Common Mistakes in Discrete Mathematics CM-1, Study notes of Discrete Mathematics

Download Common Mistakes in Discrete Mathematics CM-1 and more Study notes Discrete Mathematics in PDF only on Docsity! Common Mistakes in Discrete Mathematics CM-1 Common Mistakes in Discrete Mathematics In this section of the Guide we list many common mistakes that people studying discrete mathematics sometimes make. The list is organized chapter by chapter, based on when they first occur, but sometimes mistakes made early in the course perpetuate in later chapters. Also, some of these mis are remnants of misconceptions from high school mathematics (such as the impulse to assume that every operation distributes over every other operation). In most cases we describe the mistake, give a concrete example, and then offer advice about how to avoid it. Note that additional advice about common mistakes in given, implicitly or explicitly, in the solutions to the odd-numbered exercises, which constitute the bulk of this Guide. Solving Problems in Discrete Mathematics Before getting to the common mistakes, we offer some general advice about solving problems in mathematics, which are particularly relevant to working in discrete mathematics. The problem-solving process should consist of the following steps. (This four-step approach is usually attributed to the mathematician George Pélya (1887-1985).) 1. Read and understand the problem at hand. Play around with it to get a feeling for what is going on and what is being asked. 2. Apply one or more problem-solving strategies to attack the problem. Do not give up when one particular tactic doesn’t work. This phase of the process can take a long time. 3. Carefully write up the solution when you have solved the problem. Make sure to communicate your ideas clearly. 4. Look back on what you have done. Make sure that your answer is correct (think of creative ways to test it). Also consider other ways of solving the problem, and think about how you might generalize your results. Your bag of problem-solving strategies may include drawing a picture or diagram; looking at special cases or simpler instances of your problem; looking at related problems; searching for patterns; making tables of what you know (and in general, being organized); giving names to what you don’t know and writing equations about it (or, more generally, applying the mathematical tools you have learned in algebra and other courses); working backwards and setting subgoals (“I could solve this problem if I could do such-and-s ; using trial and error (usually in educational circles); using indirect reasoning and looking for counterexamples (“if this w true, then what would have to happen?”); jumping out of the system and trying something totally different; or just going away from the problem and coming back to it later. You will find it useful to read over this description of the problem-solving process repeatedly during your study of discrete mathematics. Here is an example of applying the problem-solving process to a problem in discrete mathematics. Suppose we want to find the number of squares (of all sizes) on a checkerboard. We understand this to mean not only the obvious 64 little squares, but also the 2 x 2 squares (of which there will be many), the 3 x 3 squares, and so on, up to the entire 8 x 8 board itself. We should draw a picture to see what’s going on here. Le with a smaller version of this problem, using a board of size 2 x 2. Then obviously there are 4 small square the entire board, for a total of 4+ 1=5. Let’s try a 3x 3 board. Here there are 9 little squares, and it is to see that there are 4 squares of size 2 x 2 (nuzzled in the upper left, the upper right, the lower le right), as well as the entire board; so the answer here is 9+ 4+1= 14. A pattern seems to be emergin, seem to be adding perfect squares to get the answer. Maybe for the 4 x 4 board there will be 16+9+4+1=30 squares in all. We draw the 4 x 4 picture and verify that this is correct. In fact, we can see exactly what is going on, the way the upper left corner of a k x k square can be in any of the first 5 — k rows and first 5 — k columns of the board, for k = 1, 2, 3, 4, and so there are (5 — k)? squares of size k in the 4x 4 board, exac tly as our sum indicated. We have now solved the problem and can write up a solution explaining why the a 8° 47746745744? 43742741? = 204. In the looking back stage (step 4) we would certainly want to notice that for an nxn board, there are 7" ares. To continue our investigation, we might want to explore such further questions as allowing rectangles rather than squares, looking at rectangular checkerboards rather than s (or counting triangles in boards made up of triangles), or moving to 3-dimensional space and counting c large cube. Notice how we followed the process outlined above and used many of the strategies listed. , or the lower uare ones bes in a CM-2 Common Mistakes in Discrete Mathemat List of Common Mistakes students or ins e for Disc ructors have items to add to the lists below, please let the author know (visit the companion ematics and Its Applications at http://www. mbhe .com/rosen). Chapter 1 Incorrectly translating English statements into symbolic form. There are many errors of this type. For example, there are difficulties with the use of the word “or” in English; be sure to differentiate between inclusive and exclusive versions (see pages 4-5 of the text). A conditional statement is quite different from a conjunction, but some speakers fail to distinguish them; to say that B will happen if A happens is quite different from ying that A and/or B will happen. Perhaps the most common mistake is confusing p > q with q— p. To y, for example, that I will go to the movie i I finish my homework me mething quite different from erting that I will go to the movie only if I finish my homework. Incorres -tly negating compound statements without using De Morgan's laws—in ¢, A(p V q) is logically equivalent to =p V 7q, or that +(p Aq) és logically equivale if it is not true that John is over 18 years old or lives away from home, then it is true that he is not over 18 years old and (not or) he does not live away from home. The correct statements are that 7(p V q) is logically equivalent to =p A-q, and that =(pA q) is logically equivalent to ap Vg. This mistake is a general instance of assuming that every operation distributes over every other operation, here that negation distributes over disjunction (or conjunc saying, for example, that 0 =p Aq. For example, Misinterpreting the meaning of the word “any” in a mathematical statement. This word is ambiguous in many situations, and so should usually be avoided in mathematical writing. If you are not sure whether the writer or “some” when the word “any” was used, get the statement clarified. As a corollary, of course, you should avoid using this word yourself. Here is an example: What would one mean if she defined a purple set of integers to be one “in which any integer in the set has at least three distinct prime divisors”? Does “am here meant “eve mean “every” here (in which case the set {30,40} is not purple), or does “any” mean “some’ case the set {30,40} is purple)? Incorres ly writing the symbolic form of an existential statement as 3x(A(x) > B(e)) B(«)). For example, the symbolic form of “There exists an e i not Se(E(«) > , where we are letting E() mean “x of thumb, cieenttal quantifiers are usually followed by conjunctions. As a rule and P(«) mean “x is prim Incorrectly writing the symbolic form of a universal statement as Vx(A(x)\B(a)) instead of Yx(A(x) > B(e)). For example, the symbolic form of “Every odd number is pri )), not Va(O(a) A P(x), where we are letting O() mean “x is odd” and P(x) mean “2 is prime.” As a rule of thumb, universal quantifiers are usually followed by conditional statements. Incorrectly putting predic 's inside predicates, such as P(O(x)). For example, if P(«) means “a is prime,” and O(«) means “a is odd,” then it would never make sense to write P(O(x)) in trying to express a statement such as “x is an odd prime” or to write Va P(O(x)) to say “all odd numbers are prime.” The notation P(O(z)) would mean that the assertion that « is odd is a prime number, and clearly an assertion isn’t any kind of number at all. Functional notation has a wonderful internal beauty and consistency to it—the thing inside the parentheses has to be what the thing outside the parentheses applies to. Failure to change the quantifier when n negation of the that no cats like liver, or that all egating a quantified proposition, especially in English. For example, the atement that some cats like liver is not the statement that some cats do not like liver Ss ats dislike liver. Overusing the term “by definition” in justifying statements in a proof. For example, Franklin Roosevelt was not the President of the United States at the start of the country’s entry into World War II in December, 1941, “by definition” ; he was the President because he had been inaugurated as such carly in 1941 and had not died or left office. Not going back to carefully check the definitions in justifying statements in a proof. For example, if one is trying to prove something about odd integers, then it is important to correctly use the meaning of that notion (that an odd integer is one that can be written as 2k +1 for some integer k:) at one or more places in the proof. Incorrectly starting a proof by assuming what is to be proved. A common occurrence of this in an earlier course is trying to prove trigonometric identities by starting with the identity and using algebra to reach A = A; this is not valid. Similarly, if we are trying to prove a set identity in Chapter 2, such as A C AU B, it would be Common Mistakes in Discrete Mathematics CM-5 example, if we try to prove that n = n +1 for all positive integers n, but this proposition is obviously not true. The catch is that the basis step (when n = 1) fails, since 14141. Failing to do more than one case in the basis step in a proof by mathematical induc such as when the ds two or more previous conditions. For example, when proving statements it usually is necessary to check the first two basis cases (say n = 1 and n = 2), on the equation fy = fn—1+ fn—2- Confusing a summation with the propositional function P(n) in an induction proof. For example, in trying to prove 14+2+3+-:-+n=n(n+1)/2 by induction, P(n) is this entire equation, not its left-hand side. Not being organized when attempting to write a recursive definition. Good advice here is to think about how you want to build up the items under discussion, step by step. The inductive rules of the definition need to be formulated to permit cach such step, and base cases are needed to get the process off the ground. Common mistakes include not including enough base (for example, the recursive definition of regular expressions in Chapter 12 requires three base cases), having conflicting cases (for example, having one clause to handle n divisible by 2 and another clause to handle n divisible by 3, and thereby not having a unique definition for those n divisible by 6), or having a function value at n depend on a function value at an input larger than n (e.g,, trying to set f(n) = f(8n +1) — 2). Ina proof by mathematical induction, writing P(k+1) incorrectly. Once P(n) is properly formulated, writing P(k +1) can usually be done more or less mechanically by plugging k +1 in for n. For example, if P(n) is the statement 24+4+6+4-++-+2n=n(n+1), then P(k+1) is 2+44+6+-+-4+2k+2(k+1) = (k+ D(k+2). In a proof by mathematical induction, making errors in basic algebra, especially in simplifying expressions. Vor example, you might have to use the fact that 2” + 2” = 2", or to simplify (n + 1)3 + 5(n + 1)”, which is best done by factoring, not by first expanding each term. Carefully check your algebraic manipulations when you have trouble with the inductive step of such a proof. on in certa: ‘uations, about the Fibona i sequence since the inductive i a In a rec about ree ive algorithm, fail ursive algorithms example, suppose that you want to write a recursive algorithm to compute a and a+! = (a)? -a, The recursive call will handle the calculation of a* to let the computer do the recursing. the reluctance to believe that the The hardest thing to overcome in thinking maller case will be handled correctly. For ” using the facts that a?" = (a*)? and so you don’t need to worry about how the computer will repeatedly recurse, all the way down to the base case, in order to do that. As e (here, a q 0 long as your recursive step and ba ) are correct, your algorithm Chapter 5 Drawing an incorrect diagram wh and drawing a diagram is almost al could also be considered a common mis g problems. Diagrams are very useful in all of mathematics, a good way to start solving a problem; thus failing to draw a diagram ake. For example, you should draw a row of six blanks (not five) as a template for constructing words of length 6 whose symbols are chosen from a set of five elements. Tree diagrams are also sometimes quite helpful. Not determining wh solving a counting problem. For example, if we are asked for the number of ways to write 7 as the sum of positive integers, then we need to know whether 3+ 2-+ 2 and 243-42 are to be considered the same way or distinct ways. Read the problem very carefully to understand what is being counted. Resolve any ambiguities ahead of time by explicitly stating any assumptions that seem to be missing from the problem formulation. er or not order mati si g whether or not repetitions are allowed in solving a counting problem. For example, if we are asked for the number of ways to choose five donuts from a shop selling eight varieties of donuts, we need to know whether we are allowed to choose more than one donut of the same variety. Read the problem very carefully to understand what is being counted. Resolve any ambiguities ahead of time by explicitly stating any assumptions that seem to be missing from the problem formulation. ‘ussion more than. once, e set under di not recognizin For example, if we count handshakes person by pers > has been counted twice, once for each of its participants. that an adjustment needs to be then we need to recognize made for double coun that each sha set under discussion more than once, not recognizing that the inclusion-exc principle is needed. For e >, if we are told that there are 26 computer science majors and 34 mathemat: majors at a certain university, then there may not be 26+ 34 = 60 people majoring in either computer science or mathematics, since these two numbers might both include the double majors. To correctly compute the jon, ete Mathemati CM-6 Common Mistakes in Dis total number of people majoring in these subjects, we would need to subtract the number of double majors from this sum. Using the pigeonhole or generalized pigeonhole principle incorrectly. It helps to explicitly identify the pigeons and the holes. For example, to find the minimum number of cards that must be chosen in order to guarantee that at least six of the same suit are picked, the holes are the suits, and the cards are the pigeons (the answer is 21). Chapter 6 © When trying to calculate the probability that one of two eve sum of the individual probabil For example, the probability that a 3 will show up if a fair die i rolled twice is not 4+ 4, the sum of the probability that the 3 occurs on the first roll and the probability that the 3 occurs on the we should calculate this as 1 minus the probability that the 3 fails to appear on either since the rolls are independent). Thus the correct answer is 1 — second roll. Instead roll (which is 8 Assuming that all events in a probabil sjoint. Doing so can lead to absurd conclusions, § as a probability greater than 1. (This is really a generalization of the previously listed mistake.) For example, to calculate the probability that someone else in your graduating class of 400 students shares your birthday (assuming that all birthdays are equally likely and ignoring February 29), you cannot argue that since each of them has a probability of 1/365 of sharing your birthday, the probability is 399/365 (probabilities « exceed 1, and in any case, this event is not a certainty). 'y calculation are an never Assuming that all events in a probability calculation are independent. For example, to calculate the probability that we get two hearts when drawing two cards from a deck of cards, without replacing the first card before drawing the second, we cannot simply note that the probabil ty of drawing a heart is #$ on each draw (that much is true) and therefore conclude that the answer is = +. Instead, we must determine that for the second draw, the probability of drawing a heart, given that we drew a heart on the first dra therefore that the probability of drawing a heart both times is 8. # = 4. © Getting misled by the subtle assumptions inherent in probability problems. The most famous example here is the Monty Hall Three Door Problem (see Example 10 in Section 6.1 of the text). Unless one is very careful about the one makes about the game host’s protocol, one cannot calculate the probability that of winning. For example, if the host (who knows where the prize lies) were to offer you a switch if and only if you had chosen the correct door, then obviously it would be wrong for you to switch when he makes the offer. A national debate about this problem raged for many months when it was popularized in a magazine article. ¢ Letting intuition interfere with reason in working with probability that among a group of 23 people, the odds favor two of them having the shows this to be true. For example, it might seem counter-intuitive calculation me birthday, but the if the . One 19 that units for variance are squares of units for the underlying random variable. For examp| of adults have a mean of 67 i and a variance are root and restate this as “the standard deviation is 3 inche spread of the distribution of heights. © Forg height should take the way to measure the Confusing p(A B) with p(B| A). For example, the probability that a person who tests positive for a disease ally has the disease is usually m positive for it. One can often use Bay additional information about the prevalence of the disease and the false positive rate for the test). maller than the probability that a person who has the di Theorem to compute the former probability, given the latter (and Chapter 7 Failing to note the need for the inclusion-exclusion principle. To believe that |AU B| = |A| + |B| is always true is related to the wishful thinking that every operation distributes (or otherwise behaves in some si agreeable way) with respect to every other operation. This equality holds only when A and B are ¢ © Confusing the signs of as we take larger and larger unions. terms when applying the inclusion-exclusion principle. Note that the signs alternate Not including all the terms when applying the inclusion-exclusion principle. If there are n sets involved, then there are nearly 2” different terms in the equation altogether. Common Mistakes in Discrete Mathematics CM-7 © Giving up too easily when trying to write down a recurrence relation to model a problem situation. Ask yourself how one can obtain an instance of the problem of size n from instances of sizes n—1 (or sometimes also smaller instar Make sure to consider all the possibilities, and make sure to include enough initial conditions. For example, if a, is the number of ways to climb n stairs if we are allowed to take them either one at a time or three at a time, then clearly a; = 1, a2 = 1, and ag = 2, and then ay = an—1 +an—3 for n > 4, since the first step could be a single step or a triple step. whi Misapplying the algorithm for solving linear homogeneous recurrence relations with constant coefficiel there are repeated roots of the equation. One needs to multiply by powers of n in th . For example, if the equation is r? — 6r + 9 = (r — 3)? = 0, then the general solution is an = o1 3" +eon 3”, g a bogus particular solution of a linear nonhomogeneous recurrence relation with constant coefficients. 3 ¢ advisable to check the solutions you obtain. For example, if you had computed that ay = 2" was lution to Gn = 2an—1 +2”, then plugging this in would show you that you must have made an getting to use the inclusion egers. For example, to count the number of solutions to « +y +2 = 58 where 0 <a <8, 0<y<10, and 0 < z < 15, one needs to count the number of solutions when the upper bound restrictions are lifted, then subtract the number of solutions in whic riction is violated, then add back the number of solu- exclusion principle when counting solutions to an equation in nonnegative in- tions in which two such restrictions are violated simultaneously, and finally subtract the number of solutions in which all three restrictions are violated. © Forgetting to worry about the first few terms of a power series. When solving a recurrence relation by using generating functions, the recurrence relation usually kicks in only for k > 1 or 2; thus the first term or two must be handled explicitly. Failing to change the variable in a power series when necessary. For example, if a power you need it to be a n replace k by k+1 throughout the summation (including the limits) and simplify 2 Deailk + alt = Oe (kh + Dak. model when solving counting problems with generating functions. You need to carefully has and Setting up the wron work out what each factor of the generating function needs to be, worrying about how much repetition is allowed and whether order matters. See, for instance, Example 12 in Section 7.4 of the text, where the proper generating function depends on whether or not we take order into ount. © Making algebraic errors in working with generating functions. When expanding a generating function to find the coefficient of x", one must of course use the d as the number of terms can grow rapidly. One solution to this problem is to use a computer algebra package such as Maple to do the algebra. For example, to multiply out (1+«+«7)(1+a?+a1+ 2°), you end up with 12 terms, which then simplify to 1+ a+ 2a? +a°+42at+a°42e%+a7+a°5. © Not knowin ributive law. The algebraic manipulations can get messy, how to use partial fraction decomposition when dealing with generating functions, or making errors in the procedure, such as forgetting to include terms of the form (a — a)" for all k such that 1<k <n when the factor (x — a)" appears in the denominator of the ‘on to be expanded. This subject is traditionally taught in calculus courses, even though it has little to do with calculus (other than the fact that it is used as a technique of integration). Therefore those students who have not yet studied enough lus (partial fractions are usually covered in the second semester), or who have taken a course in which this topic is not covered, may need to find a source of instruction for this useful tool (or rely on a computer algebra package such as Maple to perform the task). Any traditional calculus text will probably have a section from which this material can be learned or reviewed. Chapter 8 Failing to draw a picture when dealing with relations. The digraph of a relation on a set gives an excellent way to visualize what is going on. This common mistake can be generalized: Failing to draw a picture when dealing with any mathematical object. See the list of gene this section of the Guide. © Forgetting to think about pairs (a,b) and (b,a) when checking for transitivity of a relation or forming the transitive closure. In this case, one needs to have (or add) the loops (a,a) and (b,b) as well. al problem-solving strategies given in the introduction to Failing to recognize that symmetry or transitivity often hold vacuously. For example, the relation {(1,2), (1,3)}