Proof by contradiction example discrete math


). and . 1 in particular, for proof by contrapositive. Jonathan L. I know: P=n is a perfect square Q=n+2 is not a perfect square And above is in form P->Q. We find that students become so enamored with discovering a proof by approaching it “by contradiction” they often end up with direct Section 1. Shorser In this document, the de nitions of implication, contrapositive, converse, and inverse will be discussed and examples given from everyday English. Mathematical Proofs: Where to Begin And How to Write Them Starting with Linear Algebra, mathematics courses at Hamilton often require students to prove mathematical results using formalized logic. A proof provides a means for guaranteeing such claims. by Contradiction. 2 Learning Outcomes Solution: The conditional x y represents, "If Gisele has a math assignment, then David owns a car. One example from Rosen: Show that at least four days of 22 days fall on the same day. Example: Use a proof by contradiction to give a proof that √2 is irrational. We will prove by contradiction. In the examples below we use this idea to prove the impossibility of certain kinds of solutions to some equations. Show :(p!q) is equivalent to p^:q. Example 2. This could also happen with other proof methods, but this seems to be prone to it. Show that √ 2 is an algebraic number. If a 2 is even Example CD-2. P → Q is a conjecture until a proof is produced. For example, a cube and a ball are topologically equivalent, but both are not equivalent to a donut. Proof Techniques (Part 2). ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Lecture Notes What is Discrete Math? Example of continuous math – Given a fixed surface area, what are the dimensions of a cylinder that maximizes volume? Example of Discrete Math – Given a fixed set of characters, and a length, how many different positive proof and proof by contradiction. = m/n, where m, n are integers, n ≠0 2, Math Definition m and n do not have common factor. Here are a few examples: • Original practice to always ask yourself after writing a proof by contradiction if you can make the same argument   For example, the axioms of plane geometry, which you can look up online. If 5x+ 25y= 1723, then xor yis not an integer. Example 3. Rosen (Discrete Mathematics and its Applications, 2nd ed. Here is another example. [We must show that (m. 5. DISCRETE MATH. The rejection isn’t because you provide a proof that the opposite of the assumption is true. that the correct constructive definition of finite structures from discrete mathematics  Weak mathematical induction; Strong mathematical induction; Structural induction. In such a proof, we assume that \(x\) is a rational number and that \(x^2=2\text{. We then repeatedly apply (2. The topological fact used by the proof is that if you have two rubber bands lying on a plain, one of them is surrounding a nail stuck in the plane, and the other isn’t, then you cannot bring the rst rubber band Discrete Structures 22 Proof by Contradiction – Example 2 Prove that is an irrational number. We reason by contradiction and start our proof by negating the statement: 9n with 3n+2 and 3j(3n+2) (a shorthand for \3 divides (3n+2)"). 3. 3), there exists q with 3q = 3n+2, hence 3q 3n = 2. If you've assumed the statement to be false and arrive at a logical contradiction, then the statement must have been true. Best Answer: Okay, so always be clear that you are proving by contradiction. above, we used that the sum of a rational and irrational was irrational: that is proved somewhere in the text. Thus, students see the connection between the informal proof sketches they see in other courses and rigorous proofs. Proof by contrapositive inference is an example of what we call indirect inference. Note that it is usually used in a proof by contradiction; that is, construct a set S, S, S, suppose S S S is nonempty, obtain a contradiction from the well-ordering principle, and conclude that S S S must be empty. Contrapositive, Converse, Inverse{Words that made you tremble in high school geometry. Reasons 1. If our supposition in a proof by contradiction was "there exists some integer n such that the product of n and its reciprocal does not equal 1," what was our proposition? 3. Proof We will prove by contradiction. In an existence proof, we can have two types of proofs: constructive (where we provide an example) and non-constructive (indirect proof, i. ) Finding related theorems can be very helpful. For all integers n, if n3 + 5 is odd then n is even. An alternative proof is obtained by excluding all possible ways in which the propositions may fail to be equivalent. CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar. First, for contradiction, you assume the logical negation of your claim. We arrive at a contradiction when we are able to demonstrate that a statement is both simultaneously true and false, showing that our assumptions are inconsistent. Contrapositive 3. Contradiction Proofs This proof method is based on the Law of the Excluded Middle. ¥Keep going until we reach our goal. Math isn’t a court of law, so a “preponderance of the evidence” or “beyond any reasonable doubt” isn’t good enough. If you make small mistake somewhere in the process you might end up with a contradiction which wont show that the proposition is true but it will show that you made a mistake. 7, ~2 hours) Methods of mathematical argument (i. 8 p. Essentially, if you can show that a statement can not be false, then it must be true. ¥Use logical reasoning to deduce other facts. Thus, 3n + 2 is even. Mathematicians normally use a two-valued logic: Every statement is either True or False. And this proof was believed for over a decade. ) The famous proof that $\sqrt{2}$ is irrational. Download. 4. I will try to write a very very clear proof, so you see how the structure of the proof works. Proof. Squaring both sides, 2 = x2/y2, so 2y2 = x2. Keeping all this in mind, let us look at Timothy Gower's blog examples. is a rational number. Gil Williamson This example illustrates an alternative to using truth tables to establish the equiv-alence of two propositions. _____ Example: If it's raining today then the void set is a subset of every set. (for a more interesting problem, prove that there is no greatest prime number. Sometimes a proof by contradiction can be rewritten as a proof by contrapositive CSE 20 DISCRETE MATH Fall 2017 • Proofs by contradiction • Proof by cases Proof by cases Rosen 1. Contents Tableofcontentsii Listoffiguresxvii Listoftablesxix Listofalgorithmsxx Prefacexxi Syllabusxxii Resourcesxxvi Internetresourcesxxvii Lectureschedulexxviii Proof By Counterexample by L. DISCRETE MATH: LECTURE 3 3 1. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "It is a far finer gambit than any Discrete Mathematics - Propositional Logic - The rules of mathematical logic specify methods of reasoning mathematical statements. They then go on to other, more elaborate examples, to see that such contradictions are quite common occurrences and that, turning the example around a bit, "proof by contradiction" becomes a very powerful method of Contradiction definition is - act or an instance of contradicting. This video is part of a Discrete Math course Proof by Contradiction This is an example of proof by contradiction. [Hint: Assume that r = a/b is a root, where a and b are integers and a/b is in lowest terms. ○ Negations Proof by Contradiction. • Show the statement holds fo r one example but we do not have the witness example. with √2 = a/b, where . Eduardo is proving the theorem above by contradiction. Indirect Proof (Proof by Contradiction): (Assume the opposite of what you want to prove, and show it leads to a contradiction of a known  It looks “logical” to deduce that therefore, Jackson must study discrete math- ematics. Discrete Mathematics, Chapter 1. Suppose that for each a in A there is b in A such that (a,b) and is in R. Example 21 is about infinite number of primes (similar to what was done in class, but not quite the same). Mathematical proofs can be difficult, but can be conquered with the proper background knowledge of both mathematics and the format of a proof. 4 / 12  29 Dec 2015 The basic idea with a proof by contradiction is to start with something false: in this case assuming that 3n+2 is even and n is odd. For example, here is a famous proof by contradiction that p 5 is not rational 19 Achievements On August 22, 1999, a group of researchers completed the factorization of the 155 digit RSA Challenge Number. 3. A direct proof of this theorem would require that we verify that the square of every rational number is not equal to 2. CS 441 Discrete mathematics for CS M. This means there are integers x and y (y ≠ 0) with no common divisors such that = x/y. 2. Mathematical Induction What follows are some simple examples of proofs. 1. Example: Let p be “It is snowing. Learn vocabulary, terms, and more with flashcards, games, and other study tools. So x2 is Discrete math contradiction proof? Use a proof by contradiction to show that there is no rational number r for which r^(3)+r+1=0. Solution: Suppose √2 is rational. Includes elementary logic and set theory, equivalence relations, functions, counting arguments, asymptotic complexity, inductively defined sets, recursion, graphs and trees, Boolean algebra and combinatorial circuits, finite state automata, and diagonalization and countability So, Proof by Contradiction (of the premise) is a Reductio Ad Absurdum Proof. For example, 6 2, 12 2, rule of contradiction: p contradiction, p The validity of the above argument forms can all be easily verified via truth tables. Jerry is a Math major and a CSI major. 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). These rules may not mathematically look very familiar. And except for the beginning and end, to solve an indirect proof, you use the same techniques and theorems that you would use on regular proofs. , “Assume for the sake of contradiction that…”) Henry Cohn points out some dangers of proof by contradiction, and suggests strategies for avoiding those dangers. Proof By Contradiction. Proof by contradiction and the Pigeonhole Principle. The original statement is . We are going to apply the logical rules in proving mathematical theorems. Worksheet C. Hauskrecht Proofs with quantifiers • Universally quantified statements This might be my all time favorite proof by contradiction. A contradiction shows that the assumption made earlier is impossible, and therefore false. First and foremost, the proof is an argument. e. † The Principle of Induction is a defining difference between discrete and continuous mathematics. We start with the supposition that the statement is false, and use this assumption to derive a contradiction. Example: Give a proof by contradiction of the theorem “If 3n + 2 is odd, they are based in logic (e. In your case you start the proof by assuming the statement is false. Proof by Contradiction (Example 1) •Show that if 3n + 2 is an odd integer, then n is odd. • Proof : Assume that the statement is false. See Mike F. Understanding a key type of mathematical statement. Often proof by contradiction has the form CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction Direct Proofs At this point, we have seen a few examples of mathematical)proofs. However . If you can show the conclusion is a contradiction, then it is false. Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. Blerina Xhabli, University of Houston Math. H. So Discrete Math Review n What you should know about discrete math before the midterm. nThese have the following structure: ¥Start with the given fact(s). 2 Proof by Contradiction The proof by contradiction is grounded in the fact that any proposition must be either true or false, but not both true and false at the same time. And now we're going to call explicit attention to it, and think about it. You can always use other proven theorems. Negative . Lectures in Discrete Mathematics, Edward A. The goal of reading this is that you gain a comprehension of the topics at a deep level, gaining intuition. . As a first (very simple) example If that new curriculum is adopted, this course is expected to be called discrete math and theory 1 or DMT1. Contrapositive Proof Example Proposition Suppose n 2Z. MATH 230-01 Introduction to Discrete Mathematics (Ellis) Spring 2009 table like the example in class), 32, 48d (write a proof like Contradiction Proofs MATH 323 DISCRETE MATHEMATICS, Handout 2 Mathematical proof, especially proof by induction Laurence Barker, Mathematics Department, Bilkent University, version: 26 Feburary 2009. DISCRETE MATH: FINAL REVIEW 9 9 a) How do you prove a statement by contradiction? 9 b) Prove: There is no greatest integer. " Thus, to give a proof by contradiction of "If A, then B," we assume that A is true and B is false and derive a contradiction. That is our proof by induction. Math 355 - Discrete Math Notes 2. Discrete Math Structures Every mathematical statement can be negated. This result is called a contradiction. This book is designed for a one semester course in discrete mathematics 10 Methods of Indirect Proofs: Contradiction and Contraposition . Suppose that I wanted to prove that the cube root of Proof by Contrapositive Proof by contrapositive takes advantage of the logical equivalence between "P implies Q" and "Not Q implies Not P". Indirect Proof: Assume . edu) Workshop#2 { on Friday 02/21/14 There are some common techniques to approach a discrete math problem. Proof by Contradiction. We have actually seen another example indirect inference, the method of proof by contradiction. Direct proof 2. [This contradiction shows that the supposition is false and so the given statement is true. Bender and S. There is no largest real number. 331. MTH 220 Discrete Math 3: Proof Techniques (UNDER CONSTRUCTION) the proof by contrapositive, and the proof by contradiction. Chapter 5 is on Contrapositive Proof and Chapter 6 is on Proof by Contradiction. If 3n+2 is odd then . Proposition. Proof: This is a rephrasing. We shall show that you cannot draw a regular hexagon on a square lattice. Exercise 3, Contradiction. Unfortunately, there is no quick and easy way to learn how to construct a Mathematical Induction is a special way of proving things. Consider a simple example. MAT231 (Transition to Higher Math). I. 4. _____ • Trivial proof If we know Q is true then P → Q is true. There are four basic proof techniques to prove p =)q, where p is the hypothesis (or set of hypotheses) and q is the result. Gross for use with Rosen: Discrete Math and Its Applic. In principle Indirect proof is often used when the given geometric statement is NOT true. Download with Google Download with Facebook or download with email. Theorem For every , If and is prime then is odd. Mathematical proofs can themselves be represented formally as discrete structures. Scrambled proofs (Example 1, Exercises 4 and 7) Fill in the blanks (Exercises 1 and 2) Pigeonhole principle in action (Example 7, Practice Problem 4, Exercises 32, 33, and 34 Proof:Let’s rst employ direct proof. Therefore, Jerry is a Math major. Proof Proof In the language of predicate logic, the th eorem is CS 205: Discrete Math see pg. In order to do this, we shall first formally define exactly what we mean by an argument and then discuss different valid The above courses are offered at community colleges in the Los Angeles area. Show that every rational number is algebraic. ” Let q be “I will study discrete math. In fact the case of ``division into cases'' has been proven in example 2. What is the value of x after each of the following statements are encountered in a computer program, if x = 1 before the statement is reached. “I will not study discrete math or I will study English literature. That’s not true, so . Proof: A logical argument establishing the truth of the theorem given the truth of the axioms Here are some good examples of proof by contradiction: Euclid's proof of the infinitude of the primes. Example 4: Prove the following statement by contradiction: For all integers n, if n 2 is odd, then n is odd. Discrete Structures (CS 173) prove by example http:// rjlipton. Example: Give a direct proof of the theorem “If n is an odd For example: proof by contradiction in Chapter 4 with The Halting Problem; constructive proofs in Chapter 8 with "a finite projective plane of order n iff n — 1 mutually orthogonal Latin squares of order n"; complete induction in Chapter 3 with the "optimality (for suitors) of the Deferred Acceptance Algorithm". So you have the first part of an induction proof, the formula that you'd like to prove: In logic and mathematics proof by contradiction is a form of proof that establishes the truth or . ∉ Proof by Contradiction Albert R Meyer contradiction. It contains sequence of statements, the last being the conclusion which follows from the previous statements. e, and prove that a contradiction (or absurditity results). 2k + 2 = 6k+2 = 2(3k + 1). Most of the steps of a mathematical proof are applications of the elementary rules of logic. Hauskrecht Translation • General rule for translation. Discrete math, also known as “math for computer science” is usually a required course for students of computer science. With proof by contradiction, you assume the opposite and follow that line of reasoning to a true implication. For example, to prove that ot all triangles are obtuse", we give the following counter example: the equilateral triangle having all angles equal to sixty. 1. • Look for patterns corresponding to logical connectives in the sentence and use them to define elementary propositions. Farlow Jan 16 '15 at 3:09 Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a common proof technique that is based on a very simple principle: something that leads to a contradiction can not be true, and if so, the opposite must be true. -1. If we can show that pr 2 is also irrational, we have a direct proof. If 3 - n2, then 3 - n. Examples: addratirrat. 9. The negation of this statement is A good example of this is by proving that is irrational. 993 Pages. Explain fully. have no common factors (see Chapter 4). What is a proof by contradiction?. You assume it is possible, and then reach a contradiction. Contradiction Vs. a proposition of the form r /\ ¬ r. Combinatorial proof is Know when to use what technique! Sometimes I will say, "prove by contradiction" and other times you will need to figure out the best technique. assuming what is to be proved false and getting a contradiction as a result. The rest of our course covers a variety of other topics of discrete math, Proof by contradiction. aids in the proof of a theorem/proposition Corollaryis a result whose proof follows immediately from a theorem or proposition Instructor: Dr. Book of Proof, Richard Hammack. And we are done. }\) This will lead to a contradiction. Proving this directly (via constructive proof) would probably be very difficult (if not impossible). The book is divided into four parts. O. Let f be a one-to-one function as above  Discrete Mathematics - Propositional Logic - The rules of mathematical logic specify methods Example − Prove [(A→B)∧A]→B is a tautology Contradictions. (Edit: There are some issues with this example, both historical and pedagogical. 2. Show: R is an equivalence relation. cs2102: Discrete Math Final Exam Solutions (Sat, 16 Dec 2017) Here are some comments on the Final Exam: PDF (and the Original Exam). If you try to do this, you will find that if you make your hexagon very large, then you can get somewhat close to 1 Proof by Contradiction In writing a proof by contradiction, you begin by imagining a situation where the claimed result fails, and show that this is in fact contradictory. Suppose for the sake of contradiction that for every x, there is a ysuch that y2 <x. MATH 215 Discrete Mathematics Worksheets Logic and Proof Prove by contradiction that if x is rational and y is irrational then x+ y is irrational. proof by contradiction is based on the theorem p ≡¬p ⇒ false). ) Example 21 is about infinite number of primes (similar to what was done in class, but not  Induction Examples. A proof is an argument from hypotheses (assumptions) to a conclusion. METHODS OF PROOF 74 2. If it is rainy, then the pool will be closed. is a very unpro esional de nition). If from there we deduce a contradiction, that is a statement of the form C  The following proof is one of the most famous, most often quoted, and most beautiful proofs in all of mathematics. g. Although there may be more then one counterexample to any given false claim, you must always provide by a proof or argument that PatrickJMT » Algebra, Discrete Math, Linear Algebra » Proof by Induction – Example 2 Proof by Induction – Example 3; Proof By Contradiction – Calculus CSE 20 DISCRETE MATH -the set of odd positive integers Example 1 Proof: (Proof by contradiction) Assume towards a contradiction that In this post we'll expand our toolbox of proof techniques by adding the proof by contradiction. . Recall that in our proof of Euclid’s Division Theorem we began by assuming that the theorem was false. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "It is a far finer gambit than any Math 232 - Discrete Math Notes 2. Proof by contradiction is by all means a powerful technique but there is a major pitfall. Assume the opposite of the conclusion in (2). A contradiction occurs $\begingroup$ Do you understand how a proof by contradiction is actually a proof? If so, I could write up a nice, detailed solution, but I wonder if the main issue is in understanding how proofs by contradiction work and are logically valid. All proofs start with assumptions, some of which are axioms: propositions that are assumed to be true. 1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. Proof by contradiction would be saying -Q and then proving that P->-Q is not true. Show that the set of all algebraic numbers is countably infinite. Then we do only logically  Discrete Mathematics A proof by contradiction establishes the truth of a given proposition by the supposition that it is false and For example, the second of Euclid's theorems starts with the assumption that there is a finite number of primes. Homework Discrete Math, MATH 2001-001, Fall 2017 (Click here to return to the MATH-2001 course website. act or an instance of contradicting; a proposition, statement, or phrase that asserts or implies both the truth and falsity of something… Proof by contradiction (also known as reducto ad absurdum or indirect proof) is an indirect type of proof that assumes the proposition (that which is to be proven) is false and shows that this assumption leads to an error, logically or mathematically. Problem: Use proof by contradiction to prove that for all x ∈ R, if x^2 is irrational, then x is irrational. It is sometimes difficult (or impossible) to prove that a conjecture is true using direct methods. CS/Math 240: Intro to Discrete Math 2/10/2011 Solutions to Homework 2 Instructor: Dieter van Melkebeek Problem 1 (a) First consider the case where the sets A and B are disjoint. So let's do an example first to see what's going on. Work: So by proof by Here are some good examples of proof by contradiction: Euclid's proof of the infinitude of the primes. 7 pg 91 # 11. Section 1. 93 Example 3 Math 2001: Setting up Proof by Contradiction (Katherine Stange, Spring 2018) For each theorem, set up the form of a proof by contradiction. 9 Logical Equivalence • “biconditional” or “if and only if” (“iff”) or “necessary and sufficient” or “logically equivalent” Section 2. 2 February 11, 2013 2013. Learning Objectives: To prove a statement, we can assume it's negation and show this leaves to a contradiction. Proofs are valid arguments that determine the truth values of mathematical statements. Principle of You can think of the proof by (mathematical) induction as a kind of recursive proof: Instead of Proof (by contradiction) Assume that for some values of n, PHnL is false. what I need is how to solve geometric proofs im having big trouble of how to do proofs like one in the first example. However, you can readily see that not every Reductio Ad Absurdum proof is a Proof by Contradiction of the premise. 1332 > 11. 11. What makes it different is the way it begins and ends. P to show that every positive integer is the sum of an even (easy proof by Truth tables – the conditional and the biconditional (“implies” and “iff”) Just about every theorem in mathematics takes on the form “if, then” (the conditional) or “iff” (short for if and only if – the biconditional). com/2011/01/08/proofs-by-contradiction-and-other-dangers/. the format of the solution is in the attachement doc. We'll also expand on our knowledge of functions on sets, and tackle our first nontrivial theorem: that there is more than one kind of infinity. We have had good discussions on mathematical proofs, so I am planning to create a mathematical proof series that will discuss the basics such as direct proof, indirect proof, and proof by mathematical induction. A real number r is called algebraic if it is the solution of a polynomial with integer coefficients. A proposition is said to be a contradiction if its truth value is F for any assignment of truth values to its components. Links Book of Proof, chapter 5, by Richard Hammack. Proof by contradiction. is not an irrational number. Assume $\sqrt{2}$  ICS 141: Discrete Mathematics I (Fall 2014) Proof by Contradiction: Assume p ∧¬q and show this leads to a contradiction ((p∧¬q) → F). For implications, we saw how to write a direct proof and an indirect proof. For example, Given: is not a right angle. Proof that the square root of 2 is irrational. Discrete Mathematics ProofsH. 17 Aug 2017 This set of Discrete Mathematics Assessment Questions and Answers focuses on “Types of c) Counter Example b) Proof by Contradiction 7 Jul 2018 Discrete Math: A Proof By Contraposition can be formed by reversing the direction of inference and negating both terms for example : p → q Discrete Mathematics. Try to solve all of them. For reference, here is a short explanation of the difference between proof of negation and proof by contradiction. Many of the statements we prove have the form P )Q which, when negated, has the form P )˘Q. Prove: . ), chapter 3. Show it is true for the first one; Step 2. Suppose you want to use a proof by contradiction to prove the claim: 14 08/27/09 MATH 224 – Discrete Mathematics Examples Existence Proof (Non constructive): (Often non constructive proofs assume something does not exist and then show a contradiction. If you are trying to prove a negation of a statement then what you call “proof by contradiction” is unavoidable in a certain sense (there are meta-theorems in logic about this). • Show that if 3n + 2 is an odd integer, then n is odd. Mathematicians thought the proof was right until another mathematician named Heawood found a fatal flaw in the argument. This is the general form for an implication. But this is even! A contradiction! This concludes the proof. "Since the sum of two even numbers 2a and 2b One of several different ways to prove a statement in mathematics is proof by contradiction. They simply make the proof more readable. However, contradiction proofs tend to be less convincing and harder to write than In a "proof by contradiction", the idea is to show that by making the assumption that the statement is false, that you would come to a logical contradiction. February 11, Proof by Contradiction . 1 Direct Proof And this completes the proof. Direct Proof Hence, by definition of ration x is rational, which is a contradiction. Next week’s Friday Proof Quiz will be a Combinatorial Proof (count something two different ways to prove a formula). I’ve tried to make my prose and organization of the material as reader-friendly as his. For example, MATH 2450 above is comparable to MATH 10 (Discrete Structures) at Santa Monica College or MATH 272 (Methods of Discrete Mathematics) at ELAC. It is rainy. Discrete Introduction, Third Edition is an inspiring model of a textbook written for the learner of discrete mathematics rather than the teacher. Call it , say of say , from that makes YBœ+ÞÐ BßBœ+Y TÐ+ÑYou can continue 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. Example 1. For an indirect proof (see below), the last step must be a contradiction. A famous example of proof by contradiction shows that is an irrational number: We're on problem number four, and they give us a theorem. It says a triangle has, at most, one obtuse angle. Although the vacuous proof example on slide 40 is a contradiction. A statement in sentential logic is built from simple statements using the logical connectives , , , , and . An argument is a One of the most convincing was a proof using pictures by Kempe in 1879, 26 years later. Proof: Suppose m and n are any [particular but arbitrarily chosen] odd integers. A propositional form is a self-contradiction iff it is equivalent to the constant F . 5-§1. The statement \A implies B" can be written symbolically as \A → B". I think what Arnaud is saying is that you are calling two things “proof by contradiction” when only one of them deserves this name. 3: Valid and Invalid Arguments Now we have developed the basic language of logic, we shall start to consider how logic can be used to determine whether or not a given argument is valid. Invalid proof techniques, courtesy of I˘sil: Proof by obviousness \The proof is so clear it need not be mentioned!" Proof by intimidation \Don’t be stupid { of course it’s • Example: You can have free coffee if you are senior citizen and it is a Tuesday Step 1 find logical connectives CS 441 Discrete mathematics for CS M. wordpress. Below is a sample indirect geometric proof. The latter implies that n = 2k for some integer k, so that 3n + 2 = 3(2k) + 2 = 2(3k + 1). Topics: Logic, truth tables, Quantifiers Set theory, Intersections, Unions , Complements, applications to solving inequalities, cartesian products, relations, ordering, Direct proofs, proof by contradiction, proof by induction, how to approach proving, disproof by counter example Introduction to Proofs Alvin Lin Discrete Math for Computing: January 2017 - May 2017 Introduction to Proofs Rough de nitions/guidelines: A theorem is a statement that can be shown to be true. A powerful type of proof in mathematics is proof by contradiction. 3 Proofs A mathematical proof is a logical argument that justifles that a certain conclusion must follow from certain hypotheses. Summarize the proof. Assume for the sake of contradiction that there was a largest real number, In proof by contradiction (also known as reductio ad absurdum, Latin for "by reduction to the absurd"), it is shown that if some statement were true, a logical contradiction occurs, hence the statement must be false. A summary of proof techniques-- we have not covered all of these, but we have seen at least half of them! Important DISCRETE MATH TOPICS: even and odd integers AND ALL OTHER KINDS OF DIVISIBILTY Discrete Math. What comes between the first and last line of course depends on what A and B are. Hence, proof by contradiction establishes p ® q is T by assuming that the hypothesis p is T and that the conclusion q is F and by using p and ¬ q and other axioms, definitions, and previously derived theorems, derives a contradiction, i. Discrete Structures Lecture Notes Vladlen Koltun1 Winter 2008 1Computer Science Department, 353 Serra Mall, Gates 374, Stanford University, Stanford, CA 94305, USA; vladlen@stanford. G. Lecture 2: Proof . For example: Example 1. This expression worked for the sum for all of positive integers up to and including 1. Step-by-step solutions for proofs: trigonometric identities and mathematical induction. Then show that something absurd will happen. Before I start the proof, I want to say a few things. Prove that there are computational problems that cannot be solved by a computer . Seeing this I'm asking myself If I know anything that really is proof by contradiction, in the sense that there is no translation to proof of negation. The problem explicitly states, "Use proof by contradiction to show that for all odd Type this as an example (replace the [-; with [; when typing):. scientists recommended that “discrete mathematics should be part of the first two years of the standard mathematics curriculum at all colleges and universities. , homework assign-ments). The first of these two proofs illustrates proof by induction. 6 (147 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. To prove B  7 Jul 2018 Discrete Math: A Proof By Contraposition can be formed by reversing the direction of inference and negating both terms for example : p → q Covered: Proof by contrapositive, proof by contradiction; Class notes (see full notes if you Rosen (Discrete Mathematics and its Applications, 2nd ed. But every number’s square is nonnegative, so y2 0, a contradiction. the qustion is proof by contradiction is in the jpg file 2. For instance, suppose we want to prove ‘If [math]A[/math], then [math]B[/math]&#039;. Then, deductive reasoning will lead to a contradiction: two statements that cannot both be true. In this case, there are in nitely The Propositional Logic Calculator finds all the models of a given propositional formula. Example 15 is about sqrt(2) being irrational. It is a particular kind of the more general form of argument known as reductio ad absurdum. Justification Column. This Lecture Now we have learnt the basics in logic. All mathematical proofs involve some axiom(s). ○ For example, consider the statement “it is snowing outside. Example. Start studying Discrete Math: Proofs and Algorithms. These include proof by mathematical induction, proof by contradiction, proof by exhaustion, proof by enumeration, and many To prove that square root of 5 is irrational, we will use a proof by contradiction. That would mean that there are two even numbers out there in the world somewhere that'll give us an odd number when we add them. Discrete mathematics is foundational material for computer science: Many areas of computer science require the ability to work with concepts from discrete mathematics, specifically material from such areas as set theory, logic, graph theory, combinatorics, and probability theory. Therefore, Alice is either a Math major or a CSI major. 4 . You should always be able to identify how it follows from earlier statements. $\endgroup$ – Daniel W. Greek philosopher, Aristotle, was the pioneer of logical reasoning. We can have Proof by Contradiction and Contrapositive. All major mathematical results you have considered Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Now, this proof by Kempe consisted of drawing pictures of what maps have to look like. This is why many mathematicians often call a Proof by Contradiction (of the premise) a Reductio Ad Absurdum proof. Proposition 2. 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. Fall 2014. For starters, let's negate our original statement: The sum of two even numbers is not always even. Divisibility proofs (Example 6, Exercises 3, 4, 5) Fill in the blanks (Example 6, Exercises 3, 4, 5) Section 2. Assume that the statement is false. 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. Early in the season, S(N) Start studying Discrete Math. Page 21 Introduction to Discrete Mathematics for Computer Science Specialization. Let me prove that that can't happen Chapter 17 Proof by Contradiction This chapter covers proofby contradiction. ] This completes the proof. The best way to explain indirect proofs is by showing you an example. A proposition that is neither a tautology nor a contradiction is called a contingency. For the implication to be true, !p must also be false. We all like to assert things, and few of us like to say things that turn out to be false. The column labeled “justification” is analogous to the comments that appear in most good computer programs. Positive example is not a proof of universal quantification. Impossibility and an Example Proof by Contradiction Many of the most impressive results in… Here are several examples of properties of the integers which can be proved using the well-ordering principle. Shorser This proof structure allows us to prove that a property is not true by pro-viding an example where it does not hold. I am grateful to Kishalaya Saha and Anna Romanova, who bravely used these notes for their own Discrete Math What Is Indirect Proof? Proof by contradiction. Each step of the argument follows the laws of logic. 's answer and the ensuing discussion. We’ll need this method in Chapter 20, when we cover the topic of uncountability. 6. Our examples and steps show it's used to prove any statement in mathematics. is odd. CLRS is the primary suggested textbook for my Algorithms course. If an assertion implies something false, then the assertion itself must be false! Albert R This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. n Less theory, more problem solving, focuses on exam problems, use as study sheet! CS160 - Fall Semester 2015 Proof. 9 MATHEMATICAL PROOFS (INDIRECT) def: An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. Instructor: Dr. Thus, the proposition is true. So we showed , we proved our base case. Its origins date back more than 2000 years to  10 Mar 2014 Example-1 Proof: We wish to prove that whenever f(m) = f(n) Proof. I encourage all high school students o “I will not study discrete math or I will study English Example: Give a direct proof of the theorem “If n is an odd Proof by Contradiction: o “I will not study discrete math or I will study English literature. Proof techniques, proof by contradiction, mathematical induction. Proof by contradiction Po-Shen Loh 2 September 2010 1 Warm-up Putnam 2004/A1. Example: The proposition p∧¬p is a contradiction. The negation of this statement is So, I have a math problem that states: Prove by contradiction that if n is a perfect square, then n+2 is not a perfect square. Methods of Proof Lecture 3: Sep 9 2. This is an excellent way to help your grade! Clark Catalog Math 114 course description: Covers mathematical structures that naturally arise in computer science. In automated reasoning systems, proof by resolution is combined with proof by contradiction. erty. DIRECT PROOF. 64 Example 1. If every step of the proof is valid, then the only possible reason for the contradiction is that the supposition is false. We write the negated conclusion as clauses and add the clauses to the hypothesis. The text covers the mathematical 6 and 7 all combine to give an argument by contradiction] 10. So the way you prove by a contradiction, you're like, well what if this weren't true. Here you will find very diverse general questions about the different topics of discrete math. Solution:  In a Proof by Contradiction, you start by assuming that the proposition What are the best examples of self contradicting statements in maths? Proof by contradiction (also known as reducto ad absurdum or indirect proof) is an Examples. 1) until we derive a contradiction. See 5. Induction proofs allow you to prove that the formula works "everywhere" without your having to actually show that it works everywhere (by doing the infinitely-many additions). " is "A is true and B is false. We mentioned some proof paradigms that are generally applicable. Proof: Suppose not. Euclid's proof of this proposition was not by contradiction. In mathematics, a statement is not accepted as valid or correct unless it is accompanied by a proof. ) 9 a) How do you prove \8x 2D, if P(x) then Q(x)" by contraposition? 9 b) Prove: For all integers a, b, and c, if a 6jbc then a 6jb. This would prove that the statement must be true. It is not a discrete math book, but a few of the topics on this page are covered briefly in CLRS (some in the appendix, some in regular chapters). A direct proof is a sequence of statements which are either givens or deductions from previous statements, and whose last statement is the conclusion to be proved. b. But before I do that, let me continue with more examples of proof by contradiction Proving Statements with Contradiction 105 The idea of proof by contradiction is quite ancient, and goes back at least as far as the Pythagoreans, who used it to prove that certain numbers are irrational. Discrete Mathematics by Section 3. 33. Theorem: If A then B. b . Based on (1) and (3), nd a contradiction, or logical inconsistency. Solution 1. It does not rely on computers at all, but instead is a "proof by contradiction": if √ 2 WERE a rational number, we'd get a contradiction. It has only 2 steps: Step 1. p ¬p p∨¬p p∧¬p T F T F T F T F F T T F F T T F tautology contradiction Intro Rules of Inference Proof Methods Rules of Inference for Propositional Logic Which rule of inference is used in each argument below? Alice is a Math major. Example 3: Prove the following universal statement: The product of any two odd integers is odd. Fair enough. ♦ Proposition 1. 5 Methods of Proof 1. We will review both correct & fallacious inference rules, & Example: The negation of "If A, then B. Do not write a full proof. The first part introduces sets and logic and includes a chapter on counting, which gives a brief introduction to elementary combinatorial techniques. The proof that √ 2 is indeed irrational is usually found in college level math texts, but it isn't that difficult to follow. 25 for example), 12, 30 direct proof, indirect proof, proof by contradiction, proof by cases, non-constructive existence proof Rules of Inference and Logic Proofs. (a) if 2 + 3 = 6 AND 3 + 4 = 7 then x:= x + 1 (b) if 2 + 3 = 6 XOR 3 + 4 = 7 then x:= x + 1 Symmetric, Transitive, and Reflexive Relations Date: 11/10/98 at 11:30:27 From: Mike Subject: Discrete math Suppose R is a symmetric and transitive relation on A. Hira Shah. 3336: Discrete Mathematics Proof Methods and Strategy/Introduction to Sets 9/26 Section 3. 1-2. We'll also look at some examples of both types of proofs in both abstract and real-world Discrete Mathematics Inductive proofs Saad Mneimneh 1 A weird proof Contemplate the following: 1 = 1 1+3 = 4 1+3+5 = 9 1+3+5+7 = 16 1+3+5+7+9 = 25 It looks like the sum of the first n odd integers is n2. For example: Px = “4x2 + 3 is divisible by 5”. (e. 7 CPU-years in total on Discrete Mathematics. n) is odd. 5 introduces proof by contradiction. When you read or write a proof you should always be very clear exactly why each statement is valid. 3: Propositional Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. 5. 5 Proofs in Predicate Logic 4 Theorem 6Theorem 6: (Proof by Contradiction) : (Proof by Contradiction): (Proof by Contradiction) If x y, are positive integers, then x y2 2− ≠ 1. P (negation of conclusion) 2. Example: Let p be I will study discrete math. Definiton. In a direct proof, we A proof by contradiction can also be used to prove a statement that is not of the form of an implication. Prove the statement is true: Let xand ybe real numbers. 3 Proof by Contradiction In a proof by contradiction, we suppose the negation of what we are trying to prove and try to reach a contradiction. Maybe you get to a contradiction: proof by contradiction. A proof by contradiction starts with an assumption that you eventually want to reject. Proof: Assume that x is even (neg of concl). Let us go over these techniques and discuss how to apply them to new problems (e. Learn the definition of this method and observe how it is applied to proving a statement's truth value Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. _ •Proof by Contradiction Students in MATH 210 develop the ability to apply symbolic logic to clearly recognize the contradiction in the evidence provided. A conditional statement is logically equivalent to its contrapositive! (This is very useful for proof writing!) The converse of p !q is q !p. Thus, the statement to be proved must be true, because its negation is false. Contraposition and Other Logical Matters by L. What are the assumptions? 2. Then n= 2k. A vacuous proof of an implication happens when the hypothesis of the implication is always false. •Proof : Assume that the statement is false. In that case the number of elements in the union A[B is simply the sum of the number of elements in A and the number of elements in B: jA [Bj= jAj+ jBj. We have to prove the following statement: 8n 2Z, 3 does not divide 3n + 2. This is apowerful prooftechnique that can be extremely useful in the right circumstances. Proof by contradiction is often used when you wish to prove the impossibility of something. Master the math powering our lives and prepare for your software engineer or security analyst career What is a proof and how do you prove things in discrete math? What is the proof that the square root of 2 is irrational by contradiction? What is Euclid's proof by contradiction that there are an infinite number of prime numbers? How many pairs of people can you form in a group of n people? What is the formula for the nth triangle number? CME 305: Discrete Mathematics and Algorithms TA: Kun Yang (kunyang@stanford. Or if we assume it works for integer k it also works for the integer k plus 1. Fortunately, there is a powerful proof technique called proof by contradiction,. , 5th Ed. This lesson defines both direct and indirect proofs and, in turn, points out the differences between them. You should take this course if and only if. We used the book "Discrete and Combinatorial Mathematics" by Ralph Grimaldi for reference while learning about the content ourselves and for the videos. Have a great break! Problem Set Omega Highlights (Tue, 12 Dec 2017) Here are some of our favorite Problem Set Ω submissions (mostly in no particular Do not use this proof as an occasion to talk about proof by contradiction. Start the proof by assuming the statement IS true. All new quiz solution videos plus an additional graph theory section to help students master all aspects of Discrete Math! Welcome to Master Discrete Math! This is a course designed to help you master the difficult topics of Discrete Math and get you prepared for a career in computer science, actuarial science, mathematics, or even engineering! As the name suggests, this is a book on Algorithms, which is a topic that relies heavily on Discrete Math. 3 Jul 2011 In mathematics, a formal proof of a proposition is a chain of logical deductions of writing a proof, and walk you through some example proofs. Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms. Or even there is something that really only can be proved using proof by contradiction and has been showed that is the only way (seems to be a really hard thing to show). Example 1: Prove that if x is a positive integer and x = -x, then x2 = x. In practice, you assume that the statement you are trying to prove is false and then show that this leads to a contradiction (any contradiction). Build a Foundation for Your Career in IT. Math 151 Discrete Mathematics [Methods of Proof] By: Malek Zein AL-Abidin Direct Proof: A direct proof shows that a conditional statement p q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs. 6: a theorem If x2 is odd, then so is x. 2 = m 2 /n 2 3 1. Proof by Contradiction (Part 1) · Proof by Contradiction (Part 2) · Proof by Contraposition · Proof by Counter Example. And in particular, we're going to look at a proof technique now called proof by contradiction, which is probably so familiar that you never noticed you were using it. Extra explanations by topic Example: Use contradiction and W. Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and technological applications. For example, if given a proof by contradiction, could one not just create a direct implication proof via simple transformations on the proof by contradiction? Unless I am wrong already, I think this does also apply to proofs by induction if one could accept induction as a proof method at all (and not an axiom by itself). For example Observations, measurements, and experimentations are not proof If a formula works for 1 million specific examples, this is still not a proof Understanding a proof can be a daunting task. Our next example follows their logic to prove that (2 is irrational. The rejection is due to the assumption leading to a conclusion that we know is impossible. This famous problem is not of the form (p q), but we can use a slight variation of proof by contradiction. This can occasionally be a difficult process, because the same statement can be proven using For a direct proof, the last step must be the conclusion of the theorem. A preview of Chapter 4. The phrases "for all X Here is where mathematical proof comes in. edu. 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. Learn the definition of this method and observe how it is applied to proving a statement's truth value One of several different ways to prove a statement in mathematics is proof by contradiction. 29 Mar 2010 in which everything seems to be called “proof by contradiction”. 1 Eligibility. 3336: Discrete Mathematics Proof Methods 11/41 The Argument We can express the premises (above the line) and the conclusion (below the line) in predicate logic as an argument: We will see shortly that this is a valid argument. The material typically covered in this class for CS students overlaps with fields such as cryptography, logic, information theory, theoretical computer science, artificial intelligence, design of algorithms, and others, and teaches the basic language and structures used To deduce new statements from the statements whose truth that we already know, Rules of Inference are used. You very likely saw these in MA395: Discrete Methods. An important feature of the presentation is the emphasis on how not to use proof b contradiction. This is a slight oversimplification, as there are a great many proof techniquesthat havebeen developedover thepast two centuries. ] By the definition of odd numbers, we have n = 2r + 1 for some integer r m = 2s + 1 for some integer s For the most part, an indirect proof is very similar to a regular proof. Equivalently, we could just prove the logical negation of the given statement, which is the statement 9x8y: y 2 x Discrete Math Basic Proof Methods Overview of Basic Proof Methods (§1. I Proof by contradiction is a very widely used proof strategy Is l Dillig, CS243: Discrete Structures Mathematical Proof Techniques 15/38 Example I Prove by contradiction that "If 3n +2 is odd, then n is odd. The truth or falsity of a statement built with ICS 141: Discrete Mathematics I – Fall 2011 7-13 University of Hawaii Proof by Contradiction Example ! Theorem: is irrational. How to Do Math Proofs. See the course catalog for more information on these courses. Here is a template. Example 4: Discrete Math. Example: A Diophantine Equation Recall we discussed the following methods of proofs: - Vacuous proof - Trivial proof - Direct proof - Indirect proof - Proof by contradiction - Proof by cases. A proof by contradiction establishes the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that is contradictory to something that is proven to be true. You should also read Chapters 2 and 3 of the textbook, and look at the Exercises at the end of these chapters. Prove that √2 is irrational. 1 and Its Applications 4/E Kenneth Rosen TP 10 P may be a conjunction of other hypotheses. How to use contradiction in a sentence. Then there exists integers . , proof methods) can be formalized in terms of rules of logical inference. The conventional practice of rearranging it into a proof by contradiction not only adds an extra complication that serves no purpose, but also leads to confusions such as the belief that if you multiply the Section 1. Proof by Contradiction . Then reason correctly from the given information until a contradiction of a known postulate, theorem, or given fact is reached. This proof is an example of a proof by contradiction, one of the standard styles of mathematical proof. In this handout, we discuss two fundamental techniques of mathematical argument which Learn the core topics of Discrete Math to open doors to Computer Science, Data Science, Actuarial Science, and more! 4. by contradiction). In the following examples, we are given the truth values of the hypothesis and the conclusion and asked to determine the truth value of the conditional. For other examples, see proof that the square root of 2 is not rational (where indirect proofs different from the above one can be found) and Cantor's  Proof by Contradiction (Example 1). a. Comment. What are Rules of Inference for? Mathematical logic is often used for logical proofs. Basketball star Shanille O’Keal’s team statistician keeps track of the number, S(N), of successful free throws she has made in her rst N attempts of the season. Let n0 be the least. Proof by contradiction: Example. 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". Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. ! Proof:! Assume that is rational. The second section covers the basic techniques for proving conditional statements: direct proof, contrapositive proof, and proof by contradiction. This is a creative commons textbook all about proofs, that includes many of the topics we cover in cs2102. 1332 ≤ 11? If so, 1332 ≤≤ 1. Maybe you ignore the premises and do an contraposition proof. Discrete Mathematics Chapter 3 Method of Proof Proof by contradiction: proving an assertion by assuming that it is false and deriving a contradiction. Assume that the statement of the theorem is false. n. By the way, this post is something I have been meaning to write for a while. 9   4-6, proof by contradiction. 1 . • Example: PatrickJMT » Algebra, Discrete Math, Linear Algebra » Proof by Induction – Example 3 Proof by Induction – Example 2; Proof By Contradiction – Calculus Discrete Math Lecture 03: Methods of Proof 1. " Is l Dillig, CS243: Discrete Structures Mathematical Proof Techniques 16/38 Another Example I Recall:Anyrational numbercan be written in Proofs Involving Quantifiers For a given universe :Y Theorem TheoremÐaBÑTÐBÑ ÐbBÑTÐBÑ Proof Proof Let be an arbitrary member Somehow show that there is a valueB of . Show that if any one is true then the next one is true The activity’s sample proofs also illustrate how to clearly signal proof by contradiction (e. We hope everyone found the class worthwhile and enlightening. ”1 The joint ACM/IEEE Computing Curricula 2001 report provides the latest description of the discrete math knowledge 2. CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Reading 5 : Induction Instructors: Beck Hasti and Gautam Prakriya In the last reading we began discussing proofs. This is so for good reasons, but conflation of different kinds of proofs is bad mental hygiene which leads to bad teaching practice and confusion. Writing a proof can even be more daunting I kept the reader(s) in mind when I wrote the proofs outlines below. The only limitation for this calculator is that you have only three atomic propositions to choose from: p,q and r. You may earn one badge in office hour next week (any badges except Synthesis). Let's try proving that. We will focus on Counting and Combinatorial Proof all next week. However, by contradiction we have a fairly simple proof. Truth Tables, Tautologies, and Logical Equivalences. • Direct proof • Contrapositive • Proof by contradiction • Proof by cases 3. MAT231 (Transition to Higher Math) Contrapositive Proof Fall 2014 12 / 13. 1, Double Negation 3. Is. 1 Direct Proofs and Counterexamples Axiom: Proposition that is assumed to be true. ” Proof by Contradiction . This will demonstrate the statement. Although this is a counterexample, we still had to PROVE that it was in fact a counterexample and in doing so used both a proof by contradiction (this was the overall method of the proof) by a construction (of =. Recall that a number is rational if it equals a fraction of two I. Therefore a 2 must be even. Observe that any rational number r can be written as p 2pr 2 We already proved p 2 is irrational. Typically relies on the proof by contradiction – negate the existentially quantified statement and show that it implies a contradiction. Contradiction 4. Sieving: 35. The contrapositive of a conditional statement of the form p !q is: If ˘q !˘p. NEGATING QUANTIFIED STATEMENTS: The phrases "for some X" and "there exists X" mean the same thing and are called existential. Proof: This is simply a rephrasing. What is the conclusion? 3. ” ○. Discrete Mathematics - Proofs 1. The syllabus and homework for Fall 2006 Discrete Math is here. Then, by de nition of j(see De nition 1 in Section 4. There is no convenient way of doing this, so we must turn to the indirect method of proof. Then we have 3n + 2 is odd, and n is even. 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. ” . And it also works if we assume that it works for everything up to k. ♦ Coursenotes by Prof. Some are basic and have a very concrete answer, but others are actually very deep, and even "philosophical", so they may not have a unique answer. You have credit (or passed the placement test) for at least one of CS 1110, CS 1111, CS 1112, CS 1113, or CS 1120; 2. This is called the Law of the Excluded Middle. Then we have 3n + 2 is odd,  Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a common proof technique that is based on a very simple  Motivating Example. b≠ 0 . ) Information on working together on daily homework: Unless otherwise noted you may work together on the daily homework, to help each other understand and solve the problems. Then. _ I will not study discrete math or I will study English literature. So 3n+2 = 3. Then we can take x= 0, and so there must be some ysuch that y2 <0. proof by contradiction example discrete math

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