%PDF-1.5 Want to see the math tutors near you? Proceed as you would with a direct proof. Since p / q = 2 and q 0, we have p = 2q, so p2 = 2q2. Students sometimes find proofs by contradiction difficult to understand and construct (Epp, 2003; Reid & Dobbin, 1998 ). The reason is that the proof set-up involves assuming 8x,P(x) ,whichasweknowfromSection2.10isequivalentto 9x,P(x) . Peter J. Eccles Affiliation: University of Manchester. Assume that P is true. Then 9m . Well, those integers didn't work; care to keep doing that for a few hours with a few hundred other integers? The steps for proof by contradiction are as follows: Assume the . Thus, 3n + 2 is even. This was a challenging lesson. After multiplying each side of the equation by q 3, we get the equation. So we can write b=7d, with d an integer. Upload unlimited documents and save them online. 2. Be perfectly prepared on time with an individual plan. Proof. Stop procrastinating with our study reminders. Proof: Suppose not. There is no middle ground. SupposeP andQ.. Let us assume that we could find integers a and b that satisfy such an equation. Proof. What this requires is a statement which can either be true or false. This means we can replace a with 2c, as a must be even. As 7 is prime, for something squared to be a factor of 7, then the original must also be a factor of 3. Let us assume that ab is irrational, but a and b are rational. Proceed as you would with a direct proof. If a and b are integers, and we multiply each by another integer (2 and 3 respectively, in this case), then sum them, there is no possible way that this could result in being a fraction, which is what the above condition requires. Sec 3.3 Proof by Contradiction 1/5 Idea of Proof by Contradiction The method of proof by contraction is based on the fact that A two-streamed model of understanding proof by contradiction was constructed statistically. [1] This leads us to a contradiction. Everything you need for your studies in one place. /Subtype /Form Stop procrastinating with our smart planner features. Open navigation menu. If a and b are integers, and we multiply each by another integer (1 and 2 respectively), then sum them, there is no possible way that this could result in being a fraction, which is what we require. Proof by contradiction - key takeaways. ;DIe>""E^y- 4"4". If you want to read up on more types of proofs or Discrete Math topics in general a great book to easily learn and practice these topics is Practice Problems in Discrete Mathematics by Bojana Obrenic' , and . xn_q)dbnX &1L[B-9wJ-;fIkB=33yg"qMv=:{D{I7dwM5)~U[/#Ec147Y: "IvPFD'p@eT3>z\`"I8DA@D'; There are some steps that need to be taken to proof by contradiction which is described below: Step 1: In the first step, we will assume the opposite of the conclusion to be true. To prove that the statement "If A, then B" is true by means of direct proof, begin by assuming A is true and use this information to deduce that B is true. The two integers will, by the closure property of addition, produce another member of the set of integers. The famous proof that $\sqrt{2}$ is irrational. The 2 cannot be rational, so it must be irrational. stream Accordingtotheoutline,therstlineoftheproof shouldbe"Forthesakeofcontradiction,suppose a2 isevenand isnot even." Proposition Suppose a2Z.If 2 iseven,thena iseven. /Length 2072 . %~H g Je0^KNlb{+??B ?B.Y" I fD}qq& C+>=. DHi7FhjWIF?C| DRdcA`]{el1 7LzB#,4Vc{u,$C$RD&@c8 TF1yX JuW`o1X2;PW(sdwb2"6p7C*aJ65VN7;>*x/x'1c[#}eC9EjVE iFYg!AY$_8nR+4Df6qJ'!+PVUj Premises: Prove: A. Example 4: Prove the following statement by contradiction: For all integers n, if n 2 is odd, then n is odd. 3 Prove by contradiction that is irrational (Total for question 3 is 6 marks) 3 5 Prove by contradiction that the sum of a rational number and an irrational number is irrational (Total for question 5 is 6 marks) 1 Use proof by contradiction to show that there exist no integers x and y for which 6x + 9y = 1. The reasoning behind proof by contradiction is that if the assumption that the theorem is false leads to a conclusion which cannot be true, then the theorem must be true. Then 3=a/b, so a=3b. This means that as a is a factor of 3, then so is a, so we can write a=3c, with c an integer. Watch on. (3LgAN wJ%x1|d| Prove there are no integers that satisfy 4a-28b=-3. This leads us to a contradiction. of the users don't pass the Proof by Contradiction quiz! A A B. Proof by contradiction Starter 1. Assume to the contrary there is a rational number p/q, in reduced form, with p not equal to zero, that satisfies the equation. /Type /XObject Proof by contradiction examples Example: Proof that p 2 is irrational. p 2 = a b 2 = a2 b2 2b2 = a2 This means a2 is even, which implies that a is even since . Assume that :Q is true. Solution: Assume the negation, that is p 2 is ra-tional. Following the same argument as above, this means b is even, and in turn, b is even. Toillustratethisnewtechnique,werevisitafamiliarresult: Ifa2 is even,thena iseven. Consider two statements p and q. Proof by contradiction Famke Janssen 6 pages H.5.pdf University of Maryland CMSC 250 University of Maryland CMSC 250 hw06Solutions.pdf Want 2k 4k 1 j 1 J 6 K 4 pages hw06Solutions.pdf University of Maryland CMSC 250 University of Maryland CMSC 250 hw04Solutions.pdf Prime number Rational number Irrational number 2 pages hw04Solutions.pdf % 2.6 Proof by contradiction A proof by contradiction starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a result of the assumption. This means that this alternative statement is false, and thus we can conclude that the original statement is true. A statement cannot be true and false at the same time, If the statement can be proven true, then it cannot be false, If the statement can be proven false, then it cannot be true, If the statement cannot be proven true, then it is false, If the statement cannot be proven false, then it is true, Recognize and apply proof by contradiction in mathematical proofs, Develop a logical case to show that the premise is false until your argument fails by contradiction, Recognize that the contradiction in your argument demonstrates the validity of the original premise. By the same above argument, b is a factor of 7, and so is b. Proof by contradiction is based on the logic that if the converse of a statement is always false, then the statement is true. By the same above argument, b is a factor of 3, and so is b. Proof. Step 3: We use 1 as our divisor and 1 as your quotient. If x , y , z are positive real numbers. Then, show that both and Sare true, which is a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, [citation needed] and reductio ad impossibile. 3|w9Ys=e;$$tC>j}*KJEfLj 1Tm* Ih1P l_%bQ>R2y$%&JvAc.^.u$Wx&I>!Y`o&%M&h;-d("}3wo[T@(L?&|p&Z[.1~XCApA*g[v~S=UPH\xgU^p=|rSG? Answer: We assume the hypothesis x+y > 5. Truth and falsity are mutually exclusive, so that: It is that last condition of truth and falsity that is exploited, powerfully and universally, by proof by contradiction. /Filter /FlateDecode The negation of the claim then says that an object of this sort does exist. stream You may well benefit from rereading it several times, but once you do, you should feel more confident in your understanding of proof by contradiction. 44 0 obj << This means that P is a prime number, and as , this means there is a new prime, which means we now have a contradiction. This statement is a contradiction in the sense that it is false no matter what the value of q. Proof by Contradiction - Free download as PDF File (.pdf), Text File (.txt) or view presentation slides online. Come across a contradiction. taneously true and deriving a contradiction. As 3 is prime, for something squared to be a factor of 3, then the original must also be a factor of 3. - A real number x is called irrational if x 6 = p q for any integers p and q. Math 2800 Chapter 6: Proof by Contradiction Dr. Briggs. QED. contradiction proofs tend to be less convincing and harder to write than direct proofs or proofs by contrapositive. One of the most powerful types of proof in mathematics is proof by contradiction or an indirect proof. Therefore, P _Q. The working includes four parts: Derivatives of Inverse Trigonometric Functions, Initial Value Problem Differential Equations, Integration using Inverse Trigonometric Functions, Particular Solutions to Differential Equations, Frequency, Frequency Tables and Levels of Measurement, Absolute Value Equations and Inequalities, Addition and Subtraction of Rational Expressions, Addition, Subtraction, Multiplication and Division, Finding Maxima and Minima Using Derivatives, Multiplying and Dividing Rational Expressions, Solving Simultaneous Equations Using Matrices, Solving and Graphing Quadratic Inequalities, The Quadratic Formula and the Discriminant, Trigonometric Functions of General Angles, Confidence Interval for Population Proportion, Confidence Interval for Slope of Regression Line, Hypothesis Test of Two Population Proportions. >> /BBox [0 0 12.606 12.606] This is a basic rule of logic, and proof by contradiction depends upon it. 2. The value of c is unimportant, but it must be an integer. We follow these steps when using proof by contradiction: Assume your statement to be false. Mathematical Induction: Proof by Induction. W X v89MJ4}CIfj~yO$:Y8|e=f}xZj4;%bH_,a.6&| HF`u!,b],FL~K_s7om({`}IBOK-yCJTMM.Wy c0{N}S>T| y\11ugonbDL4_|m8~ )*~}.~1:dP 3k,7UbfS7+ L-!t:x%"x0?by`/dL$z' ,\S a) Let \ (n\) be an integer. Now it is time to look at the other indirect proof proof by contradiction. Prove that if X Y = X then X Y = . Hence a contradiction, and so 3 is irrational. % Forthesakeofcontradiction,suppose a2 . The first step is to assume the statement is false, that the number of primes is finite. Instead, it might be easier to prove that something related must be False. Recall that a and b cannot both be even, so b must be odd. The steps for a proof by contradiction are: Step 1: Take the statement, and assume that the contrary is true (i.e. From here we . A proof by contradiction is sometimes . Chapter Book contents. Generally, the false statement that is derived in a proof by contradiction is of the form q q. The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. Thus, there are no integers a and b such that 3a-9b=1. Let be an integer.. To prove: If is even, then is even. Then n2 is an odd integer. 4. Take the usual definition of a prime as a natural number greater than 1 divisible only by itself and 1. The sum of the integers is a fraction! ZDM. Assume the contrary. Show author details. /Length 15 Here is a template. Thus, there are no integers a and b such that -7a+14b=4, Prove there are no integers that satisfy 10a+20b=5, Let us assume that we could find integers a and b which satisfy such an equation. Prove there are no integers that satisfy 3a-9b=1. Write a=c/d, and b=e/f, with c,d,e,f , d,f0. "U$;)a63C6%_lej[Gj[VWuU^:o;uR}'O:);cpW Statement p: x = a/b, where a and b are co-prime numbers. Thus, if ab is irrational, then at least one of a and b are also irrational. Will you pass the quiz? Prove that if ab is irrational, then at least one of a and b are also irrational. Proof by Contradiction. That is a contradiction: two integers cannot add together to yield a non-integer (a fraction). You could spend days, weeks, years stumbling around with specific numbers to show that every integer you try works in the statement. That is, P =)Q. Lemma: Given integers a and b, with a > 1, if ajb then a 6j(b + 1). Let us assume that we could find integers a and b which satisfy such an equation. Use proof by contradiction when it is difficult or impossible to prove a claim directly, but the converse case is easier to prove. Rational Numbers - A real number x is called rational if there exists integers p and q, where q 6 = 0 and x = p q. 3 0 obj (prove by contradiction) 6. 8. Thisgivesusaspecic x forwhich P( ) istrue,andoftenthatisenough Step 2: Start an argument from the assumed statement and work it towards the conclusion.Step 3: While doing so, you should reach a contradiction.This means that this alternative statement is false, and thus we . This may look tricky, so we will now look through some examples to get your head around this concept. Suppose not; i.e., suppose p 3 2Q. Example: Prove by contradiction that if x+y > 5 then either x > 2 or y > 3. Then, we have p 3 /q 3 + p/q+ 1 = 0. Then we can write c / d + b = e / f. This implies . 104 Proof by Contradiction 6.1 Proving Statements with Contradiction Let's now see why the proof on the previous page is logically valid. Remember that the negation of \ (p q\) is \ (p q\). Continue the proof using direct proof methods.. Proof by Contrapositive and Contradiction 1. xW[o6~ xyeCknFCC" \CRJ Step 1: We write 3 as 3.00 00 00. !iH:mS!,sHE"A*|z?n%Q|'9~!(/m ZGx%imgwKe-KeeJd$pb^Tbq$CxUb"6c3E2[fBjN. +_\7?o>kYg/H>U@Q;qb]Q[Gp( Consider \textcolor {blue} {L}+2 L + 2. Learn faster with a math tutor. Prove by contradiction that 2 3 is an . This leads us to a contradiction. 48! It's based on >FcmLBy>yup|q'CvfD\BvI?)->%"h;V@x7i~ 1) Assume that the opposite of what youre trying to prove is true. An irrational number cannot be expressed as a fraction or ratio. Find a tutor locally or online. Prove that if the sum of two primes is prime, Q 12: Suppose a 2Z, If a2 is not divisible by 4 , then a is odd. It will be at the start of class, largely short answer, about 15 minutes long. The statement we are trying to prove must have only two possible outcomes. Every prime number has two positive factors 1 and itself, so either (k 1) = 1 or (k + 1) = 1. First, assume that the statement is not true and that there is a largest even number, call it \textcolor {blue} {L = 2n} L = 2n. By registering you get free access to our website and app (available on desktop AND mobile) which will help you to super-charge your learning process. )9l|HHs&YqVEc^Mr7pfP@OCvS7W1UkL~we[n_ER4:jXh Free and expert-verified textbook solutions. Notice in its simplified form at least one term of the fraction is odd. /Filter /FlateDecode Example 1: Proof of an infinite amount of prime numbers. Step 2: Find a number whose square is less than or equal to the number 3. Have all your study materials in one place. Theorem: If A then B. Discrete Math 1.7.3 Proof by Contradiction 33,189 views Mar 11, 2018 486 Dislike Share Kimberly Brehm 34.6K subscribers Please see the updated video at https://youtu.be/b-kFWP9a2tw The full. (Proof by Contradiction.) It is powerful because it can be used to prove any statement, in several fields of mathematics. Since a contradiction is always false, your assumption must be false, so the original statement P must be true. Proof: By contradiction; assume 2is rational. Thus, we can write b = 2d, d. Here are some good examples of proof by contradiction: Euclid's proof of the infinitude of the primes. Proof by Contradiction (Example 1) Show that if 3n + 2 is an odd integer, then n is odd. To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false. Suppose there is some greatest even integer, and call this n. Any even integer can be written as the product of 2 times another integer, so let us say that n=2k, k. From this assumption, p 2 can be writ-ten in terms of a b, where a and b have no common factor. Let's say that there are only n prime numbers, and label these from to . Proof. This means that gcd (a, b) = gcd (2c, 2d) 1. As a + b is rational, we can write a + b = e / f, e, f, f 0, and the fraction in its lowest terms. ;nShcb}@l'lL'1C:Bq680&secYv~,x.#343n-^v0onA4_}Xm`*yAstc$l;N-D,0u*O9oTKQMl6a5,M oetXh{Rdd%`r&} fo'l=bGaC7_Gn:mgafpnY|vBy;_y\vb@yi*UxM-](V~W~X PL?p This means that there must be an infinite number of prime numbers. Let us assume that we could find integers a and b which satisfy such an equation. Since q2 is an integer and p2 = 2q2, we have that p2 is even. xXKo7W:,kc-mlYjKrR P^.gK1~^(LY2L6l;gnda;^UXp.\XTlZf#6Q3#!qio0kbUA*uE#c{i.H{[%.YQ`Pr$clOD~)zFy@Ct(&$6 zyL+h8=u$H.)G," >> If a number is odd, then we can write it as 2k + 1. Identify your study strength and weaknesses. assume the statement is false). Often proof by contradiction has the form . 19 0 obj Presumably we have either assumed or already proved P to be true so that nding a contradiction implies that :Q must be false. Proof: a valid argument that shows that a theorem is true. (Review of last lesson) Prove that the square of an odd number is always odd. Local and online. Frontmatter. Step 4: We carry down a pair of zero. We can then divide through by 7, to give -a+2b=4/7. 2.Prove that each of the following statements is true. a contradiction of the original assumption. Before we begin this proof, we need to know that any natural number greater than 1 (so ) has a prime factor. The statement is easier to prove by a direct method as we have seen in Theorem 20.1. Now you are able to recognize and apply proof by contradiction in proofs, develop a logical case to show that the premise is false, until your argument fails by contradiction, and recognize the contradiction in your argument that demonstrates the validity of the original premise. For example: Claim 51 There is no largest even integer. Then, if a = 2b, we have 4c = 2b => b = 2c. [We must deduce the contradiction.] We subtract 1 from 3 and get a reminder of 2. This textbook is very comprehensive. Simple examples of proof by contradiction The rst example is just to show you the idea of proof by contradiction. In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Many of the statements we prove have the form P )Q which, when negated, has the form P )Q. Notes Proof by contradiction assumes that the opposite is true and then a series of logical arguments are followed which lead to an incorrect conclusion i.e. This means that a / b is a fraction in its lowest terms. We can then divide both sides by 5 to give 2a + 3b = 1/5. Let X and Y be sets. This shows that x has two factors. A Famous and Beautiful Proof Theorem: 2 is irrational. Proof by contradiction has 3 steps: 1. On the analysis of indirect proofs Example 1 Let x be an integer. Let us assume that 2 is rational. 'Assumption: 2 is a rational number.' 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