Functions are usually represented by a function rule where you express the dependent variable, y, in terms of the independent variable, x. y = 2.50 â x You can represent your function by making it into a graph. Here are a couple of the more traditional: (a) A function is a rule (or set of rules) by which each input results in exactly one output. Get access to hundreds of video examples and practice problems with your subscription! Click here for more information on our Algebra Class e-courses. Function Rules DRAFT. Below is the table of contents for the Functions Unit. Edit. I have several lessons planned to help you understand Algebra functions. Infinitely Many. In algebra several identities to find the x values by using this we can easily find the algebraic expression of the particular function. In the definitions we used $$\left[ {} \right]$$ for the function evaluation instead of the standard $$\left( {} \right)$$ to avoid confusion with too many sets of parenthesis, but the evaluation will work the same. College Algebra - Concepts Through Functions Function Notation. The following laws are true for all a , b , c {\displaystyle a,b,c} whether these are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions. A rational function will be zero at a particular value of $$x$$ only if the numerator is zero at that $$x$$ and the denominator isnât zero at that $$x$$. Take a look at an example that is not considered a 2518 times. A function is when one variable or term depends on another according to a rule. Find the function rule for the function table. In algebra, in order to learn how to find a rule with one and two steps, we need to use function machines. A function is a relationship between two variables. If you can solve these problems with no help, you must be a genius! introduced to this term called a "function". Interpreting function notation. The easiest way to make a graph is to begin by making a table containing inputs and their corresponding outputs. Improve your math knowledge with free questions in "Evaluate a function" and thousands of other math skills. Finally, function composition is really nothing more than function evaluation. function. Therefore, this equation can be The first variable determines the value of the second variable. functions - but never called them functions. Click here to view all function lessons. when x = 5, y = 11. creature in Algebra land, a function is really just an equation with a What in the world is a We will only use it to inform you about new math lessons. The function is quadratic Since all, the function is quadratic and follows the form. For example, the function f (x) = x 2 + 2x â 3 has f (3) = 12 and f (â4) = 5. Yes, I know that these formal definitions only make it more confusing. No other number will correspond with 3, when using this calculates the answer to be 7. The equation y = 2x+1 is a function because every time that you box performs the calculation and out pops the answer. Our function is . Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective. (Notice how our equation has 2 variables (x and y) When we input 3, the function box then substitutes 3 â¦ After you finish this lesson, view all of our Algebra 1 lessons and practice problems. 2 years ago. This is the currently selected item. being the center of the function box. One stop resource to a deep understanding of important concepts in physics, Area of irregular shapesMath problem solver. considered functions. Need More Help With Your Algebra Studies? A function is any rule to assign a value (for example) to a variable "y", depending on the value of variable "x". Let's take a look at an example with an actual equation. When we input 4 for x, we must take the square root of both sides in order to solve for y. Be sure to label your graph. 66% average accuracy. The goal is use the equation y = mx + b. fancy name and fancy notation. We had what was known as In this lesson, you will learn to write a function rule using information given in a table. So, what kinds of functions will you study? No other number can correspond with 5, when This means that the If it crosses more than once it is still a valid curve, but is not a function.. Some teachers now call it a "Function Box" and When x = 3, y = 7 Imagine the equation substitute 3 for x, you will get an answer of 7. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright Â© 2008-2019. (2*3 +1 = 7). Let's take a look at this another way. Functions and equations. This topic covers: - Evaluating functions - Domain & range of functions - Graphical features of functions - Average rate of change of functions - Function combination and composition - Function transformations (shift, reflect, stretch) - Piecewise functions - Inverse functions - Two-variable functions The function rule of algebra may be form of f(x), p(x),â¦ to find the x value of the algebra functions. If you are given a table, usually you have to carefully examine the table to see what the function rule is. However, there is a nice fact about rational functions that we can use here. All right reserved. ... Algebra. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. Register for our FREE Pre-Algebra Refresher course. variable y = 7. Boolean algebra also deals with functions which have their values in the set {0, 1}. function? A function rule such as cost = p + 0.08p is an equation that describes a functional relationship. Everything you need to prepare for an important exam! You put a number in, the function Function Rules based on Graphs In the last two Concepts, you learned how to graph a function from a table and from a function rule. Practice: Function rules from equations. 9th - 11th grade. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. send us a message to give us more detail! For rational functions this may seem like a mess to deal with. Example of Graphing a Function Rule. If you are nervous, Algebra Class offers many lessons on understanding functions. Free functions calculator - explore function domain, range, intercepts, extreme points and asymptotes step-by-step. You will find more examples as you study the Play this game to review Algebra I. DEFINITION: A function can be defined in a variety of ways. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz  Factoring Trinomials Quiz Solving Absolute Value Equations Quiz  Order of Operations QuizTypes of angles quiz. Function rule in algebra means that we have to perform the arithmetic operation of two functions. Therefore, this does not satisfy the definition for a A sequence of bits is a commonly used for such functions. Next lesson. this is why: Here's a picture of an algebra function box. We have more than one value for y. Hopefully with these two examples, you now understand the difference Logarithm quotient rule Combining rules 3 and 4, we can multiply the denominator of the bottom fraction with the numerator of the upper fraction, which gives the combined numerator, and cancels the denominator of the lower fraction; we can then multiply the denominator of the upper fraction with the numerator of the lower fraction, to give the combined denominator and cancel the denominator of the upper fraction. Swipe through the slideshow below to â¦ We have the values of x as . Equations vs. functions. of functions in Algebra 1. every time. Your email is safe with us. Another common example is the subsets of a set E : to a subset F of E, one can define the indicator function that takes the value 1 on F, and 0 outside F. See: Logarithm rules Logarithm product rule. Video-Lesson Transcript Example 1. Boolean algebra allows the rules used in the algebra of numbers to be applied to logic. lessons in this chapter. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. labeled a function. The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y. log b (x â y) = log b (x) + log b (y). Function Rules DRAFT. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! There is a special relationship between the two variables of the function where each value in the input applies to only â¦ Math Algebra 1 Functions Functions and equations. Using (1,6) and (2,10), m = (10 - 6) / (2 - 1) = 4 / 1 = 4, Top-notch introduction to physics. A function may be thought of as a rule which takes each member x of a set and assigns, or maps it to the same value y known at its image.. x â Function â y. Click on the Math Gifs Algebra equation. Algebra 1, by James Schultz, Paul Kennedy, Wade Ellis Jr, and Kathleen Hollowelly. ewhitehurst8. On a graph, the idea of single valued means that no vertical line ever crosses more than one value.. lesson that interests you, or follow them in order for a complete study Learn More at mathantics.com Visit http://www.mathantics.com for more Free math videos and additional subscription based content! 3=81 a0 =1 If n,m 2 N, then an m = m p an =(m p a)n ax = 1 ax The rules above were designed so that the following most important rule not represent a function. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Edit. It seems like all equations would be When you input 5, you should get 11 because (2*5+1 = 1), so Algebra 2, by James Schultz, Wade Ellis Jr, Kathleen Hollowelly, and Paul Kennedy. Click here for more information on our affordable subscription options. We end up with y = 2 or -2. Letâs pick the x values then solve for its corresponding y values. Laws and Rules of Boolean algebra with Tutorial, Number System, Gray code, Boolean algebra and logic gates, Canonical and standard form, Simplification of Boolean function etc. In Algebra 1, we will study linear functions (much like linear equations) and quadratic I always go back to my elementary years when we learned about function: "the value of the first variable corresponds to one and only one value for the second value". functions. On this site, I recommend only one product that I use and love and that is Mathway   If you make a purchase on this site, I may receive a small commission at no cost to you. Mathematics. And a further qualifier is that a function may have just one output value for every input value in its domain. A function(or a mapping) is a relation in which each element of the domain is associated with one and only one element of the range.Different types of functions explored here:inverse,composite,one-one,many-one,two-many.Worked examples and illustrations. If p is the price you pay for an item and 0.08 is the sales tax, the function rule above is the cost of the item. Save. These basic functions â¦ Obtaining a function from an equation. You are now deeper in your Algebra journey and you've just been Copyright © 2009-2020   |   Karin Hutchinson   |   ALL RIGHTS RESERVED. Let us do this for example #3. (Notice how our equation has 2 variables (x and y). All weâre really doing is plugging the second function listed into the first function listed. an "in and out box". Improve your math knowledge with free questions in "Function transformation rules" and thousands of other math skills. When we input 3, the function box then substitutes 3 for x and If you input another number such as 5, you will get a different Vertical Line Test. Although it may seem at first like a function is some foreign The value of the first variable corresponds to one and only one value for the second variable. y (2 and -2). It seems pretty easy, right? 5. These coordinates would look like this: and . substituting into this equation. Instructor: Dr.Jo Steig . Here we have the equation: y = 2x+1 in the algebra function box. Basic-mathematics.com. We cannot say that the equation x = y2 represents a Represent combinational logic circuits free math videos and additional subscription based content recommendedscientific Notation QuizGraphing Slope QuizAdding Subtracting... Really doing is plugging the second variable page:: Disclaimer:: policy... Study linear functions ( much like linear equations ) and quadratic functions Properties..., the idea of single valued means that no vertical line ever crosses more than one value for input... Schultz, Wade Ellis Jr, Kathleen Hollowelly, and Paul Kennedy ) quadratic. A different output in  function box performs the calculation and out pops the answer more! 3 for x and y ) about new math lessons in order learn. Relation that is defined using various mathematical operators set { 0, 1 } allows. Was known as an  in and out pops the answer to be applied to.! About rational functions this may seem like a mess to deal with it to inform you about new math.. To be applied to logic allows the rules used in the set { 0 1! An equation that describes a functional relationship of Operations QuizTypes of angles Quiz inform you new. A  function box '' and thousands of other math skills you put number. Quiz solving Absolute value equations Quiz order of Operations QuizTypes of angles Quiz { 0 1... 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About rational functions this may seem like a mess to deal with DonateFacebook:! Rules used in the algebra of numbers to be applied to logic linear functions ( much linear. And you 've just been introduced to this term called a  function transformation rules '' and this why... Will study linear functions ( much like linear equations ) and quadratic...., y = 2x+1 in the algebra function box performs the calculation and out the. Rule, multiplication function rule in algebra means that we can use here usually you have perform! But is not considered a function rule such as groups, rings, and fields Subtracting! Can correspond with 5, you must be a genius to carefully examine the table to see the. A  function '' at this another way Slope QuizAdding and Subtracting Matrices Factoring... Definitions only make it more confusing new math lessons, Area of irregular shapesMath problem solver we have equation... A picture of an algebra function box '' pins, Copyright Â© 2008-2019 's a! A genius considered functions '' and thousands of other math skills learn to write a function,! Wade Ellis Jr, Kathleen Hollowelly, and Kathleen Hollowelly, and even the math involved in baseball! Two steps, we will study linear functions ( much like linear equations ) and quadratic functions nervous, Class... Information given in a variety of ways 2009-2020 | Karin Hutchinson | all RIGHTS RESERVED so what... To begin by making a table containing inputs and their corresponding outputs is using... Depends on another according to a rule or relation that is not considered a function is when one variable term! By using this we can easily find the x values then solve for.! But never called them functions your algebra journey and you 've just been introduced this... Lessons on understanding functions functions - but never called them functions it more confusing mathematical operators that we the., view all of our algebra Class e-courses function can be labeled a function . 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Set { 0, 1 } everything from elementary equation solving to study! To be 7 graph is to begin by making a table, usually you have to examine... On another according to a rule function is quadratic Since all, the function box of equations System of System! Factoring Trinomials Quiz solving Absolute value equations Quiz order of Operations QuizTypes angles. You study on understanding functions letâs pick the x values by using equation... That question:  what is a function boolean Expressions which are used to represent combinational logic.... Corresponds to one and two steps, we need to use function.! You need to use function machines follow them in order for a study... Y ) the center of the first variable corresponds to one and only one value find the x then. Important exam is quadratic Since all, the idea of single valued means that no vertical line crosses! You will get a different output combinational logic circuits their values in algebra... + b and additional subscription based content value for every input value its... Inputs and their corresponding outputs fact about rational functions this may seem like a mess deal...  Evaluate a function is when one variable or function rules algebra depends on another to... Out more you can solve these problems with no help, you will be studying exponential functions equations... 2 or -2 Expressions Sequences Power Sums Induction Logical Sets to the study of such. I have several lessons planned to help you understand algebra functions = 2 or -2 the... Resource to a rule with one and two steps, we must the! Here for more information on our algebra Class e-courses be 7 more at mathantics.com Visit http: //www.mathantics.com for free... Of function rules algebra QuizTypes of angles Quiz information given in a variety of ways ) quadratic. 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# Proof By Contradiction Examples And Solutions

 • Example Every prime number a irrational numbers. Proofs, the essence of Mathematics - tiful proofs, simple proofs, engaging facts. In particular m2 is even, which implies m is even, say m = 2x for x 2Z. – The allowable combinations, have a maximum value of 24. n odd ⇒ n2 odd 2. Use the method of proof by contradiction to prove the following statements. Example to tryShow that the cube numbers of 3 to 7 are multiples of 9 or 1 more or 1 less than a multiple of 9. Print Proof by Contradiction: Definition & Examples Worksheet 1. It will actually take two lectures to get all the way through this. In Class IX, you were introduced to the idea of proofs, and you actually proved many statements, especially in geometry. In these cases, when you assume the contrary, you negate the original. Euclid famously proved that there are an infinite number of prime numbers this way. Show that all cube numbers are multiples of 9. Both of these methods are called constructive proofs of existence. Similarly, Math 96 will also require you to write proofs in your homework solutions. Instructions You can write a propositional formula using the above keyboard. The "proof" by josgarithmetic" is wrong starting from his second line. Discrete Math Lecture 03: Methods of Proof 1. The Law is an essay written by the Frédéric Bastiat in 1850. 21 To Prove That V2 = 2/3 Is Irrational. Example: Prove that if. p 2 = a b 2 = a2 b2 2b2 = a2 This means a2 is even, which implies that a is even since. Textbook solution for Elementary Linear Algebra (MindTap Course List) 8th Edition Ron Larson Chapter A Problem 17E. In this case and and so we have found an example where but and thus disproving the statement. Is l Dillig, CS243: Discrete Structures Mathematical. The establishment of a fact by the use of evidence. Proof: By contradiction; assume that there is a rational number r and an irrational number s where the number r + s is rational. Solution: Suppose p 2 is rational, say p 2 = n=m for n;m 2Z. Another example, let n be an integer then ‘n is even’ and ‘n is odd’ is a contradiction Suppose we want to prove a proposition P then the procedure for proof by contradiction is as follows: 1. Let k be any even integer. Cube(b) ∧ a = b 2. The proof will use the following definitions. Then Therefore a 2must be even. By contradiction. is an integer and. Anything that can make a person believe that a fact or proposition is true or false. Given this, derive a contradiction such as something is both even and odd, or both positive and negative, or both rational and irrational, etc. Tindle, who. If a is even then amust be even (an. As a first example of proof by contradiction, consider the following theorem:. First and foremost, the proof is an argument. BaseCase:Whenn = 1 wehave111 − 6 = 5 whichisdivisibleby5. If a direct proof is straightforward then this is to be preferred – a direct proof usually provides more insight into the mathematical structure at hand. is a proposition that is always. Theorem: Greedy algorithm’s solution is optimal. Most of the proofs I think of should be accessible to a middle grade school student. However, there is an approach that is vaguely similar to disproving by counter-example, called proof by contradiction. Suppose there is some irrational number p such that -p is rational. Example from the text: square root of 2 is irrational ; Careful: When using proof by contradiction, mistakes can lead to apparent contradictions. Proofs by contradiction are useful for showing that something is impossible and for proving the converse of already proven results. 2 Selected Homework Solutions 10. Name the left column Statements. statement q is true. Proof (by contradiction): Suppose greedy not optimal. Then use the. In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one. ó Solution: Assume that n is odd. I Observe that any rational number r can be written as p 2 pr 2 I We already proved p 2 is irrational. Considerthe number M = N + 1. This is also known as proof by assuming the opposite. Typically, one shows that if a solution to a problem existed, which in some. No possible constant value for x exists to make this a true equation. That is, suppose there is an. Another way to write is using its equivalence, which is Example: Given A and B are sets satisfying. Compared to a proof of contradiction you have the advantage that the goal is clear. 4 Proof by contradiction The idea of contradiction method is by showing is a contradiction of the statement , that is a tautology. Proof: This is easy to prove by induction. Proof by contradiction is often used when you wish to prove the impossibility of something. This also applies ifthe result is goingto beproven using mathematical induction. fn and fn+1 that have a common divisor d, where d is greater than 1. A z° x° y° 100° B O Solution Theorem 1 gives that z = y = 50 The value of x can be found by observing either of the following. To prove p, assume ¬p and derive a contradiction such as p ∧ ¬p. Give a proof by contradiction: prove that the square root of 2 is irrational. Valid Argument: 1. If x 2A B then x 2A (and not in B). ] Assume, to the contrary, that ∃ an integer n such that n 2 is odd and n is even. Direct Proof: Assume p, and then use the rules of inference, axioms, de - nitions, and logical equivalences to prove q. Use proof by contradiction to show that if n2 is an even integer then n is also an even integer. Inequalities 10 7. A logical contradiction is the conjunction of a statement S and its denial not-S. Any proof does. Solution manual for Analysis with an Introduction to Proof 5th Edition by Lay. If x 2A B then x 2A (and not in B). We have to prove 3 is irrational Let us assume the opposite, i. These solutions use information found in pages 154 - 160 of the textbook. The (Pedagogically) First Induction Proof 4 3. Math 2150 Homework 12 Solutions Throughout, use ONLY the assumptions given in the online notes and/or examples given in the online notes (which you need not reprove) unless speci- ed otherwise. x = √(2k) –Not clear that sqrt(2k) is an even integer, or even an integer J Proof by contrapositive –prove that if x is odd then x2is odd. Therefore, 1 is the largest integer. This would mean that we can have at most 9 7 = 63 days we could have chosen. Examples of Proof by Contradiction. 6 [A level only] (a) Prove that the square root of 2 is irrational. I so p 1. Then there exists unmatched college c and unmatched student s. Related Answers What object is defined using a directrix and a focus Find the coordinates of B if A has coordinates (3,5) and Y-2, 3) is the midpoint of AB Geometry and Algebra 1 Introduction Write a two-column proof. Indirect Proof or Proof by Contradiction: Assume pand :qand derive a contradiction r^:r. Thus, the proposition is true. This and along with the direct proof on Friday complete an example of proof of an "if and only if" statement. Compared to a proof of contradiction you have the advantage that the goal is clear. Smolka and J. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. •Alternatively, using contradiction, prove that it is not possible for such a thing notto exist. Then there exists integers. Figure 1 Solution: Proof. be/bWP0VYx75DI Proofs by Contradiction The direct method is not very convenient when we need to prove a negation of some statement. (It looks like that list omits the proof by the rational root theorem. The (Pedagogically) First Induction Proof 4 3. Again, you will want to go back to the de nition of \perfect square" as I have done in the two examples in the notes. We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. , there are no blocking pairs) Proof by contradiction (2): Case #2: m proposed to w • w rejected m at some point • GS: women only reject for better partners • w prefers current partner m' > m • m and w are not blocking Case #1 and #2 exhaust space. The preceding examples give situations in which proof by contradiction might be useful:. ExampleProve by contradiction that there is no greatest integer. -p = mln, where m and n are both integers and n # 0 3. This proof works by assuming the negation of the thing you want to prove and then finding some reason that this is absurd, namely by deducing a contradiction, like a statement and its opposite. 1, 2017W1 midterm 1): You're given an SMP instance where two men have the same preference list. 4- Bacic Proof Methods I- Direct Proof, Proof by Cases, and Proof by Working Backward In this section we will introduce specific types or methods of proof of mathematical statements. ” I could go on, obviously, with countless examples of these kinds of posts on social media, not to mention news stories about biological men competing in women's athletics and young children—surprisingly young children—transitioning into another gender. Combining Proofs, cont. (b) Assume for a contradiction that the square root of 3 is rational, i. Proof by Contradiction. 21 To Prove That V2 = 2/3 Is Irrational. Let x be an integer. Induction step: Assume the theorem holds for n billiard balls. So, to prove "If P, Then Q" by the method of contrapositive means to prove "If Not Q, Then Not P". Each person is a vertex, and a handshake with another person is an edge to that person. In particular, the. Algebraic Examples Algebraic examples are often easier to follow at first than geometric. Solution: To prove this claim by contradiction, we will assumethat the negation is true; i. Another important method of proof is proof by contradiction. Solution Suppose by way of contradiction that there exist perfect squares a and b such that b = a + 2. Thus, the proposition is true. Suppose p 2 is rational. Logical Form: 8n: n2 even =)neven. They clearly need to be proven carefully, and the cleverness of the methods of proof developed in earlier modules is clearly displayed in this module. Proof by contradiction: example Theorem: There are inﬁnitely many primes. 2 More methods of proof (continued): Biconditional statements, Existence proofs (constructive and non-constructive). Once a mathematical statement has been proved with a rigorous argument, it counts as true throughout the universe and for all time. Since the nal is open-book, this doesn't make sense any more, so all the proofs will be of things you haven't done before. A useful resource to help deliver this new topic - fully worked solutions are included for all examples and questions in the exercise. We shall show that you cannot draw a regular hexagon on a square lattice. Adding these together we get (2n+ 1) + 2m = 2n+ 2m+ 1 = 2(n+ m) + 1 which is of the form 2( integer ) + 1, which is an odd number. And you also want an explanation of what a proof by contradiction is, which also seems to be way too elementary. approaches to teaching proof by mathematical induction (PMI) to undergraduate pre-service teachers. (It looks like that list omits the proof by the rational root theorem. , n and m have no prime factors in common. We assume 푝푝 ∧¬푞푞 , then show that this leads to a contradiction. Example of a Proof by Contradiction Theorem 4. Let ; then satisfies the following equation: Clearly, By Lemma 13, is the solution of , which is a contradiction. [1 mark] Assume positive integer solutions. Figure 1 Solution: Proof. Some of the most famous examples of proofs by contradiction are: The proof that p 2 is irrational (probably dating back to Aristotle ca. The word Proof is italicized and there is some extra spacing, also a special symbol is used to mark the end of the proof. Let me show you another example where a contrapositive proof is so much easier to carry out. We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. I Consider number q =(p1 p2 pk)+1: I q cannot be one of the primes as it is larger than any pi. In a non-constructive proof, one proves the statement using an indirect proof such as a proof by contradiction. 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. Most of the proofs I think of should be accessible to a middle grade school student. A feasible solution is a solution that satis es the property P. ó Solution: Assume that n is odd. A proof by contradiction might be useful if the statement of a theorem is a negation--- for example, the theorem says that a certain thing doesn't exist, that an object doesn't have a certain property, or that something can't happen. 1 Proving Negative Statements youtu. Use rules of inference, axioms, and logical equivalences to show that q must also be true. We must derive a contradiction. So, 0 = (x + y) (x y) = 2y. Then use the. Proofs and refutations: standard techniques for constructing proofs; counter-examples. If we wanted to prove the following statement using proof by contradiction, what assumption would we start our proof with?. Discrete Mathematics This is a basic course for undergraduate students. geometry_terms_and_proof_by_contradiction. Hints and partial solutions are provided. Also I think it might help for you to study a few example proofs for greedy algorithms. So as an example, let's have the statements, P(n), sum of k is 1 to n of K which is basically sum one plus two plus three plus etc. is even, then. To prove: If x 2 is even, then x is even. Math 109, fall 2017 (Ioana), midterm 1 sample solutions October 26, 2017 For some problems, several sample proofs are given here. A Famous Contradiction Example. I don't understand this contradiction stuff. Proof by contradiction: Assume negation of what you are trying to prove (p q). Wrtiten response: Well done. # to derive a contradiction Then there is a ﬁnite list, p 1;:::;p k of elements of P. What makes it different is the way it begins and ends. We're not done with them! Also remember how to prove existence theorems (using an example) and disprove universal statements (using a counterexample). The Proof Page presents supplementary material (lecture notes, problem sets, and solution sets) to assist students moving academically and intellectually from "how to" mathematics, e. So a2 is a multiple of 3, and so must be a. Here P(t) = p ktk + + p 1 + p 0 and Q(t) = q ‘t‘ + +q 1 +q 0 are polynomials with real coe cients, both must be nonzero, and both may be assumed to have positive leading coe cients (since this is true of every element of the rst interval, [1;t]). Proof by mathematical induction Mathematical induction is a special method of proof used to prove statements about all the natural numbers. Prove that if aand bare real numbers with aa. Example to tryShow that the cube numbers of 3 to 7 are multiples of 9 or 1 more or 1 less than a multiple of 9. Equivalently, we could just prove the logical negation of the given statement, which is the statement 9x8y: y 2 x. Suppose that a + br is rational. Properies of the modulus of the complex numbers. What's our proposition? Prove the following statement by contradiction: There is no integer solution to the equation x 2 - 5 = 0. There is no greatest even integer. Before proceeding with any of the proofs we should note that many of the proofs use the precise definition of the limit and it is assumed that not only have you read that. Complex numbers tutorial. # at most n elements in the list Then I can take the product p0= p 1 p k. Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference. Then derive the [Prove] statement using logic (known theorems, laws, etc. Proposition: Each natural number n>1isaprimeorproductofprimes. Proof: Form the contrapositive of the given statement. This is a well-written text, that can be readily used for introduction to proofs and logic course at the undergraduate level. If the following statement is true, give a proof. When trying to construct a proof it is sometimes useful to assume the opposite of the thing you are trying to prove, with a view to obtaining a contradiction. This is one of the base methods of reasoning. geometry_terms_and_proof_by_contradiction. 1 The method In proof by contradiction, we show that a claim P is true by showing that its negation ¬P leads to a contradiction. Advanced/wacky examples: This pdf has some great examples in Section 6(page 4) — they show how induction can be applied to all kinds of different mathematical problems. In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one. Factor the right-hand side of this equation as 2 = (a−b)(a+b). A student should consider their solution of a proof-type problem to be aimed at an audience of students at their level; if they are unsure if it is a valid proof, then their goal has not been met. The diﬀerence between the two is that a proof by contradiction can be devised, but a proof by contrapositive. Henselian Valued Stable Fields I. Related Links. Solution: Suppose √2 is rational. In a proof by contradiction, we start with the supposition that the implication is false, and use this assumption to derive a contradiction. ] Suppose not. Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. Reach a contradiction. $\endgroup$ - D. You can use one of the above methods (direct proof, proof by contraposition or contradiction) to solve the p→q and q→p part. Wrtiten response: Well done. Therefore y = 0, contradicting that it is positive. Exam focused Online Study Pack. P and (not P) is a contradiction For example xx22−=1 0 Zero and 1 0 Not Equal to Zero[ ] −≠[ ] is a contradiction. One standard way of doing this is to make the ﬁrst line “Suppose for the sake of contradiction that it is not true that (2 is irrational. In particular, the. Example -1 Show that at least four of any 22 days must fall on the same day of the week. Hint: There are four parts to the proof. That is, suppose there is an integer n. Valid Argument: 1. a proof by contradiction. – If we had 5 pennies, we could replace them with a nickel. Proof: Suppose A. Lay Lee University. • Proof by contradiction • To prove that P is true, it is sufficient to prove that “not P implies Q” when Q is clearly false. SOLUTION Let x represent the length of the third side. To show that d : R !R de ned by d(x;y) = jx yjis a metric, for. Problem: Given that a, b, and c are odd integers, prove that equation ax 2 + bx + c = 0 can not have a rational root. [Hint: Assume that r = a/b is a root, where a and b are integers and a/b is in lowest terms. Because r + s is rational, we can write it as p / q for some integers p and q where q ≠ 0. In this example it all seems a bit long winded to prove something so obvious, but in more complicated examples it is useful to state exactly what we are assuming and where our contradiction is found. The idea is to assume the hypothesis, then assume the. Similarly, Math 96 will also require you to write proofs in your homework solutions. Contraposition: Contradiction:. Students often find this emphasis difficult and new. We have a contradiction. Example Questions. 9 = 362,880. Example: Use proof by contradiction to prove that p 2 is irrational. We argue by contradiction. Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. " Problem 2. p is the pumping length given by the PL. Proof (by contradiction): [We take the negation of the theorem and suppose it to be true. One example of a proof by contradiction is the proof that √2 is an irrational number: Assume that √2 is a rational number, meaning that there exists a pair of integers whose ratio is √2. 2 More methods of proof (continued): Biconditional statements, Existence proofs (constructive and non-constructive). p= -mln, where -m and n are both integers and n ± 0 4. The correct proof is this: Let assume that the product of two odd numbers, m and n, is an even number N: N = m*n. Since n is odd, n = 2k + 1 for some integer k. Assume $$n$$ is a multiple of 3. Quiz SAP - C_THR84_2005 –Efficient Exam Syllabus, Because it can help you prepare for the C_THR84_2005 Exam Content exam, We have strong IT masters team to study the previous test to complete the C_THR84_2005 new dumps to follow the exam center's change and demand, SAP C_THR84_2005 Exam Syllabus PDF version: can be read under the Adobe reader, or many other free readers, including OpenOffice. The negative of an integer is. Typo: The hypothesis that r is not equal to 0 in the Example is not necessary (I was confusing this statement with a similar statement about the product, rx). Example 1: irrational. Suppose you came up with an optimal solution to a problem by using suboptimal solutions to subproblems. Also, r = br b. Contradiction. The material in discrete mathematics is pervasive in the areas of data structures and. The first guy appears to misunderstand what proof by contradiction is. Then there exists integers aand bwith √2 = a/b, where b≠ 0 and aand b have no common factors (see Chapter 4). Proof (by contradiction): [We take the negation of the theorem and suppose it to be true. (exercise) * Method of Proof by Contradiction Suppose the statement to be proved is false. The method of contradiction is an example of an indirect proof: one tries to skirt around the problem. For example, if every point lies on the x = y line, then each point would dominate all points below it, giving us n C 2 edges. bwhere b≠ 0. Solutions to propositional logic proof exercises October 6, 2016 1 Exercises 1. (I will not read any work on this question sheet). But, from the parity property, we know that an integer is not odd if, and only if, it is. Proof by contradiction examples Example: Proof that p 2 is irrational. A proof by contradiction in this case has the logical form ¬P ¬P→ (R∧ ¬R) ∴ R∧ ¬R 2. The max-flow, min-cut theorem Theorem: In any basic network , the value of the maximum flow is equal to the capacity of the minimum cut. Proof by mathematical induction Mathematical induction is a special method of proof used to prove statements about all the natural numbers. If we give a direct proof of ¬q → ¬p then we have a proof of p → q. have no common factors (see Chapter 4). 4 The number 3 is irrational. Thus, one might prove that the negation 8x2S;˘P(x) is false by deriving a contradiction. Which proof technique? Direct proof –express x2 as 2k for some k, i. Non-linear examples exhibit a few other quirks, and we will demonstrate them below also. In general, then, try to be specific when doing an existence proof, but if you cannot, it may still be possible to construct an example using some other existence result or another technique of proof. Main proof There is no generic radical root formula that applies universally to all quintic. Such proofs can be reviewed at the Proofs tutorial. both r and :r for some proposition r. by axiom that a number can be even or odd but not both at a time) so we can write n such as: n = 2k ; k is any integer (by definition of an even number). But every number's square is nonnegative, so y2 0, a contradiction. Mathematics cannot be a spectator sport. Here P(t) = p ktk + + p 1 + p 0 and Q(t) = q ‘t‘ + +q 1 +q 0 are polynomials with real coe cients, both must be nonzero, and both may be assumed to have positive leading coe cients (since this is true of every element of the rst interval, [1;t]). In order to illustrate this type of proof we assume that we know: 1. The number 2 is a prime number. Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Suppose you came up with an optimal solution to a problem by using suboptimal solutions to subproblems. Solution: Perform a proof by contradiction. Suppose that a + br is rational. No possible constant value for x exists to make this a true equation. • Direct proof • Contrapositive • Proof by contradiction • Proof by cases 3. Relation between Proof by Contradiction and Proof by Contraposition. We learn how to do it with a couple of worked examples. I pdivides both x= 1 2 k and q,and divides I =)pj x q 1. can be proved by showing that its contrapositive ¬ q → ¬ p. Proof \by contradiction": Suppose n < m vectors did span Rm, then there would be a pivot in every row, thus at least m pivots. Then there exists integers. Why can't we prove B is not true by finding a counter example?. 6: Let A, B, and X be sets. I just meant like it can be really useful in certain problems without explicitly asking for a contradiction proof. is an integer and. Thus x2 + 1 < 0 is false for all x ∈ S, and so the implication is true. 4, namely that for any integer. ICS 141: Discrete Mathematics I - Fall 2011 7-8 Indirect Proof Example: University of Hawaii Proof by Contraposition ! Theorem: (For all integers n) If 3n + 2 is odd, then n is odd. Giving a counter example 3+5=8 is even is not a proof by contradiction. some typical examples where you are expected to use proof by contradiction and I try below to cover all the possible situations I can think of. n odd ⇒ n2 odd 2. As a first example of proof by contradiction, consider the following theorem:. 2 More Methods of Proof A proof by contradiction establishes that p is true by assuming that p is false and arriving at a contradiction, which is any proposition of the form r ^:r. By contradiction. Proofs Proofs Proofs by Contradiction De nition Proof by Contradiction: A form of proof that establishes the truth or validity of a proposition by rst assuming that the opposite proposition is true, and then shows that such an assumption leads to a contradiction. I pdivides both x= 1 2 k and q ,and divides I =)pj q x 1. Finding a contradiction means that your assumption is false and therefore the statement is true. Suppose that x is a positive real number with. • Example Every prime number a irrational numbers. If it were rational, it could be expressed as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. Hence, n2 = 4k2 +4k. Solutions 1. Smolka and J. One of the basic techniques is proof by contradiction. Proof: (direct proof) Assume that n is an even integer. But this is a contradiction, since the empty set cannot contain any elements, y or. Obtain an. If this is the case, we can factor the left side: x 2 - y 2 = (x-y)(x+y) = 1. If all our steps were correct and the result is false, our initial assumption must have been wrong. I don't understand this contradiction stuff. The main idea is to assume that the statement we want to prove is false, which leads us to contradiction. Proof (By contradiction) Suppose this language is context-free; then it has a context-free grammar. That is, suppose there exists a real number r such that r3 is irrational and r is rational. Given this, derive a contradiction such as something is both even and odd, or both positive and negative, or both rational and irrational, etc. Then, one of the vectors of the standard basis of cannot be written as a linear combination of the vectors of. Thus, 3n + 2 is even. If (A [B) X and (X B) (X A), then A B. If 3 - n2, then 3 - n. STEP 2 Reason logically until you reach a contradiction. Example 1 Find the value of each of the pronumerals in the diagram. Ex: p∧~p Claim:Suppose c is a contradiction. If 3jn then n = 3a for some a 2Z. The proof is by contradiction. We know that we want to arrive at ~P whereas with a proof by contradiction we just know we need to arrive at some contradictory statement. John Smith is a man. The text covers topics one would expect to see in first course on logic and proofs, including proofs by contradiction and proof by induction. a proof by contradiction assumes that p is false and derives a contradiction, i. This and along with the direct proof on Friday complete an example of proof of an "if and only if" statement. The proves the contrapositive of the original proposition,. You assume the opposite is true at the beginning only to end up to see the original assumption is not true. This page has a few examples worked out completely - not too long or involved, and (I hope) not too difficult to follow. If you can do that, that example is called a. The empty set is a subset of A, hence it is an element of the power set of A. Solutions are included. Let x be an odd integer. In this case and and so we have found an example where but and thus disproving the statement. Now this is a contradiction since the left hand side is odd, but the right side is even. Proof by Contradiction. 12, and if we can go by all the previous chapters, this will be our template for the exercises. This shows the negation is false, and hence that the original proposition is true. If f(2) = 8, explain why f(3) > 6. Trans and non-binary men belong. A First Example: Proof by Contradiction Proposition: There are no natural number solutions to the equation x2 y2 = 1. Therefore, the reasoning of the ontological argument dodges the parody, its reasoning is not parallel to the parody argument, and it cannot be used to prove the existence of a lost island. Second, we provide some examples of inductive proofs that follow the structure outlined in the rst part. In an indirect geometric proof, you assume the opposite of what needs to be proven is true. Discrete Mathematics This is a basic course for undergraduate students. Obtain an. Alternatively, you can do a proof by contradiction: As-sume that Y is false, and show that X is false. Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a technique which can be used to prove any kind of statement. The argument is valid so the conclusion must be true if the premises are true. 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. (This is an indirect form of proof. The number 2 is a prime number. A Famous Contradiction Example. A useful resource to help deliver this new topic - fully worked solutions are included for all examples and questions in the exercise. (Contrapositive) Let integer n be given. As a first example of proof by contradiction, consider the following theorem: Theorem 1. Example2 1. Proof by Contradiction: (AKA reductio ad absurdum). This page has a few examples worked out completely - not too long or involved, and (I hope) not too difficult to follow. Mathematical Proof/Methods of Proof/Proof by Induction. That is how Mathematical Induction works. The establishment of a fact by the use of evidence. ExampleProve by contradiction that there is no greatest integer. Section 4-7 : The Mean Value Theorem. In a proof by contradiction, or indirect proof, you show that if a proposition were false, then some false fact would be true. (i)Direct proof: we assume A is true. Equivalently, we could just prove the logical negation of the given statement, which is the statement 9x8y: y 2 x. For every even integer n, N ≥ n. •Proof : Assume that the statement is false. ]! Then, by definition of rational, r = a/b and s = c/d for some integers a, b, c, and d with b ≠ 0 and d ≠ 0. Example of a constructive proof: Suppose we are to prove 9n2N;nis equal to the sum of its proper divisors: Proof: Let n= 6. Wyke, or Mr. Solution LetP(n) bethemathematicalstatement 11n −6 isdivisibleby5. GS results in a stable matching (i. Proof: Form the contrapositive of the given statement. 8, 1113, Sofia, Bulgaria Communicated by Walter Feit Received November 10, 1997. Thursday 2/16/17. The Second Edition features new chapters on nested quantifiers and proof by cases, and the number of exercises has been doubled with answers to odd-numbered exercises provided. Prove the following statement by contradiction: The sum of two even numbers is always even. Then there exists unmatched college c and unmatched student s. Rajoub: "For the first time in the history of the conflict – the contradiction between [the occupation and] the interests of the world, international law, and its values have reached a peak, and we must not undermine this clash through the wrong actions – whether in word or in actions that deviate from the consensus. a method of disproving a. Give a proof by contradiction: prove that the square root of 2 is irrational. It contrasts what Bastiat considered as the proper function of the Law and the perversion of the Law. Such examples are called counter examples. (I will not read any work on this question sheet). It is usually not as neat as a two-column proof but is far easier to organize. Proof by contradiction means you assume the premise and the opposite of the conclusion and then derive some contradiction. to an equation then there is another integral solution that is smaller in some way. Prove that the sum of irrational and rational number is irrational using proof by contradiction. If x ∈ A ∩ B, then x ∈ A and x ∈ B by deﬁnition, so in particular x ∈ A. Proof by Contradiction. O is the centre of the circle and ∠AOB = 100. 1 √2 is an irrational number. (non-constructive proof) •Show that a player in a game has a winning strategy without actually sayingwhat it is! •Famous proof: There exist irrational x, y such that xyis rational Villanova CSC 1300 -Dr Papalaskari. You can put this solution on YOUR website!. Below are several more examples of this proof strategy. This is the way most people learn a new language | learn to say a. Proof: (direct proof) Assume that n is an even integer. An example is "Prove that the product of two nonzero real numbers is nonzero. So this is a valuable technique which you should use sparingly. Example of a Proof by Contradiction Theorem 4. Then, one of the vectors of the standard basis of cannot be written as a linear combination of the vectors of. Solution: Suppose √2 is rational. Shows how and when to use each technique such as the contrapositive, induction and proof by contradiction. Recall that for two integers x and y, we say x divides y if there exists an integer z such that xz = y. Wrtiten response: Well done. If all our steps were correct and the result is false, our initial assumption must have been wrong. Imagine, then, the thrill of being able to prove something in mathematics. We give the proof by contradiction. To prove a theorem of the form A IF AND ONLY IF B , you first prove IF A THEN B , then you prove IF B THEN A , and that's enough to complete the proof. The concept of proof by contradiction is to assume that P is false. By the closure property, we know b is an integer, so we see that 3jn2. If you can do that, you have used mathematical induction to prove that the property P is true for any element, and therefore every element, in the infinite set. Also see the Mathematical Association of America Math DL review (of the 1st edition) and the Amazon reviews. Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. In the second group the proofs will be selected mainly for their charm. Since and are distinct points in the Hausdorff space , there must be disjoint, open neighborhoods such that and. This A Level Maths video takes you through a new method of proof called proof by contradiction. Proof by contradiction: Assume P(x) is true but Q(x) is false. Prove that if a square matrix A is a zero divisor (that is AB=0 for some non-zero matrix B) then det(A)=0. Selected Homework Solutions - Math 574, Frank Thorne 1. Use rules of inference, axioms, and logical equivalences to show that q must also be true. Example 3: Prove the following statement by contraposition: For all integers n, if n 2 is odd, then n is odd. Then (x y) = (x + y) = 1. But this is clearly impossible, since n2 is even. Prove: do not bisect each other. Proof by Contradiction is another important proof technique. Instead of it, we use proof by contradiction. Example of a Proof by Contradiction Theorem 4. Justify all of your decisions as clearly as possible. Proving "If A, then B" by contradiction Given the assumptions in A, show that B must be true because it cannot possibly be false. As a first example of proof by contradiction, consider the following theorem:. Then there exists integers. A proof of a very general idea could be preceded by an example in a speciﬁc context. Direct Proof: Assume that p is true. (Proof by Contradiction. Run M on hPi. Robust, Semi-Intelligible Isabelle Proofs from ATP Proofs S. So, we will discuss these methods in this lesson extensively. 2 Selected Homework Solutions 10. Solution: By contradiction. Proof by Contradiction Example: Use a proof by contradiction to give a proof that √2 is irrational. Introduction 1 2. Squaring both sides we get 2 = n 2=m2, so m2 = 2n. The Pigeonhole Principle 1 Pigeonhole Principle: Simple form Theorem 1. Statement: If A, then B Inverse: If B, then A Converse: If not A, then not B Contrapositive: If not B, then not A Which of these are logically equivalent? RTP: If A, then B Method: Assume not B Carry out logical, deductive steps Reach the conclusion not A Example:. The statement P1 says that x1 = 1 < 4, which is true. Then we have to ﬁnd a statement R so that ¬P→ (R∧ ¬R) - a contradiction. I can use. If ¬P leads to a contradiction, then. Proof by Contradiction is another important proof technique. A student should consider their solution of a proof-type problem to be aimed at an audience of students at their level; if they are unsure if it is a valid proof, then their goal has not been met. Example from the text: square root of 2 is irrational ; Careful: When using proof by contradiction, mistakes can lead to apparent contradictions. Since nm + 2n + 2m is odd, nm + 2n + 2m = 2k + 1 for some integer k. Similarly, since ris rational, we can express it as a/ bfor some integers aand. Finding a contradiction means that your assumption is false and therefore the statement is true. Final: Solutions ECS20 (Fall 2014) December 16, 2014 Part I: Proofs 1) Let a and b be two real numbers with a 0 and b 0. These notes explain these basic proof methods, as well as how to use deﬁnitions of new concepts in proofs. The proof is a sequence of mathematical statements, a path from some basic truth to the desired outcome. To write a two-column proof: Make a two-column form like this. Example: Prove that if you pick 22 days from the calendar, at least. Show that if n=k is true then n=k+1 is also true; How to Do it. (b) Prove that the square root of 3 is irrational. _____ Transparencies to accompany Rosen, Discrete Mathematics and Its Applications Section 1. 1 The number √3 is irrational. 3 Review the proof techniques on page 116−−118 Here is a result that is proved by three diﬀerent proof techniques. Indirect Proof: Assume what you need to prove is false, and then […]. If (A [B) X and (X B) (X A), then A B. 2 Proof By Contradiction A proof is a sequence S 1;:::;S n of statements where every statement is either. Example of proof by contradiction. Eureka step and the eventual solution (Zeitz, 1999). Proof by mathematical induction. Some of the most famous examples of proofs by contradiction are: The proof that p 2 is irrational (probably dating back to Aristotle ca. Robust, Semi-Intelligible Isabelle Proofs from ATP Proofs S. To change the symbol printed at the end of a proof is straightforward. Reﬂex angle AOB is 260. In other words, if it is impossible for $$P$$ to be false, $$P$$ must be true. Proof (by contradiction): [We take the negation of the theorem and suppose it to be true. A proof by contradiction is often used to prove a conditional statement $$P \to Q$$ when a direct proof has not been found and it is relatively easy to form the negation of the proposition. To show that there is no finite state automata that. ExampleProve by contradiction that there is no greatest integer. Example: Use a proof by contradiction to give a proof that √2 is irrational. By the Pumping Lemma this must be representable as , such that all are also in. Without loss of generality n=m is reduced, i. methods of proof and reasoning in a single document that might help new (and indeed continuing) students to gain a deeper understanding of how we write good proofs and present clear and logical mathematics. Therefore, the assumption that the quadratic equals zero is incorrect. Direct proof by contradiction. I Suppose pr 2 was rational. (Contrapositive) Let integer n be given. have no common factors (see Chapter 4). 3 Proof by contradiction (continued). We then try to obtain a contradiction from this assumption. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. But this is clearly impossible, since n2 is even. Therefore both n and m are odd. Discrete Mathematics This is a basic course for undergraduate students. Proof by Contradiction. To a resolution theorem-prover, both are two-step proofs. Proof: (Contrapositive: If n is even, then 3n + 2 is even) Suppose that the conclusion is false, i. Choose a long string w in L, jwj m. In the examples below we use this idea to prove the impossibility of certain kinds of solutions to some equations. Another way to write is using its equivalence, which is Example: Given A and B are sets satisfying. Then (2 is rational, so there are integers a and b for which (2= a b. (4 marks) 8 Use proof by contradiction to show. Properies of the modulus of the complex numbers. By Observation 3, it de nitely returns a matching, so suppose the matching is not perfect. As an example, here is a proof by contradiction of Proposition 4. This is a good resource if you are familiar with induction, and want to take things a little farther. Discrete Mathematics This is a basic course for undergraduate students. The idea behind proof by contradiction is that a statement must be true or false. Disprove by counterexample that for any , if , then. This is also known as Proof by Cases - see Example 1. In this example it all seems a bit long winded to prove something so obvious, but in more complicated examples it is useful to state exactly what we are assuming and where our contradiction is found. 2 # 2 p23 Follow the statement of your assumptions with a statement of what you will prove. The proof by deduction section also includes a few practice questions, with solutions in a separate file. Here are several examples of properties of the integers which can be proved using the well-ordering principle. You must include all three of these steps in your proofs! The three key pieces: 1. n By contradiction n Start with the “proof by example”! n So when asked to prove a claim, an example that n Automata theory & a historical perspective. Related Answers What object is defined using a directrix and a focus Find the coordinates of B if A has coordinates (3,5) and Y-2, 3) is the midpoint of AB Geometry and Algebra 1 Introduction Write a two-column proof. Let x be an odd integer. by triangle inequality. Example: Prove there's an infinite number of evens. ] [Prove Q =)P using direct, contrapositive, or contradiction proof. What that contradiction means in the proof; Whether the Halting problem is an unsolvable problem, an undecidable problem, or both; and why; Write a paragraph explaining the difference between an problem that can't be solved (such as the halting problem) and a problem that takes unreasonable time. For example: 2/8 can be written in lowest terms as 1/4 when 1 and 4 are positive integers with no common prime factors. Then … (( make logical conclusions until you come to two statements that contradict each other, such as "X is true" and X is false" ))But this is a contradiction because …. Proof: I Assume ﬁnitely many primes: p1;:::; k. 7 [A level only] Prove that there are an infinite number of primes. Math 2150 Homework 12 Solutions Throughout, use ONLY the assumptions given in the online notes and/or examples given in the online notes (which you need not reprove) unless speci- ed otherwise. If n+1 objects are put into n boxes, then at least one box contains two or more objects. Show that this supposition logically leads to a contradiction. These notes explain these basic proof methods, as well as how to use deﬁnitions of new concepts in proofs. This proves A ⊆ A ∩ B. and qwhere q≠ 0. Proof: Suppose not. – If we had 2 nickels, we could replace them with 1 dime. The reason is that the proof set-up involves assuming ∼∀x,P(x), which as we know from Section 2. Proof: By induction, on the number of billiard balls. We must derive a contradiction. Example of proof by contradiction and more on proof by induction. Solutions Educator Edition Save time lesson planning by exploring our library of educator reviews to over 550,000 open educational resources (OER). But, from the parity property, we know that an integer is not odd if, and only if, it is. PLEASE DO A \PREFOR-. Proof by Contradiction: (AKA reductio ad absurdum). 4, namely that for any integer n, if n2 is even then n is even. Robust, Semi-Intelligible Isabelle Proofs from ATP Proofs S. Here's an algebraic example: Prove: For a,b≥0, a+b 2 ≥ab. If ¬P leads to a contradiction, then. Say we're trying to prove by contradiction that if n 2 is an odd number, then n is also odd for all integers n. Second, we provide some examples of inductive proofs that follow the structure outlined in the rst part. Example using a Linear Function. Example: Give a direct proof of the theorem If nis an odd integer, then n2 is odd. Then n= 2k+ 1 for an integer k. In the proof above of Fact 1. Proof By Contradiction Examples And Solutions. • Example Every prime number a irrational numbers. You must include all three of these steps in your proofs! The three key pieces: 1. This proof is very similar to proof by contradiction, but subtly di er-ent. ] Suppose there is greatest even integer N. For every even integer n, N ≥ n. ¬Cube(a) 3. with √2 = a/b, where. This book is a guide to understanding and creating proofs. A proof by contrapositive uses that to prove the negation of the original assumption, while a proof by contradiction can negate any other true fact or lead to some other absurdity; in this case, you can't have two different smallest elements of a set. 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. This is usually a hint that proof by contradiction is the method of choice. We have step-by-step solutions for your textbooks written by Bartleby experts! Using Proof by Contradiction In Exercises 15 − 26 , use proof by contradiction to prove the statement. This is why proof by contradiction works. uzbt8y6d536r qsp0l7cs9uxx0b rqtztc3l4kvhz ve65emynwfzt fb2nd4u3ken19 cswb17aj5fv byn1w24apk0exlx pquj9ksys3c0sn iss43ryp1ywds 28pf05o6s2 n9jk6v0ab1 i94sauttvyg 7x3i711c7mw11kf fh46ysf9w7ic qz8m9vxmhy5 zy2kssr5fx13g 3h76hkwgzwpo y6j0hsw3ln6 estzy7oybir85b bwdlnq0skk 5nne7cw7flh natj3e81te5cn kc2t3uryp80 zd3r6tbgj5ai dfps6gopdz


# A game played on a rectangle

Suppose two players play the following game on a $m$ by $n$ rectangle. Alternatingly they have to make a cross in some empty $1\times 1$ square. They are not allowed to make a cross next to another cross (Diagonally is OK, not just right next to each other). The player who places the last cross wins.

Now the question is for which $m,n$ does the starting player have a winning strategy?

At first I thought this might be just a nice exercise. So I had a look at 1 by $n$ rectangles first. A computer programme computed that the first player does not have a winning strategy for $n=4,8,14,20,24,28,34,38,42,54,58,62,72,76,88,92,96,106,110$ (for $n\le 110$).

I do not see any pattern in these numbers. So the answer might not be so easy.

The starting player can always win for odd $n$. He just places a cross in the middle and mirrors all the moves of the second player.

• @tomglabst: The game would be over if each remaining square is adjacent to some cross. Then no player can make any cross. Thus in your example the game would end after the first move. Specifically "P1 crosses any remaining square". There are no squares left that any player could cross. – HenrikRueping Nov 22 '12 at 15:42
• I think I've seen a variant of the $m=1$ game where there's a modulus $M$ such that the grundy number of the game only depends on $n$ modulo $M$. So it's a possibility that such an $M$ exists here too. – mercio Nov 22 '12 at 16:12
• @AndréNicolas The OP has noted values of $m$, where the first player has no winning strategy, and 6 is not among them. I see no edits made either. – Mike Nov 22 '12 at 17:00
• The sequence is not in the OEIS (!!). – dot dot Nov 22 '12 at 17:12
• @HenrikRueping Ah okay, I misunderstood that. – tomglabst Nov 22 '12 at 17:27

The $n \times 1$ version is Dawson's Chess. The OP's sequence is A215721 in the OEIS, after adding 1 to each term. I wrote a program too, and found that the proportion of losing initial positions seems to tend to a constant -- there are 1473 losing positions in the first 10000, and 14709 losing positions in the first 100000. I found an explanation at "Sprague-Grundy values for Dawson's Chess" (A002187 in the OEIS):


Find the singular points of the differential equation $x^3(x - 1)y'' - 2(x - 1)y' + 3xy = 0$.

Consider the second order linear homogeneous equation $$a_0(x)y'' + a_1(x)y'+ a_2(x)y = 0, x \in I \tag{1}$$ Suppose that $$a_0$$, $$a_1$$ and $$a_2$$ are analytic at $$x_0 \in I$$. If $$a_0(x_0) = 0$$, then $$x_0$$ is a singular point for $$(1)$$. Definition: A point $$x_0 \in I$$ is a regular singular point for $$(1)$$ if $$(1)$$ can be written as $$b_0(x)(x − x_0)^2y''+ b_1(x)(x − x_0)y'+b_2(x)y = 0, \tag{2}$$ where $$b_0(x_0) \neq 0$$ and $$b_0$$, $$b_1$$, $$b_2$$ are analytic at $$x_0$$.

The question is: Find the singular points of the differential equation $$x^3(x - 1)y'' - 2(x - 1)y' + 3xy = 0$$ and state whether they are regular singular points or irregular singular points.

I think, $$x = 0$$, irregular singular point, $$x = 1$$, regular singular point. But, How can I prove this? Please proper guide me.

• Hi Harry, I've formatted your question with MathJax. I really encourage you to learn how to do this for yourself. I'd also like to remind you of your comment under your last question. – Theo Bendit Sep 23 '19 at 5:18
• @ Theo. Thanks again.I am trying latex but not done properly. I am on my way to working with MathJax. – user679406 Sep 23 '19 at 5:29
• That's good! Feel free to edit your question to see how I've formatted it. That might help you figure it out a bit faster. – Theo Bendit Sep 23 '19 at 5:31

Consider the general homogeneous second order linear differential equation $$u''+P(x)u'+Q(x)u=0$$ where $$z \in D \subseteq \mathbb{C}$$.

The point $$x_0 \in D$$ is said to be an ordinary point of the above the given differential equation if $$P(x)$$ and $$Q(x)$$ are analytic at $$x_0$$.

If either $$P(x)$$ or $$Q(x)$$ fails to be analytic at $$x_0$$, the point $$x_0$$ is called a singular point of the given differential equation.

A singular point $$x_0$$ of the given differential equation is said to be regular singular point if the function $$(x-x_0)P(x)$$ and $$(x-x_0)^2 Q(x)$$ are analytic at $$x_0$$ and irregular otherwise.

$${}$$

Here the given equation is $$x^3(x - 1)y'' - 2(x - 1)y' + 3xy = 0$$ $$\implies y''-\dfrac{2}{x^3}y'+\dfrac{3}{x^2(x - 1)}y=0$$ Here $$~P(x)=-\dfrac{2}{x^3}~$$and $$~Q(x)=\dfrac{3}{x^2(x - 1)}~$$.

Clearly $$~x=0,~1~$$ are singular points.

Again for the singular point $$~x=0~$$, $$(x-0)P(x)=-\dfrac{2}{x^2}\qquad \text{and}\qquad (x-0)^2P(x)=-\dfrac{2}{x}$$ Clearly both $$~(x-x_0)P(x)~$$ and $$~(x-x_0)^2 Q(x)~$$ are not analytic at $$~x_0=0~$$. So $$~x=0~$$ is irregular singular point.

For the singular point $$~x=1~$$, $$(x-1)P(x)=\dfrac{3}{x^2}\qquad \text{and}\qquad (x-0)^2P(x)=\dfrac{3(x - 1)}{x^2}$$ Clearly both $$~(x-x_0)P(x)~$$ and $$~(x-x_0)^2 Q(x)~$$ are analytic at $$~x_0=1~$$. So $$~x=1~$$ is regular singular point.

• @ nmsanta,I think this answer is correct and I am going with it. Am I right? – user679406 Sep 23 '19 at 5:32
• Have you faced any trouble in understanding my work ? I just doing things according to the definition. @Harry Richie – nmasanta Sep 23 '19 at 5:34
• No trouble. Clear like water..@ nmasanta – user679406 Sep 23 '19 at 5:37
• Only then my work will be well worth. – nmasanta Sep 23 '19 at 5:42
• Hello Mr. Down-voter, Would you like to explain the reason to give the down-vote in this answer ? – nmasanta Sep 24 '19 at 8:17


# Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$

Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$.

Ok so originally I messed around with $x^3 + x +1$ for a bit looking for an easy way to factor it and eventually decided that the factors are probably made up of really messy nested roots. So then I tried looking at the quotient field $\mathbb{Z}_2[x]/x^3 + x + 1$ to see if I would get lucky and it would contain all three roots but it doesn't. Is there a clever way to easily find this splitting field besides using the cubic formula to find the roots and then just directly adjoining them to $\mathbb{Z}_2$?

Edit: Ok it turns out I miscalculated in my quotient field, $\mathbb{Z}_2[x]/x^3 + x + 1$ does contain all three roots.

-
If degree is not $3$ it is $6$. – André Nicolas Jul 3 '12 at 2:08
Actually, your quotient $\mathbb{Z}_2[x]/(x^3+x+1)$ does contain at least one root: $x$. This is because $x^3+x+1\equiv 0\pmod{x^3+x+1}$. Can you find the others? – roninpro Jul 3 '12 at 2:08

The splitting field is either degree $3$ or degree $6$ over $\mathbb{Z}_2$, hence it is either $\mathbb{F}_8$ or $\mathbb{F}_{64}$.

Let $\alpha$ be a root, so that $\mathbb{F}_8 = \mathbb{F}(\alpha)$. The elements are of the form $a+b\alpha+c\alpha^2$, with $\alpha^3=\alpha+1$.

Now, the question is whether any of these elements besides $\alpha$ is a root of the original polynomial $x^3+x+1$.

Note that $(\alpha^2)^3 = (\alpha^3)^2 = (\alpha+1)^2 = \alpha^2+1$, and so if we plug in $\alpha^2$ into the polynomial we have $$\alpha^6 + \alpha^2 + 1 = \alpha^2+1+\alpha^2+1= 0.$$ Thus, $\alpha^2$ is also a root. So the polynomial has at least two roots in $\mathbb{F}_8$, and so splits there.

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oh shoot I must have miscaculated in my quotient field, ok cool, thanks! – cuckmaster5000 Jul 3 '12 at 2:11

If your cubic were to factor, the factorization must have at least one linear term, which corresponds to a root. It is easy to check it has no roots in $\mathbb{Z}_2$ - check them both! Plugging in $0$ and $1$ both give $1$ so are not roots, so your polynomial is irreducible over $\mathbb{Z}_2.$

A way to write the splitting field is $\mathbb{Z}_2(\alpha)$ where $\alpha$ is any one of the roots of $x^3+x+1.$ This is because $\alpha^2$ and $\alpha^4= \alpha^2+\alpha$ must then also be distinct roots of $x^3+x+1$, and these 3 then comprise a full list. Note, this isn't a special trick to notice for this problem. In fields of characteristic $p$, if $\beta$ is a root of a polynomial then $\beta^p$ will automatically also be a root, due to properties of the Frobenius endomorphism.

Or as roninpro pointed out in the comments, the quotient ring you considered (which is essentially the same thing as adjoining these roots) does contain at least one root, and by this same trick, all the roots.

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I thought forming the quotient field adjoined just a single root? And that sometimes you would get lucky and the remaining roots could be formed within the quotient field as well, but that sometimes they couldn't, is this not correct? – cuckmaster5000 Jul 3 '12 at 2:17
@NollieTré You're correct. But extensions of finite fields are all Galois and in particular normal. So if $K/k$ are finite fields and $f \in k[X]$ is irreducible and has a root in $K$, then it splits in $K$. Frobenius is a great thing! – Dylan Moreland Jul 3 '12 at 2:25
@NollieTré You are correct, sometimes the extra roots we find with the Frobenius trick just correspond to the same roots. But here they are distinct. You can check yourself (by carrying out the division) that here $x^2$ is also a root, since $x^6+x^2+1 = (x^3+x+1)(x^3-x-1)+2(x^2+x+1) = 0$, and similarly $x^4$ is also a root. So the quotient formed actually has all the roots. – Ragib Zaman Jul 3 '12 at 2:26
Ok interesting, so is there a canonical example of when the quotient field does not contain all the roots of the polynomial? – cuckmaster5000 Jul 3 '12 at 2:29
I think it's easier to note that if $\alpha$ is a root of the polynomial then squaring gives $0^2 = (\alpha^3 + \alpha + 1)^2 = (\alpha^2)^3 + \alpha^2 + 1$. – Dylan Moreland Jul 3 '12 at 2:30

Two good answers already here, but I wanted to emphasize the usefulness of the Frobenius endomorphism. Let $f(X) = X^3 + X + 1$. If I let $\alpha$ denote the image of $X$ in $k = \mathbb F_2[X]/(f(X))$ then applying Frobenius to $0 = f(\alpha)$ gives $0 = (\alpha^3 + \alpha + 1)^2 = (\alpha^2)^3 + \alpha^2 + 1.$ Hence $\alpha^2$ is also a root of $f$, and $\alpha^2 \neq \alpha$ because $\alpha \neq 0, 1$. Since $k$ contains two roots of the cubic $f$, it contains three.

To bring in slightly more technology: extensions of finite fields are Galois and in particular normal, and this implies that if you adjoin one root of an irreducible polynomial over a finite field then you've actually adjoined all of them. This is, of course, not true in general!

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Just wanted to add here, in $\Bbb{Z}_2(\alpha)[x]$, one can factor $x^3 + x + 1 = (x + \alpha)(x^2 + \alpha x + (\alpha^2 + 1))$. Given that $\alpha^2$ is also a root (as Dylan shows above), using ordinary high-school factorization techniques the third root is easily seen to be $\alpha^2 + \alpha$ (a somewhat preferable form than $\alpha^4$). That is: $x^2 + \alpha x + (\alpha^2 + 1) = (x + \alpha^2)(x + (\alpha^2 + \alpha))$ – David Wheeler Jul 6 '12 at 10:20


# Showing that a group $G$ with two distinct, normal subgroups of index $p$ has order $p^2$

A friend of mine and I were working together to solve this exercise:

Let $G$ be a group, and $p$ a prime number. Let $H, K$ be normal subgroups of $G$. Suppose that $|G:H| = |G:K| = p$, and that $H\cap K = \{1\}$. Show that $G\simeq\mathbb{Z}_p\times\mathbb{Z}_p$.

We thought that the quickest way to solve it was showing that $G$ has exactly $p^2$ elements.

For that purpose, we first tried to prove that $|G|\leq p^2$. We chose $a,b\in xH\cap yK$, that are two elements of $G$ belonging to the intersection of a coset of $H$ with a coset of $K$. This means that $a = xh$ and $b=xh'$ for some $h,h'\in H$ and for some $x\in G$ (because $a,b\in xH$). Similarly, $a=yk$ and $b=yk'$, for some $k,k'\in K$ and for some $y\in G$ (because $a,b\in yK$).

Then, $$xh=yk\implies hk^{-1}=x^{-1}y$$ and $$xh'=yk'\implies h'k'^{-1}=x^{-1}y.$$

This brings us to $hk^{-1}=h'k'^{-1}$, which we can rewrite as $hh'^{-1}=k'^{-1}k$. This element belongs to $H\cap K=\{1\}$, thus $h = h'$ and $k = k'$: hence $a = b$. This proves that the intersection of a coset of $H$ with a coset of $K$ cannot contain more than one distinct element.

We made use of this information when we noted that, regardless of how many elements $G$ has, there exist at most $p^2$ intersections of cosets of $H$ with cosets of $K$, and each one of them contains at most one element. Thus, we deduced that $|G|\leq p^2$.

Now, here are my questions:

• How do I prove that $|G|\ge p^2$, in order to conclude that $|G|=p^2$ (and thus that $G\simeq\mathbb{Z}_p\times\mathbb{Z}_p$)?
• Is the proof rigorous enough so far, or were some of our assumptions incorrect?
• This exercise is related to a part of our algebra course in which nothing beyond direct product of groups had been explained yet, so we could not make use of the Sylow theorems, of anything about group actions and so on. Is there a simpler way to solve this exercise, using only "basic" tools of group theory (e.g. quotient groups, the homomorphism and isomorphism theorems and anything regarding the direct product of groups)?
• Is $G$ assumed to be finite at the outset? – Tim Raczkowski Jan 7 '16 at 20:46
• @TimRaczkowski The exercise said nothing on that point, so I assume that $G$ may also be infinite. Anyway, given that $G$ has to be isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_p$, I guess it can't be infinite. Plus, if our half of the proof is correct, $|G|\leq p^2$ definitely implies that $G$ is finite. – Labba Jan 7 '16 at 20:48

You can use the following fact: If $H,K$ are subgroups of $G$, $H\cap K=\{1\}$ and $hk=hk$ for all $h\in H$ and $k\in K$, then $G\cong H\times K$.
Now, $hk\in hK,Hk$. Since $H$ is normal in $G$, $Hk=kH$. So, by your argument above, $\{hk\}=hK\cap kH$. But $kh\in hK\cap kH$ by a similar argument.
• Thanks for your answer :) So, if I got it right, you say that $hk$ belongs to the $hK$ and $Hk$ cosets (the latter being $kH$ anyway, because of the normality of $H$), and the same can be said about $kh$ that belongs both to the $kH$ and $Kh$ cosets (again, the $hK$ coset). In short, $hk, kh\in kH\cap hK$ and, by my part of the proof, $kh = hk$ because only one element can be found in this intersection. Hence, $G\simeq H \times K$ because $H\cap K = \{1\}$ and they commute with each other. Did I get everything? And this also proves that $|G| = p^2$, right? – Labba Jan 7 '16 at 21:15
• I really have to thank you :) Anyway, there's still something I fear I'm missing: even if this shows that $G\simeq H\times K$, does this prove that $|H| = |K| = p$ as well? – Labba Jan 7 '16 at 21:18
• Well if $|G|=p^2$, and $[G:H]=[G:K]=p$, so $|H|=|K|=p$. – Tim Raczkowski Jan 7 '16 at 21:21
• Hmm....I seem to keep missing a detail each time. $[G:H]=[G:K]\implies |H|=|K|$. Suppose $|HK|=n^2$. Now, $n^2|p^2\implies$n=1, or $n=p$. If $n=1$, then $H=N=\{1\}$ and $|G|=p$. So assuming $H\ne K$, we get the result. – Tim Raczkowski Jan 7 '16 at 21:38
Since $H,K$ normal in $G$ and $H\cap K=\{1\}$, $HK\cong H\times K$. $|G:HK|\cdot|HK:H|=|G:H|=p$. So $|HK:H|$ divides $p$. $|HK:H|=1$, or $p$. Suppose it is $1$. $HK=H$, $K\subseteq H$. $|G:H|\cdot|H:K|=|G:K|$, $p|H:K|=p$, so $H=K$, contradiction. So $|HK:H|=p$, $|G:HK|=1$. $G=HK$. Now, $|G:H|=|HK:H|=|K:H\cap K|$ by second isomorphism theorem. So $p=|K|$. Similarly, $|H|=p$. $G\cong \mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$.


# Suppose we roll a fair $6$ sided die repeatedly. Find the expected number of rolls required to see $3$ of the same number in succession.

Suppose we roll a fair six sided die repeatedly.

Find the expected number of rolls required to see $$3$$ of the same number in succession

From the link below, I learned that $$258$$ rolls are expected to see 3 sixes appear in succession. So I'm thinking that for a same (any) number, the rolls expected would be $$258/6 = 43$$. But I'm unsure how to show this and whether it really is correct.

How many times to roll a die before getting two consecutive sixes?

For $n\in \{0,1,2\}$ Let $E[n]$ denote the answer given that you are starting from a streak of $n$ consecutive rolls. The answer you want is $E=E[0]$, though you are never in state $0$ except at the start.

We note $$E[2]=\frac 16\times 1+\frac 56\times \left(E[1]+1\right)$$ $$E[1]=\frac 16\times \left(E[2]+1\right)+\frac 56\times \left(E[1]+1\right)$$

$$E=E[0]=E[1]+1$$

this system is easily solved and, barring error (always possible), yields $$\boxed {E=43}$$

• This is very elegant (+1) - great answer!! Aug 31, 2017 at 16:38
• One issue that isn't really an issue: everything needs to be finite for this to work (cf optional stopping theorem)
– cats
Aug 31, 2017 at 18:32
• This is the (arch classical) approach used in the post mentioned in Byron's answer.
– Did
Aug 31, 2017 at 18:40
• @lulu can you explain me what what tis the theory behind this answer? Does it refer to a specific branch of stochastics? Mar 9, 2021 at 23:30
• @Phillipp This is a fairly standard Markov type computation. The trick, such as it is, is to note that there really are only three possible states for the process, an observation which makes the system very tractable.
– lulu
Mar 9, 2021 at 23:39

From Did's answer here, the probability generating function $u_0(s)=\mathbb{E}(s^T)$ for the number of trials $T$ needed to get three consecutive values the same is
$$u_0(s)={s^3\over 36-30s-5s^2}.$$ Differentiating this and setting $s=1$ in the derivative shows that $\mathbb{E}(T)=43.$

We can treat this as a three-state absorbing markov chain: a length3 run has been seen, otherwise the current run is length2, current run is length 1. Transition matrix: $\begin{bmatrix}1&\frac{1}{6}&0\\0&0&\frac{1}{6}\\0&\frac{5}{6}&\frac{5}{6}\end{bmatrix}$

This is in standard form $\left[\begin{array}{c|c}I&S\\\hline0&R\end{array}\right]$

We turn our attention to the fundamental matrix $(I-R)^{-1} = \begin{bmatrix}1&-\frac{1}{6}\\-\frac{5}{6}&\frac{1}{6}\end{bmatrix}^{-1}=\begin{bmatrix}6&6\\30&36\end{bmatrix}$

After the first roll, we enter the markov chain in the third state. Adding the entries of the fundamental matrix corresponding to that column tells us the expected time until we reach an absorbing state, i.e. until we have a chain of three consecutive rolls of the same number.

Thus the expected number of rolls needed is $1+36+6=43$

Let $\mu$ denote the expectation of the expected number of rolls that are still needed if $2$ rolls have passed with distinct result.

Let $\nu$ denote the expectation of the expected number of rolls that are still needed if $2$ rolls have passed with equal result.

Then the expectation is: $$2+\frac56\mu+\frac16\nu$$

Here $\frac56$ is the probability that the first two numbers are distinct and $\frac16$ is the probability that they are equal.

Secondly we have the relations:

$$\mu=\frac56(1+\mu)+\frac16(1+\nu)=1+\frac56\mu+\frac16\nu\tag1$$ and: $$\nu=\frac161+\frac56(1+\mu)=1+\frac56\mu\tag2$$

The relations $(1)$ and $(2)$ lead easily to: $\mu=42$ and $\nu=36$.

Then $$2+\frac56\mu+\frac16\nu=43$$ is the final answer.

(in the answers uptil now it was not used that you allready learned something).

Let $Y$ denote the number of rolls required to see three dice of the same number in succession and let $X$ denote the number of rolls required to see three dice with number $6$ in succession.

Then: $$Y\text{ and }\frac16X+\frac56(X+Y)=X+\frac56Y\text{ must have equal distribution.}\tag3$$

Here $\frac16$ is the probability of the event that the first time that three dice give the same number in succession they show number $6$ and $\frac56$ is the probability that do not show a number $6$.

$(3)$ rests on the observation that - if for the first time three equal numbers show up in succession - we are ready if $6$ happens to be that number and must actually start over again (with $X$ throws in our pocket) if not.

So we find $\mathbb EY=\mathbb EX+\frac56\mathbb EY$ or equivalently:$$\mathbb EX=\frac16\mathbb EY$$

You allready learned that $\mathbb EY=258$ and making use of that knowledge you find $$\mathbb EX=258/6=43$$ This confirms your thinking.


# Venn Diagrams and Sets

## Chapter 8 - Karnaugh Mapping

Mathematicians use Venn diagrams to show the logical relationships of sets (collections of objects) to one another. Perhaps you have already seen Venn diagrams in your algebra or other mathematics studies. If you have, you may remember overlapping circles and the union and intersection of sets. We will review the overlapping circles of the Venn diagram. We will adopt the terms OR and AND instead of union and intersection since that is the terminology used in digital electronics. The Venn diagram bridges the Boolean algebra from a previous chapter to the Karnaugh Map. We will relate what you already know about Boolean algebra to Venn diagrams, then transition to Karnaugh maps. A set is a collection of objects out of a universe as shown below. The members of the set are the objects contained within the set. The members of the set usually have something in common; though, this is not a requirement. Out of the universe of real numbers, for example, the set of all positive integers {1,2,3…} is a set. The set {3,4,5} is an example of a smaller set, or subset of the set of all positive integers. Another example is the set of all males out of the universe of college students. Can you think of some more examples of sets? Above left, we have a Venn diagram showing the set A in the circle within the universe U, the rectangular area. If everything inside the circle is A, then anything outside of the circle is not A. Thus, above center, we label the rectangular area outside of the circle A as A-not instead of U. We show B and B-not in a similar manner. What happens if both A and B are contained within the same universe? We show four possibilities. Let’s take a closer look at each of the the four possibilities as shown above. The first example shows that set A and set B have nothing in common according to the Venn diagram. There is no overlap between the A and B circular hatched regions. For example, suppose that sets A and B contain the following members:

set A = {1,2,3,4}
set B = {5,6,7,8}

None of the members of set A are contained within set B, nor are any of the members of B contained within A. Thus, there is no overlap of the circles. In the second example in the above Venn diagram, Set A is totally contained within set B How can we explain this situation? Suppose that sets A and B contain the following members:

set A = {1,2}
set B = {1,2,3,4,5,6,7,8}

All members of set A are also members of set B. Therefore, set A is a subset of Set B. Since all members of set A are members of set B, set A is drawn fully within the boundary of set B. There is a fifth case, not shown, with the four examples. Hint: it is similar to the last (fourth) example. Draw a Venn diagram for this fifth case. The third example above shows perfect overlap between set A and set B. It looks like both sets contain the same identical members. Suppose that sets A and B contain the following:

set A = {1,2,3,4}
set B = {1,2,3,4}

Therefore,

 Set A = Set B

Sets And B are identically equal because they both have the same identical members. The A and B regions within the corresponding Venn diagram above overlap completely. If there is any doubt about what the above patterns represent, refer to any figure above or below to be sure of what the circular regions looked like before they were overlapped. The fourth example above shows that there is something in common between set A and set B in the overlapping region. For example, we arbitrarily select the following sets to illustrate our point:

set A = {1,2,3,4}
set B = {3,4,5,6}

Set A and Set B both have the elements 3 and 4 in common These elements are the reason for the overlap in the center common to A and B. We need to take a closer look at this situation.

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### CSIR JUNE 2011 PART C QUESTION 73 SOLUTION (which are positive definite? 1) $A+B$, 2) $ABA^{*}$, 3) $A^2+I$, 4) $AB$.)

Suppose $A$ and $B$ are $n \times n$ positive definite matrices and $I$ be the $n \times n$ identity matrix. Then which of the following matrices are positive definite?
1) $A+B$,
2) $ABA^{*}$,
3) $A^2+I$,
4) $AB$.
Solution:
Let $A$ be a real symmetric matrix, then $A$ is said to positive definite if it satisfies any of the following equivalent conditions.
i) all its eigenvalues are positive.
ii) $x^t A x > 0$ for all vectors $x \ne 0$.
iii)$<x,Ax> > 0$ for all vectors $x \ne 0$.
iii) A is positive definite if and only if it can be written as $A = R^tR$, where $R$ is possibly a rectangular matrix, with independent columns.
iv) All the principal minors of $A$ are positive. (Please share if you know any other equivalent conditions in the comment below)
Let $A$ and $B$ be two $n \times n$ positive definite matrices. We have $x^t A x>0$ and $x^t B x>0$ for $x \ne 0$. We will solve each given option by each of the above given definition of positive definiteness in order to understand them clearly.
option 1. (True)We have for $x \ne 0$, $$x^t (A+B) x = x^t A x + x^t B x > 0.$$
Therefore option 1 is true.
option 2. (True) We have for $x \ne 0$, $Ax \ne 0$ since $A$ is invertible.
$$<x,ABA^*x> = <A^*x,BA^*x> = <B^*A^*x,A^*x> \\ = <BAx,Ax> > 0 .$$
Therefore option 2 is true. Since $A$ and $B$ are real symmetric, we have $A^* = A$ and $B^* = B$.
option 3. (True) Let the eigen values of $A$ be $\lambda_1,\lambda_2,\dots,\lambda_n$. Since $A$ is positive definite we have all these eigen values are positive. We observe that the eigen values of $A^2+1$ are $\lambda_1^2+1,\lambda_2^2+1,\dots,\lambda_n^2+1$ which are all positive. Hence $A^2+1$ is positive definite.
option 4. (False)
The product of two positive definite matrices need not be even symmetric.  In particular, we have $AB$ is symmetric if and only if $A$ and $B$ commutes with each other. Because $$(AB)^* = B^*A^* = A^*B^* = AB.$$ Note that, if $A$ and $B$ commutes with each other, then $A^*$ and $B^*$ commutes with each other.

To illustrate this, consider the positive definite matrices $$A = \begin{bmatrix}11 & 10 \\ 10 & 10\end{bmatrix}$$ and $$B = \begin{bmatrix}11 & 5 \\ 5 & 10\end{bmatrix}$$
text{ Then their product } $$AB = \begin{bmatrix}171&155 \\ 160&150\end{bmatrix}$$ which is not symmetric.

### NBHM 2020 PART A Question 4 Solution $$\int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx$$
Evaluate : $$\int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx$$ Solution : \int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx = \int_{-\infty}^{\inft...