method for use in number theory: contrapositive proofs. In a proof by contradiction or (Reductio ad Absurdum) we assume the hypotheses and the negation of the conclu-sion, and try to derive a contradiction, i. First, we formulate the contrapositive: • is so is . Since we have proved the contrapositive: ¬(n even) : ¬(n2 even) egy to try is a proof of the contrapositive. Do not write a full proof. Proof: Case 1: x>0,…. Second, it forces you to write out the Examples of Proof by Contraposition I Prove:If n 2 is odd, then n is odd. 58 Proof by Contrapositive: Assume n is an even integer. Q is has no solutions so Q is has at least one solution. For example 2#= would be a ﬁne contradiction, as would be 4|, provided that you could deduce them. Note the following. ~ q → ~ p: If 15 is not a prime number, then 15 is not an odd number. So, to prove "If P, Then Q" by the method of contrapositive means to prove "If Not Q, Then Not P". Proof by Contradiction (Example 1) •Show that if 3n + 2 is an odd integer, then n is odd. The sum of two rational numbers is rational. Use both a direct proof and a proof by contrapositive to show that if n is even, then 3n+ 7 is odd. Proof by contrapositive instead assumes and proves an opposite and inverse argument to the theorem in order to prove the theorem valid: proves p → c by showing that ¬c → ¬p. Suppose ne Z. positive and proof by contradiction. above by contradiction. The implication $P \rightarrow Q$, its contrapositive, converse, and inverse are all listed below: $P \rightarrow Q$: If Bob is hungry then Bob will eat lunch. 1. Then x = 2a, for some integer a. • Then: 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1) Example. Proof by Contraposition Often times, in order to show p!q, it will be easier to prove :q!:p, the contrapositive. Hence, n^2 = 4k^2 = 2 (2k^2) and n^2 is even (i. , we will prove if q is not divisible by 3, then q2 is not divisible by 3. 1 ¬p. '' Any sentence and its contrapositive are logically equivalent (theorem 1. For example, logical statement if A is true, then The contrapositive is the assertion \If Q is false, then P is false". Recall that an implication \(P \imp Q\) is logically equivalent to its contrapositive \( eg Q \imp eg P\text{. 1. In general, your contradiction need not necessarily be of this form. You will find in most cases that proof by contradiction is easier. Call this integer n. and they give us a theorem. This proof must be done by contradiction not by contrapositive. Cahit unveiled his 12-page solution in August 2004, but here is his proof of Lemma 4: “Details of this lemma is left to the reader (see Fig. There is no integer that is both even and odd. 2, we know that if q is not divisible by 3, then q2 1 (mod 3). Mathematical Induction What follows are some simple examples of proofs. Then x = 2m for some integer m. We have shown that if x is not odd, then B is not even, and therefore if B is even, x is odd. Example: 8x 2R(2x = (x+ 1) + (x 1)). e. (In each case, you should also think about how a direct or contrapositive proof would work. What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. Result: If x,y ∈ℝ, then x2/3+(3y2)/4 ≥ xy. , Suppose that (A∪B) 6= B. proof by induction, proof by contradiction) will be covered later. This related implication is usually proved directly. Cite. wish to prove. Saying here to indirect proof by contradiction, because it often leads to beat someone in these angles will not congruent. Note: there is no “if” or “then” clause, and the statement sounds negative. That is, suppose there is an integer n We explain Converse, Inverse, and Contrapositive of an If-Then Statement with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. Start of proof: Let \(n\) be an integer. So we assume and have opposite parity. First, it communicates to the reader what they should expect in the proof: you're not going to prove the original statement, and instead that you're going to prove the contrapositive. Pf: Suppose n is an even integer. is congruent to BC, they give us that. " I A statement and its contrapositive are logically equivalent. In addition to the Law of Contrapositive, indirect proofs often use two other common laws of logic: the Law of Ruling out Possibilities, and the Law of Indirect Reasoning (Modus Tollens). To prove: If x 2 is even, then x is even. A contrapositive proof is just a direct proof of the negation. ” A proof that proves the contrapositive is called a proof by contraposition. Suppose ne Z. We have phrased this method as a chain of implications p)r 1, r 1)r 2, :::, r Proof techniques. Proof: Part 1: P )Q. Basically, instead of proving "p implies q", you say, well what if q were not true, and then you get a contradiction. In some cases, you use this to prove a conditional statement by replacing it with its contrapositive. ) 2 Showthatthissuppositionleadstoalogicalcontradiction. 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. 3 Mathe-matical Writing Contrapositive Proof of Conditional Statements How to prove P )Q: Recall this statement is equivalent to ˘Q )˘P Proposition: If P, then Q. By Theorem 1. logical equivalent Examples Irrationality of the square root of 2. Contrapositive definition is - a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition or theorem and interchanging them. An extension to general roots of natural numbers is made via contrapositive. This is usually a hint that proof by contradiction is the method of choice. Such a proof might look like the following. Example Prove that if n is an integer such that n 3 + n 2 + n is odd, then n is odd. Q. ⇒either (A∪B) * Bor B* (A∪B). (an indirect form of proof). Converse: If you are happy, then you drink Pepsi. Then 2m+1 = 3. There exists an integer k so that n = 2k. Examples: (1) Prove that \if 3n+ 2 is odd, then n is odd". Proof by contrapositive: to prove P )Q, prove the contrapositive ˘Q )˘P. All the implications in Implications can be proven to hold by constructing truth tables and showing that they are always true. Start 1 0 A B 1 C 0 0,1 contrapositive proof, we assume :Q, and then work to deduce that :P follows. Let be a nonempty collection of natural numbers. I Then, n 2= (2 k)2 = 4 k2 = 2(2 k ) I Thus, n 2 is also even. 1 Con-trapositive Proof 5. com Now he wants to prove it using the contrapositive, just as we saw last week. 1. How should we proceed to prove this statement? A direct proof would require that we begin with n3 +5 being odd and conclude that n is even. We discuss the difference between contrapositive and dir Proof by contraposition is a type of proof used in mathematics and is a rule of inference. First sentence(s): Let n;m 2 Z. Example Theorem: For every odd integer n, if n is odd, then n2 is odd. i. Squaring, we have n2 = (3a)2 = 3(3a2) = 3b where b = 3a2. Show that ¬p logically follows 6 Example!Thm. Proof: Supposenot. Then . p ∧ ¬ p. We see that 3n+1=2k, but it’s not clear how that helps. More specifically, the contrapositive of the statement "if A, then B " is "if not B, then not A. Contrapositive proofs can have a di erent avor than direct proofs because one starts with very di erent assumptions in each. , a proposition of the form r∧¬r. Proof. We deﬁned a rational number to be a real number that can be written as a fraction a b The contrapositive of an the implication \A implies B" is \Not B implies not A", written \∼B →∼A". Prove that for any real number x > 1 and any positive integer x, (1 + x)n 1 + nx. It is still a direct proof method. 13 = 2\left ( 7 \right) - 1 13 = 2(7) − 1. Simple inductive proof based on: • Every state has exactly one transition on 1, one transition on 0. statement 2. 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. 1 Proving a universal statement Now, let’s consider how to prove a claim like Proof by Contradiction (Example 1) •Show that if 3n + 2 is an odd integer, then n is odd. Proof by contradiction. Here is a contrapositive proof of the same statement: Proposition Suppose \(x \in \mathbb{Z}\). In that proof we needed to show that a statement P:(a, b∈Z)⇒(2 −4 #=2) was true. Therefore B is not even. implication is also true. A good example is linear dependence, which only means that a set is not linearly independent. Here is another approach to writing a proof of Theorem I. contrapositive from cognitive and dida ctical points of view. You will find in most cases that proof by contradiction is easier. 2. Domino k-1 " 0 did fall, but k-1 will knock over domino k. 10/10/2014 17. Here’s another example. , 11 works as well). The contrapositive of a conditional statement of the form p !q is: If ˘q !˘p. If n2 + 6n+ 5 is even, then n is odd. apply definition of odd number from definition – found k such that n2 = 2k + 1 Proof by contrapositive Proof. g. We will prove the contrapositive, (A∪B) 6= B | {z } ∼q ⇒A* B | {z } ∼p. Example: proof by contrapositive Prove the following. Assume and are positive and that BC BC B CÞ For example, in the proofs in Examples 1 and 2, we introduced variables and speci ed that these variables represented integers. Then … (( insert sequence of logical arguments here; probably will involve the definitions of the objects involved )) Therefore, Not A. Then n = 2k for some integer k. That is, write the rst sentence (or two) of the proof (that is, the assumptions), and the last sentence of the proof (that is, the conclusion). 3 Contradiction A proof by contradiction is considered an indirect proof. The statement, “all birds fly” happens to be false. The Contrapositive: The contrapositive is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. ii. property. Therefore, proving that P → Q is equivalent to proving that ¬ Q →¬ P. 22 yx Suppose a ≠ 2. ¬p 3, 4, modus tollens 6. You will find in most cases that proof by contradiction is easier. Proof by Contrapositive (Cont…) 9/19/2014 18 Result 3. Thus x5-3x4+2x3-x2+4x-1<0-1<0, as desired. Let the hypothesis be given. Since a conditional statement is equivalent to its contrapositive, we may also prove a conditional by proving the contrapositive instead. The set of natural numbers has no greatest element. Last sentence(s): Therefore n and m are not both negative. Proof by contrapositive Prove: If n2 is an even integer, then n is even . Proof by contradiction State the contrapositive, and then prove it. Switching the hypothesis and conclusion of a conditional statement and negating both. 1. In this example, 5 is not the only number that works (e. Go prove the contrapositive. [We must deduce the contradiction. Proof: A number qis odd if there exists an integer msuch that q= 2m+ 1. Theorem 1 (Example Theorem). Suppose ne Z. 2 Example: Prove the number three is an odd number. Proof by Contradiction. 1 – Conditional Statements CONDTIONAL ⇔ CONTRAPOSITIVE – Both have the same truth value. Proof by Contrapositive Walkthrough: Prove that if a2 is even, then a is even. If a statement X is of the form “if A then B ", the contrapositive of X is the statement “if B is false then A is false”. EXAMPLES Direct statement: If you drink Pepsi, then you are happy. Prove thatBC is rational ÐaBÑÐaCÑÐB !•C !•B CÑÊÐbDÑÐD •B D CÑ Proof Let and be real numbers. Instead, we show that the assumption that root two is rational leads to a contradiction. Example. Relation between Proof by Contradiction and Proof by Contraposition As an example, here is a proof by contradiction of Proposition 4. Then 6j(a b), so 6x = (a b) for Since the converse of the inverse is the contrapositive, the converse and the inverse also have the same truth value. Thus x is even, so \(x = 2a\) for some integer a. The twist here is that the proof of B is a proof by contradiction (and in the form which is not allowed in constructive mathematics). P. It is sometimes difficult (or impossible) to prove that a conjecture is true using direct methods. Assume n is even (i. 2. 1 It is a logical law that IF A THEN B is always equivalent to IF NOT B THEN NOT A (this is called the contrapositive, and is the basis to proof by contrapositive), so A ONLY IF B is equivalent to IF A THEN B as well. Example – Law of Detachment Since we have a proof that , we have which arrives at the contradiction we want. Share. 6 in the course of proving that \(\sqrt{2}\) wasn't rational. In case you've forgotten we needed the fact that whenever \(x^2\) is an even number, so is \(x\). i. The second proposition is called the contrapositive of the rst proposition. *** Anytime you are asked for the . 2 Con-gruence of Integers 5. The reason is that if it is hard or impossible to prove a statement by direct proof, then the workaround is to prove its contrapositive. , not odd). Whenever a conditional statement is true, its contrapositive is also true and vice versa. Therefore, 3n+2 is an integer multiple of 2. The number must be either complex (including i) or rational. For example instead of showing directly p â†’ q, one proves its contrapositive ~q â†’ ~p (one assumes ~q and shows that it leads to ~p). Thus, there are integers and for which and . Ex. Proof by Contrapositive (Direct Proof) If A, then B [Note A ! B ˘ B !˘ A] Proof (by contrapositive) 1. p → q: If 15 is an odd number, then 15 is a prime number. Example \(\PageIndex{2}\) If I am at Disneyland, then I am in California. That is, we can write “p implies q” as “not q implies not p” to get the equivalent claim: If Y has property B then X has property A. orF example, if ( A) all people with driver's licenses are ( B) at least 16 years old, then Thus, holds for n = k + 1, and the proof of the induction step is complete. The basic concept is that proof by con-trapositive relies on the fact that p !q and its contrapositive :q !:p are logically equivalent, thus, if p(x) !q(x) is true for all x then :q(x) !:p(x) is also true for all x, and vice versa. Suppose the contrapositive is true. We say that these two statements are logically equivalent. ÿ When proving a conditional, one can prove the contrapositive statement instead of the original – this is called a contrapositive proof. ] Suppose not. Hmm, in everyday life Proof by contrapositive example. Proof. (n2 even) : (n even) By the contrapositive: This is the same as showing that ¬(n even) : ¬(n2 even) If n is odd, then n 2 is odd. s) > 100, then r > 10 or s > 10. Derive contrapositive form: (¬q→ ¬ p) 2. The contrapositive proof of a logical condition or statement is done by taking negative of both the logic and conclusion of the logical statement. 4. If n is odd, then nº is odd. Explicitly state the contrapositive of what we want to prove. and derive a contradiction such as . Proof: (Proof by Contrapositive) Assume that x <0. 2 Proving the contrapositive. 4 Proof by Contrapositive Proof by contraposition is a method of proof which is not a method all its own per se. These notes explain these basic proof methods, as well as how to use deﬁnitions of new concepts in proofs. 2. (Proving the contrapositive) Let n be an integer. Example 2. Section 3. Here we have to prove that if , then = . (2) Prove that if n = ab, where a and b are positive integers, then a p n or b p n. If 3 - n2, then 3 - n. 2. The blue area cannot be in T. 2. Similarly, a statement's converse and its inverse are always either both true or both false. Contrapositive Proofs. Example: Contrapositive proof Prove hypothetical syllogism. An argument of this type is called a proof by contraposition or an indirect proof. Proof. q →s Premise 3. Prerequisite knowledge: section 2. Proof. (c) Proof by contrapositive Proof by contrapositive, also known as proof by contraposition is based on the logical equivalence between a conditional sentence and its contrapositive. This is false. A proof that proves the contrapositive is called a proof by contraposition. " in contrapositive form. More advanced methods (e. Example Prove that if n is an integer such that n 3 + n 2 + n is odd, then n is odd. For example consider the first implication "addition": P (P Q). Examples \(\forall x \in \mathbb{Z}, 3 mid x^2 \Rightarrow 3 mid x\) 8 Contrapositive 29 nearly always be an example of a bad proof! Tea or co ee? Mathematical language, though using mentioned earlier \correct English", di ers A Famous Contradiction Example. Direct proof or proof by contraposition? Proof (proof by contraposition): Assume n is an odd integer. Proof by contradiction often works well in proving statements of the form ∀ x,P( ). 6. (In each case, you should also think about how a direct or contrapositive proof would work. If n is odd, then n = 2k+1 for some integer k. The blue area is the area not in S. Contrapositive: If you aren't happy, then you don't drink Pepsi. Result: Let x ∈ Z. (In each case, you should also think about how a direct or contrapositive proof would work. Contrapositive 3. If the xed set U is understood, it may be omitted from the quanti er. Proof: Assume n is even. If x 4 − x 3 + x 2 ≠ 1, then x ≠ 1. Proof by Contraposition Examples Example 1 : Prove the following statement by contraposition: If a product of two positive real numbers is greater than 100, then at … By the closure 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 Motivating Example Proposition For all integers n, if n3 +5 is odd then n is even. This method sets out to prove a proposition P by assuming it is false and deriving a contradiction. For a quadratic polynomial to have roots its PQÞ ¬QPÞ¬ ¹Ï 49 8 c > fx x x c() 2 7=++2 P49 8 c > ¬P 49 8 c £ For example, P1: 3 is odd and P2: 57 is prime, the contrapositive of the implication P1⇒P2: If 3 is odd, the 57 is prime is the implication (~P2)⇒(~P1): If 57 is not prime, then 3 is even. p, assume ¬ p. The contrapositive is: If xor yis even, then x2(y2 2y) is even. Therefore, there exists an integer k such that n = 2k. n2 = (2k)2 = 4k2 = 2(2k2) Since 2k2 is an integer, n2 is even. Example. The statement P =)(˘P) is a contradiction. 1. Then, there exists integer k such that n = 2 k. If \(7x + 9\) is even, then x is odd. Assume that \(a\) and \(b\) are even. On the other hand, proof by 2. Claim 10 For any integers a and b, a + b ≥ 15 implies that a ≥ 8 or b ≥ 8. Thus, 3n + 2 is even. 3. The contrapositive of (n2 is an even integer) → (n is an even integer) is ¬(n is an even integer) → ¬(n2 is an even integer) which is equivalent to (n is an odd integer) → (n2 is an odd integer) Proof Methods: direct proof, contrapositive, contradiction, proof by cases. Example. The general pattern here is the following: 1. e. ” Extending our truth table above to include the contrapositive form, we have H C If (H), then (C) not C not H If (not C), then (not H) True True True False False True True False False True False False Examples The contrapositive is " If an object does not have color, then it is not red. Statement of the contrapositive 2. Proof by contradiction: to prove P )Q, assume that P is true and Q is false, and Proof By Contradiction. Therefore B = x2 ≠ 4x + 3 = 4m2 ≠ 8x + 3 = 2(2m2 ≠ 2m + 1)+1. We'll prove the contrapositive, namely, that if a collection of natural numbers has no least element, then it must be empty. – = , where b ≠ 0. In logic the contrapositive of a statement can be formed by reversing the direction of inference and negating both terms for example : p → q = -p ← -q = -q → -p See full list on jeremykun. I What is the contrapositive of this statement? I Proof:Suppose n is even. Suppose by contradiction that there is a greatest even integer. It gives a In the above proof we got the contradiction (bis even) ∧∼( is even) which has the form C∧∼. In1996, Robertson, Sanders, Seymour, and Thomasproduced a rigorous proof that still relied on computers. Assume givens and ¬𝑄 Example Prove: there is a rational number between any two positive real numbers. Proof: Theorem 1 : An integer n is even if and only if n2 is even. If we give a direct proof of ¬q → ¬p then we have a proof of p → q. Statement B: The Contrapositive of A. Conclusion: By the principle of induction, it follows that is true for all n 4. Proof: If n is even, then n2 is even is true by Lemma 1. As we see in this example, sometimes it is easier to prove a stronger statement then what is being asked. This is false. State the contrapositive, and then prove it. argument. Contradiction. 1 Proving Statements with Contradiction Let’s now see why the proof on the previous page is logically valid. Modus Tollendo tollens: \the way that denies by denying" Rules of Inference Contrapositive The tautology (p ! q) ! (: q ! : p) is called the contrapositive . Proof: This follows immediately by proposition 1 by a change of variables. Then 3n+ 7 = 2y for some y 2Z. It is also known as indirect proof. See first page of the notes. As an example, here is a proof by contradiction of Proposition 4. This is called a proof by contrapositive. Proof by Contradiction: (AKA reductio ad absurdum). Two common examples are proof by contradiction and proof by contrapositive. The proves the contrapositive of the original proposition, Proof by Contraposition Examples . Let n;m 2 Z. n2 = (2k)2 = 4k2 = 2(2k2) Since 2k2 is an integer, n2 is even. 6. Use the contrapositive the prove that if then has no solutions. Subsection Proof by Contrapositive. Determine if each resulting statement is true or false. (i. also holds. Contrapositive Statement: From Columbus, take I-70 east then turn left to I-77 north to Cleveland. g. Use the method of proof by contradiction to prove the following statements. If n2 is an even integer, then n is an even integer. Prove by contrapositive: Let x 2Z. Proof by contradiction is probably the easiest to go with. Use the method of proof by contradiction to prove the following statements. We assume p^˘q and come to some sort of contradiction. Therefore, proving that P → Q is equivalent to proving that ¬ Q →¬ P. The most important feature of the contrapositive (~Q)⇒(~P) is that it is logically equivalent to P⇒Q. Suppose ne Z. Although a direct proof can be given, we choose to prove this statement by contraposition. Contrapositive Proof Example: If n2 is an odd integer, then n is odd. Proof by contrapositive. is true , it follows that the contrapositive. ) [Given:] 3. Figure out what the negative of these are. This would be a more interesting theorem, but the point remains: when doing an existence proof, be as concrete as possible. I will assume that x is odd and y is even without loss of generality, since x and y are commutative. There exists an integer k so that n = 2k. A proof by contraposition (contrapositive) is a direct proof of the contrapositive of a statement. Suppose that 3n+ 7 is even. Proof: We That's how you do a proof by contrapositive. Proof: Suppose not. Theorem If n is a positive integer such that n mod(4) is 2 To conclude this article on proof by contrapositive, I will write an example using odd and even numbers. ∗ Example: Prove that an integer that is not divisible by 2 cannot be divisible by 4. proof, examples/counter-examples, and proof by contrapositive. Contrapositive, Converse, Inverse{Words that made you tremble in high school geometry. To help set up a contrapositive proof, here’s a rephrasing of the theorem: Theorem 7. EXAMPLE 2. This existential quanti er means that there exists a (or there is at least one) value of x in U for which P(x) is true. (Note that the inverse is the contrapositive of the converse. \Suppose not B" [Show not A] 2. If p Thus, the contrapositive provides an alternative way of viewing the definition. In 1891, Georg Cantor assumed that there is a set T that contains every infinite-lengt The contrapositive of this statement is: “if a² ≠ b² + c² then the triangle in not right-angled at ‘A’”. Write x = 2a for some a 2Z, and plug in: x2 6x+ 5 = (2a)2 6(2a) + 5 = 4a2 12a+ 5 Example 1. Answers Learning objective: prove an implication by showing the contrapositive is true. e. A couple definitions about prime numbers are mentioned in one proof. Hence, to prove pq , we sometimes prefer to prove qp . Follow Contrapositive help understanding these specific examples from Graph Theory. Example proofs; Contrapositive Published on Dec 20, 2011 Theorem Prove that if a n − 1 is prime, then a = 2, where a and n are positive integers, with n ≥ 2. Examples \(\forall x \in \mathbb{Z}, 3 mid x^2 \Rightarrow 3 mid x\) One of the most famous proofs of all time - ironically, one that is incorrectly cited as an example of proof by contradiction[1] - is a proof by contraposition. Suppose the statement "if A then B" is true. 2 Equivalent Statements 7. State the contrapositive in English: _____ Now use a direct proof of the contrapositive. In Contrapositive proof of p→q Procedure: 1. This proof, and consequently knowledge of the existence of irrational numbers, apparently dates back to the Greek philosopher Hippasus in the 5th century BC. Since we have shown that ¬ p → F. We should try to use direct proof first. Therefore the statement is true using the technique of proof by contrapositive. Example: Prove that for an integer n, if n^2 is odd, then n is odd. Then n2 + 6n+ 5 = 4k2 + 12k + 5 = 2(2k2 + 6k + 2) + 1 is odd. This statement has the form p!(r_s). Proof. }\) There are plenty of examples of statements which are hard to prove directly, but whose contrapositive can easily be proved directly. Suppose that x is even. Hence the biconditional statement n is even if and only if n2 is even is true. So 3n+ 7 = 3(2x) + 7 = 6x+ 6 + 1 = 2(3x+ 2) + 1, where 3x+ 2 2Z. Problem 8. 1 3 = 2 ( 7) − 1. Logically, Theorem I is exactly the same as this theorem, which is called the contrapositive of Theorem I. Start by announcing that we're going to use a proof by contrapositive so that the reader knows what to expect. The logical steps in the proof are essentially the same for the argument by contradiction and the contrapositive. This is all that proof by contrapositive does. •Proof –Assume that x is a real number and that x is not irrational. This is even because it is divisible by 2, since 2 divides 4. If n is odd, then nº is odd. Counterexamples and the Contrapositive. Theorem II. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. 3n+2 is even. If n is odd, then nº is odd. We would need to find a single example of one of these conditions, any one of which would be a counterexample: A living woman who does not eat food, or; A woman who eats food but who is not alive, or; A nonliving woman who eats food, or; A woman who does not eat food but The contrapositive is: if x2 10x+ 25 = 0 then x = 5. Contrapositive Proof Example: If n2 is an odd integer, then n is odd. Use the method of proof by contradiction to prove the following statements. Because the statements 5{13 were true, a counter-example cannot actually be found. In these cases, we may be able to provide a proof of (¬Q → P'). p →s 1, 5, contrapositive method MSU/CSE 260 Fall 2009 16 Example: Contradiction proof That's how you do a proof by contrapositive. 5 Another example Here’s another claim where proof by contrapositive is helpful. Follow Contrapositive help understanding these specific examples from Graph Theory. So proof by contrapositive is in some sense "at least as hard to formulate" as proof by contradiction. 1. Proof 5. You will find in most cases that proof by contradiction is easier. ] Assume, to the contrary, that ∃ an integer n such that n 2 is odd and n is even. g. For example, to show that the square root of two is irrational, we cannot directly test and reject the infinite number of rational numbers whose square might be two. The contrapositive of this is. By saying that the two propositions are equivalent we mean that A to Z Directory – Virginia Commonwealth University argument. 3 no . ¬s Assumption 2. In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap the roles of the hypothesis and conclusion. Proof. Pf: Suppose n is an even integer. (In each case, you should also think about how a direct or contrapositive proof would work. 2 Proof by contradiction In proof by contradiction, you assume your statement is not true, and then derive a con-tradiction. Contrapositive Proof Example Proposition Suppose n 2Z. You very likely saw these in MA395: Discrete Methods. Cite. Two Examples: The Supremum of a Closed Interval and Its Interior (an Open Interval) Consider an example on the real number line (under the standard order, of course). 1 Direct Proof The proof by contrapositive is based on the fact that an implication is equivalent to its contrapositive. 4, namely that for any integer n, if n2 is even then n is even. Direct Proof ; Proof by Contradiction; Proof by Contrapositive ; If, and Only If ; Proof by Mathematical Induction . 3. We actually have proved a stronger statement, that :Q ,:P. Proof: We prove this by contrapositive. Using the Contrapositive Every w gets the DFA to exactly one state. Proof. That's how you do a proof by contrapositive. " Problem 9. (iii) He is a university student. e. For example, instead of proving \x being an integer implies that x is a real Example. A proof by contrapositive of P ⇒ Q is a direct proof of ∼ Q ⇒∼ P. Start of proof: Let \(a\) and \(b\) be integers. Even if the original assertion is true, the converse may well be false. Example 1. It turns out that any conditional proposition ("if-then" statement) and its contrapositive are logically equivalent. What you *are* allowed to use in a logic proof is the contrapositive. So to prove for an arbitrary x, try to prove . D. Thus 3n + 2 is even, because it equals 2j for an integer j = 3k + 1. Proof: Suppose that p Proof by contradiction • We want to prove p q • To reject p q show that (p ¬q ) can be true • To reject (p ¬q ) show that either q or ¬ p is True Example: Prove If 3n + 2 is odd then n is odd. Method of Proof by Contradiction Procedure 1 Supposethestatementtobeprovedisfalse. If 0<𝑎<𝑏 then 𝑎2<𝑏2. Now then, we compute the sum , which is an odd integer by definition. Another proof is an indirect proof may start with certain hypothetical situations and then continue to eliminate the uncertainties in each of these situations until an inescapable conclusion is forced. Recall that the conditional is logically equivalent to its contrapositive . Suppose ne Z. 1 we use this method to prove that if the product of two integers, mand n, is even, then mor nis even. n = 2k+1 (k is integer) n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 Assume integer m = 2k2 + 2k. Contrapositive proof: Assume that x and y have different parity (~Q). If nº is odd, then n is odd. Unpublishing the sum of the above graph by contrapositive and logic. If you take our advice above, you will rst try to give a direct proof of this statement: assume mnis even and try to prove mis even or nis even. Determine what the A and B are in the proof you are asked to do. Proof (by contradiction): [We take the negation of the theorem and suppose it to be true . The following is an example of proof by contrapositive: Theorem: If x 2 is odd, then x is odd. 2. Here we try to prove it in two other ways. Therefore, proving that P → Q is equivalent to proving that ¬ Q →¬ P. Proof: Suppose ˘Q . We will add to these tips as we continue these notes. statement 1. Example Prove that if n is an integer such that n 3 + n 2 + n is odd, then n is odd. It should read \For all x 2R, x2 1 0. For example, the converse of ‘If P, then not-Q’ is ‘If not-P, then Q. This happens in example 1, for instance { living in the United States does not imply living in Texas. • is has no solutions so is has at least one solution. e. Since k is an integer and sums and products of integers are integers, 3k+1 must be an integer. (Supposethenegationof thestatementistrue. A word of warning: you must not confuse the contrapositive of a conditional with that conditional’s converse. proof x2 = p2/q2 is the quotient of integers, so x2 is rational Contrapositiveconclusion x2 is rational An indirect proof is a nonconstructive proof. Then 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1). Proposition: 8a;b 2Z, a b mod 6 if and only if a b mod 2 and a b mod 3. Try a direct proof first. Solution. By contraposition, If n2 is even, then n is even. Don Knuth was not a cornhusker. 4 (Non-) Construc-tive Proofs Proving If-And-Only-If Statements Outline: Proposition: P ,Q. F. If statement B is false then statement A is false. 3 Existence and Uniqueness Proofs 7. The contrapositive of p --> q is ~q --> ~p. Let m= 1. 3. n2 = 2m + 1 So, n2 is odd. Example: 9x 2Z(x > 5). Suppose, the closed interval from to. Therefore ˘P. From here we must Contrapositive: If Jennifer does not eat food, then Jennifer is not alive. –Since x is real and not irrational, then it is rational. For example, I. Why does this work? Example: Theorem: Prove that if n is an integer and 3n + 2 is odd, then n is odd. Suppose ne Z. Proof by contrapositive takes advantage of the logical equivalence between "P implies Q" and "Not Q implies Not P". Suppose ne Z. To show that X is true, we assume A and attempt to deduce B. Prove that if is not even, then n is not even. If 5x-7 is even, then x is odd. Use this packet to help you better understand conditional statements. (n2 even) → (n even) • By the contrapositive: This is the same as showing that 1. Cite. Proposition 2. Let x be an integer. One more quick note about the method of direct proof. Then x5<0, x4 >0 ⇒ -3x4 <0 x3<0, x2>0 ⇒ -x2<0 and 4x<0. contrapositive proof example