DISCRETE MATH.+ITS APPLICATIONS CUSTOM
8th Edition
ISBN: 9781307447118
Author: ROSEN
Publisher: MCG
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Textbook Question
Chapter 5.1, Problem 49E
rcises 49-51 present incorrect proofs using mathematical induction. You rill need to identify" an error in reasoningin each exercise.
49.What is wrong with this "proof" that all horses are the same color?
LetP(n) be the proposition that all thehorses in a set ofnhorses are the same color.
Basis Step: Clearly,P(1) is true.
Inductive Step: Assume thatP(k) is true, so that all the horses in any set of Morses are the same color. Consider anyk +1 horses; number these as horses 1,,2,3,...,k,k+1. Now the firstk ofthese horses all must have the same color, and the last taf these must also have the same color. Because the set of the first horses and the set of thelast horses overlap, allk+1 must be the same color. This shows thatP(k+1\)is true and finishes the proof by induction.
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Theorem 2.6 (The Minkowski inequality)
Let p≥1. Suppose that X and Y are random variables, such that E|X|P <∞ and
E|Y P <00. Then
X+YpX+Yp
Theorem 1.2 (1) Suppose that P(|X|≤b) = 1 for some b > 0, that EX = 0, and
set Var X = 0². Then, for 0 0,
P(X > x) ≤e-x+1²²
P(|X|>x) ≤2e-1x+1²²
(ii) Let X1, X2...., Xn be independent random variables with mean 0, suppose
that P(X ≤b) = 1 for all k, and set oσ = Var X. Then, for
x > 0.
and
0x) ≤2 exp
Σ
k=1
(iii) If, in addition, X1, X2, X, are identically distributed, then
P(S|x) ≤2 expl-tx+nt²o).
Theorem 5.1 (Jensen's inequality)
state without proof the Jensen's Ineg.
Let X be a random variable, g a convex function, and suppose that X and g(X) are
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Chapter 5 Solutions
DISCRETE MATH.+ITS APPLICATIONS CUSTOM
Ch. 5.1 - re are infinite]y many stations on a train route....Ch. 5.1 - pose that you know that a golfer plays theho1e of...Ch. 5.1 - P(n) be the statement...Ch. 5.1 - P(n) be the statementthat 13+ 23+ ... + n3=...Ch. 5.1 - ve...Ch. 5.1 - ve that1.1!+2.2!+...n.n!=(n+1)!1whenevernis a...Ch. 5.1 - ve that3+3.5+3.52+...+3.5n=3(5n+11)/4whenevernis a...Ch. 5.1 - ve that22.7+2.72...+2(7)n=(1(7)n+1)/4whenevernis a...Ch. 5.1 - a)Find a formula for the sum of the firstneven...Ch. 5.1 - a) Find a formula for 112+123++1m(n+1) by...
Ch. 5.1 - a) Find a formula for 12+14+18+...+12n by...Ch. 5.1 - ve that j=0n(12)=2n+1+(1)n32n whenevernis a...Ch. 5.1 - ve that1222+32...+(1)n1n2=(1)n1n(n+1)/2whenevernis...Ch. 5.1 - ve that for every positive...Ch. 5.1 - ve that for every positive integern,...Ch. 5.1 - ve that for every positive integern,...Ch. 5.1 - ve thatj=1nj4=n(n+1)(2n+1)(3n2+3n1)/30whenevernis...Ch. 5.1 - P(n) be the statement thatn!< nn, where n is an...Ch. 5.1 - P(n)be tie statement that 1+14+19+...+1n221n,...Ch. 5.1 - ve that3nn!if n is an integer greater than6.Ch. 5.1 - ve that2nn2ifnis an integer greater than 4.Ch. 5.1 - Prob. 22ECh. 5.1 - which nonnegative integersnis2n+32n?Prove your...Ch. 5.1 - ve that1/(2n)[1.3.5..(2n1)]/(2.4....2n)whenevernis...Ch. 5.1 - ve that ifhi,then1+nh(1+h)nfor all nonnegative...Ch. 5.1 - pose that a and b are real numbers with o< b< a....Ch. 5.1 - ve that for every positive integern,...Ch. 5.1 - ve thatn27n+12is nonnegative whenevernis an...Ch. 5.1 - Prob. 29ECh. 5.1 - ve that H1+H2+...+Hn=(n+1)HnnCh. 5.1 - mathematical induction in Exercises 31-37 to prove...Ch. 5.1 - mathematical induction in Exercises 31-37 to prove...Ch. 5.1 - mathematical induction in Exercises 31-37 to prove...Ch. 5.1 - mathematical induction in Exercises 31-37 to prove...Ch. 5.1 - mathematical induction in Exercises 31-37 to prove...Ch. 5.1 - mathematical induction in Exercises 31-37 to prove...Ch. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - mathematical induction in Exercises 38-46 to prove...Ch. 5.1 - mathematical induction in Exercises 38-46 to prove...Ch. 5.1 - mathematical induction in Exercises 38-46 to prove...Ch. 5.1 - Prob. 43ECh. 5.1 - mathematical induction in Exercises 38-46 to prove...Ch. 5.1 - mathematical induction in Exercises 38-46 to prove...Ch. 5.1 - mathematical induction in Exercises 38-46 to prove...Ch. 5.1 - Exercises 47 and 48 we consider the problem of...Ch. 5.1 - In Exercises 47 and 48 we consider the problem of...Ch. 5.1 - rcises 49-51 present incorrect proofs using...Ch. 5.1 - Exercises 49-51 present incorrect proofs using...Ch. 5.1 - rcises 49-51 present incorrect proofs using...Ch. 5.1 - pose thatmandnare positive integers withm >nandfis...Ch. 5.1 - Prob. 53ECh. 5.1 - mathematical induction to show that given a set...Ch. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - 57.(Requires calculus) use mathematical induction...Ch. 5.1 - pose that A and B are square matrices with the...Ch. 5.1 - Prob. 59ECh. 5.1 - Prob. 60ECh. 5.1 - Prob. 61ECh. 5.1 - w that n lines separate the plane into (n2+n+ 2)/...Ch. 5.1 - A=(a1+a2+...+an)/nG= and the geometric mean of...Ch. 5.1 - Prob. 64ECh. 5.1 - Prob. 65ECh. 5.1 - Prob. 66ECh. 5.1 - Prob. 67ECh. 5.1 - Prob. 68ECh. 5.1 - pose there arenpeople in a group, each aware of a...Ch. 5.1 - pose there arenpeople in a group, each aware of a...Ch. 5.1 - Prob. 71ECh. 5.1 - pose there arenpeople in a group, each aware of a...Ch. 5.1 - Prob. 73ECh. 5.1 - etimes ire cannot use mathematical induction to...Ch. 5.1 - Prob. 75ECh. 5.1 - etimes we cannot use mathematical induction to...Ch. 5.1 - nbe an even integer. Show that it is people to...Ch. 5.1 - Prob. 78ECh. 5.1 - .Construct a ling using right triominoes of the 8...Ch. 5.1 - ve or disprovethatall checkerboards of these...Ch. 5.1 - w that a three-dimensional2n2n2ncheckerboard with...Ch. 5.1 - w that annncheckerboard with on square removed can...Ch. 5.1 - w that acheckerboard with a corner square removed...Ch. 5.1 - Prob. 84ECh. 5.1 - Prob. 85ECh. 5.2 - Use strong induction to show that if you can run...Ch. 5.2 - strong induction to show that all dominoes fall in...Ch. 5.2 - P(n)be the statement that a postage ofncents can...Ch. 5.2 - P(n)be the statement that a postage of n cents can...Ch. 5.2 - a)Determine which amounts of postage can be formed...Ch. 5.2 - a)Determine which amounts of postage can be formed...Ch. 5.2 - ch amount of money can b formed using just two...Ch. 5.2 - pose that a store offers gift certificates in...Ch. 5.2 - song induction to prove that2is irrational. [Hint:...Ch. 5.2 - Assume that a chocolate bar consists ofnsquares...Ch. 5.2 - sider this variation of the game of Nim. The game...Ch. 5.2 - . Use strong induction to show that every positive...Ch. 5.2 - A jigsaw puzzle is put together by successively...Ch. 5.2 - Supposeyou begin with apile ofnstones and split...Ch. 5.2 - Prob. 15ECh. 5.2 - ve that the first player has a winning strategy...Ch. 5.2 - strong induction to show that if a simple polygon...Ch. 5.2 - strong induction to show that a simple po1gonPwith...Ch. 5.2 - Prob. 19ECh. 5.2 - Prob. 20ECh. 5.2 - the proof ofLemma 1we mentioned that many...Ch. 5.2 - rcises 22 and 23 present examples that show...Ch. 5.2 - Prob. 23ECh. 5.2 - Prob. 24ECh. 5.2 - pose thatP(n) is a propositional function....Ch. 5.2 - pose that ifp(n) is a propositional function....Ch. 5.2 - w that if the statement is for infinitely many...Ch. 5.2 - bbe a fix integer and a fixed positive integer....Ch. 5.2 - Prob. 29ECh. 5.2 - d the flaw with the following "proof" thatan=1 for...Ch. 5.2 - w that strong induction is a valid method of proof...Ch. 5.2 - Prob. 32ECh. 5.2 - Prob. 33ECh. 5.2 - ve that (math) for all positive integerskandn,...Ch. 5.2 - Prob. 35ECh. 5.2 - well-orderingproperty can be used to show that...Ch. 5.2 - a be an integer and b be a positive integer. Show...Ch. 5.2 - Prob. 38ECh. 5.2 - you u se th e well - ord ering pr operty to pr o v...Ch. 5.2 - Prob. 40ECh. 5.2 - w that the well-ordering property can be proved...Ch. 5.2 - w that principle of mathematical induction and...Ch. 5.2 - Prob. 43ECh. 5.3 - Findf(1),f(2),f(3), andf(4) iff(n) is defined...Ch. 5.3 - Findf(1),f(2),f(3),f(4), andf(5)iff(n)is defined...Ch. 5.3 - LetP(n) bethestatementthata postage ofncents can...Ch. 5.3 - Prob. 4ECh. 5.3 - Determine which amounts of postage can be formed...Ch. 5.3 - Determine which amounts of postage can be formed...Ch. 5.3 - e a recursive definition of the...Ch. 5.3 - Give a recursive definition of the sequence...Ch. 5.3 - Fbe the function such thatF(n) is the sum of the...Ch. 5.3 - en a recursive definition ofsm(n), the sum of the...Ch. 5.3 - e a recursive definition ofPm(n), the product of...Ch. 5.3 - Exercises 12—19fnis the nth Fibonacci 12.Prove...Ch. 5.3 - Exercises1219fnis the nth Fibonacci number....Ch. 5.3 - Exercises 12—l9fnis the nth Fibonacci *14.Show...Ch. 5.3 - Prob. 15ECh. 5.3 - Prob. 16ECh. 5.3 - Exercises 12-19fnis thenthFibonacci number....Ch. 5.3 - Exercises 12-19fnis thenthFibonacci number. 18....Ch. 5.3 - Prob. 19ECh. 5.3 - e a recursive definition of the if functions max...Ch. 5.3 - Prob. 21ECh. 5.3 - Prob. 22ECh. 5.3 - Prob. 23ECh. 5.3 - e a recursive definition of a)the set of odd...Ch. 5.3 - e a recursive definition of a)the set of even...Ch. 5.3 - Sbe the set of positive integers defined by Basis...Ch. 5.3 - Sbe the set of positive integers defined by Basis...Ch. 5.3 - Sbe the subset of the set of ordered pairs of...Ch. 5.3 - Sbe the subset of the set of ordered pairs of...Ch. 5.3 - e a recursive definition of each ofthesesets of...Ch. 5.3 - e arecursive definition of each of these sets of...Ch. 5.3 - ve that in a bit string, the string 01 occurs at...Ch. 5.3 - ine well-formed formulae of sets, variables...Ch. 5.3 - Prob. 34ECh. 5.3 - Give a recursive definition of the...Ch. 5.3 - d the reversal of the following bit strings....Ch. 5.3 - e a recursive definition of the reversal of a...Ch. 5.3 - structural induction to prove that(w1w2)R=w2Rw1R.Ch. 5.3 - Prob. 39ECh. 5.3 - the well-ordermg property to show that ifxandyare...Ch. 5.3 - n does a swing belong to eset Aof bit stings...Ch. 5.3 - ursively define the set of bit strings that have...Ch. 5.3 - Prob. 43ECh. 5.3 - Prob. 44ECh. 5.3 - structural induction to show thatn(T)>&[I)+inhere...Ch. 5.3 - Prob. 46ECh. 5.3 - Prob. 47ECh. 5.3 - generalized induction as was doneinExample 13to...Ch. 5.3 - A partition of a positive integer nis amy to...Ch. 5.3 - Prob. 50ECh. 5.3 - sider the Mowing inductive definition of a version...Ch. 5.3 - Prob. 52ECh. 5.3 - Prob. 53ECh. 5.3 - sider the Mowing inductive definition of a version...Ch. 5.3 - sider the Mowing inductive definition of a version...Ch. 5.3 - Prob. 56ECh. 5.3 - sider the Mowing inductive definition of a version...Ch. 5.3 - Prob. 58ECh. 5.3 - Prob. 59ECh. 5.3 - Prob. 60ECh. 5.3 - Prob. 61ECh. 5.3 - rcises 62-64 deal with iterations of the logarithm...Ch. 5.3 - rcises 62-64 deal with iterations of the logarithm...Ch. 5.3 - Prob. 64ECh. 5.3 - Prob. 65ECh. 5.3 - f(n)=n/2.Find a formula forf(k)(n).What is the...Ch. 5.3 - Prob. 67ECh. 5.4 - ce Algorithm 1when it is givenn= 5 as input, That...Ch. 5.4 - Prob. 2ECh. 5.4 - Prob. 3ECh. 5.4 - Prob. 4ECh. 5.4 - ce Algorithm 4 when it is given In=5,n= 11, andb=3...Ch. 5.4 - ce Algorithm 4 when it ism=7,n=10, andb=2 as...Ch. 5.4 - Prob. 7ECh. 5.4 - e a recursive algorithm for finding the sum of the...Ch. 5.4 - Prob. 9ECh. 5.4 - e a recursive algorithm for finding the maximum of...Ch. 5.4 - Prob. 11ECh. 5.4 - ise a recursive algorithm for...Ch. 5.4 - e a recursive algorithm for...Ch. 5.4 - Give a recursive algorithm for finding mode of a...Ch. 5.4 - ise a recursive algorithm for computing the...Ch. 5.4 - ve that the recursive algorithm for finding the...Ch. 5.4 - Prob. 17ECh. 5.4 - ve that Algorithm 1 for computingn! whennis a...Ch. 5.4 - Prob. 19ECh. 5.4 - Prob. 20ECh. 5.4 - Prob. 21ECh. 5.4 - ve that the recursive algorithm that you found in...Ch. 5.4 - ise a recursive algorithm for computing for...Ch. 5.4 - ise a recursive algorithm to finda2n, whereais a...Ch. 5.4 - Prob. 25ECh. 5.4 - the algorithm in Exercise 24 to devise an...Ch. 5.4 - does the number of multiplication used by the...Ch. 5.4 - many additions are used by the recursive and...Ch. 5.4 - ise a recursive algorithm to find thenthterm of...Ch. 5.4 - ise an iterative algorithm to find the nth term of...Ch. 5.4 - Prob. 31ECh. 5.4 - ise a recursive algorithm to find the nth term of...Ch. 5.4 - Prob. 33ECh. 5.4 - the recursive or the iterative algorithm for...Ch. 5.4 - Prob. 35ECh. 5.4 - Prob. 36ECh. 5.4 - e algorithm for finding the reversal of a bit...Ch. 5.4 - Prob. 38ECh. 5.4 - Prob. 39ECh. 5.4 - ve that the recursive algorithm for finding the...Ch. 5.4 - Prob. 41ECh. 5.4 - Prob. 42ECh. 5.4 - Prob. 43ECh. 5.4 - a merge sort to sort 4.3,2,5, i, 8, 7, 6 into...Ch. 5.4 - Prob. 45ECh. 5.4 - many comparisons are required to merge these pairs...Ch. 5.4 - Prob. 47ECh. 5.4 - What theleast number comparisons needed to merge...Ch. 5.4 - ve that the merge sort algorithm is correct.Ch. 5.4 - Prob. 50ECh. 5.4 - Prob. 51ECh. 5.4 - quick sort is an efficient algorithm. To...Ch. 5.4 - Prob. 53ECh. 5.4 - Prob. 54ECh. 5.4 - Prob. 55ECh. 5.5 - ve that the program segment y:=1z:=x+y is correct...Ch. 5.5 - ify that the program segment ifx0thenx:=0 is...Ch. 5.5 - ify that the progr am segment is correct with...Ch. 5.5 - Prob. 4ECh. 5.5 - ise a rule of inference for verification of...Ch. 5.5 - the rule of inference developed in Exercise 5 to...Ch. 5.5 - Prob. 7ECh. 5.5 - Prob. 8ECh. 5.5 - Prob. 9ECh. 5.5 - Prob. 10ECh. 5.5 - Prob. 11ECh. 5.5 - Prob. 12ECh. 5.5 - a loop invariant to verify thattheEuclidean...Ch. 5 - Can you use theprinciple of mathematical induction...Ch. 5 - a) For which positive integersnis iin+ 17 S b)...Ch. 5 - Which amounts of postage can be formed using only...Ch. 5 - e two different examples of proofs that use strong...Ch. 5 - a) State the well-ordering property for the set of...Ch. 5 - Prob. 6RQCh. 5 - Prob. 7RQCh. 5 - Prob. 8RQCh. 5 - Prob. 9RQCh. 5 - Prob. 10RQCh. 5 - Prob. 11RQCh. 5 - Prob. 12RQCh. 5 - Prob. 13RQCh. 5 - Prob. 14RQCh. 5 - Prob. 15RQCh. 5 - Prob. 16RQCh. 5 - Prob. 1SECh. 5 - Prob. 2SECh. 5 - mathematica1 induction to show...Ch. 5 - Prob. 4SECh. 5 - Prob. 5SECh. 5 - mathematical induction to show...Ch. 5 - Prob. 7SECh. 5 - d an integ N such that2nn4whenevernan integer...Ch. 5 - Prob. 9SECh. 5 - Prob. 10SECh. 5 - Prob. 11SECh. 5 - Prob. 12SECh. 5 - Prob. 13SECh. 5 - Prob. 14SECh. 5 - Prob. 15SECh. 5 - Prob. 16SECh. 5 - Prob. 17SECh. 5 - Prob. 18SECh. 5 - mulate a conjecture about which Fibonacci nubs are...Ch. 5 - Prob. 20SECh. 5 - Prob. 21SECh. 5 - w thatfn+fn+2=ln+1whenevernis a positive integer,...Ch. 5 - Prob. 23SECh. 5 - Prob. 24SECh. 5 - Prob. 25SECh. 5 - Prob. 26SECh. 5 - Prob. 27SECh. 5 - (Requires calculus)Suppose that the...Ch. 5 - w ifnis a positive integer withn>2, then...Ch. 5 - Prob. 30SECh. 5 - Prob. 31SECh. 5 - (Requires calculus) Use mathematical induction and...Ch. 5 - Prob. 33SECh. 5 - Prob. 34SECh. 5 - Prob. 35SECh. 5 - mathematical induction to prove that ifx1,x2,...Ch. 5 - mathematical induction to prove that ifnpeople...Ch. 5 - pose that for every pair of cities in a country...Ch. 5 - Prob. 39SECh. 5 - Prob. 40SECh. 5 - Prob. 41SECh. 5 - Prob. 42SECh. 5 - Use mathematical induction to show that ifnis a...Ch. 5 - Prob. 44SECh. 5 - Prob. 45SECh. 5 - Prob. 46SECh. 5 - Prob. 47SECh. 5 - Prob. 48SECh. 5 - Prob. 49SECh. 5 - w thatnplanes divide three-dimensional...Ch. 5 - Prob. 51SECh. 5 - Prob. 52SECh. 5 - Prob. 53SECh. 5 - Prob. 54SECh. 5 - Prob. 55SECh. 5 - Prob. 56SECh. 5 - Prob. 57SECh. 5 - Prob. 58SECh. 5 - Prob. 59SECh. 5 - d all balanced string of parentheses with exactly...Ch. 5 - Prob. 61SECh. 5 - Prob. 62SECh. 5 - Prob. 63SECh. 5 - Prob. 64SECh. 5 - e a recursive algorithm for finding all balanced...Ch. 5 - Prob. 66SECh. 5 - Prob. 67SECh. 5 - Prob. 68SECh. 5 - Prob. 69SECh. 5 - Prob. 70SECh. 5 - Prob. 71SECh. 5 - Prob. 72SECh. 5 - Prob. 73SECh. 5 - Prob. 74SECh. 5 - Prob. 75SECh. 5 - Prob. 76SECh. 5 - Prob. 77SECh. 5 - Prob. 1CPCh. 5 - Prob. 2CPCh. 5 - Prob. 3CPCh. 5 - Prob. 4CPCh. 5 - Prob. 5CPCh. 5 - Prob. 6CPCh. 5 - Prob. 7CPCh. 5 - Prob. 8CPCh. 5 - Prob. 9CPCh. 5 - Prob. 10CPCh. 5 - en a nonnegative integern,find the nth Fibonacci...Ch. 5 - Prob. 12CPCh. 5 - Prob. 13CPCh. 5 - Prob. 14CPCh. 5 - en a list of integers, sort these integers using...Ch. 5 - Prob. 1CAECh. 5 - Prob. 2CAECh. 5 - Prob. 3CAECh. 5 - Prob. 4CAECh. 5 - Prob. 5CAECh. 5 - Prob. 6CAECh. 5 - Prob. 7CAECh. 5 - pare either number of operations or the needed to...Ch. 5 - cribe the origins of mathematical induction. Who...Ch. 5 - lain how to prove the Jordan curve theorem for...Ch. 5 - Prob. 3WPCh. 5 - cribe a variety of different app1icaons of the...Ch. 5 - Prob. 5WPCh. 5 - e die recursive definition of Knuth’s up-arrow...Ch. 5 - Prob. 7WPCh. 5 - lain how the ideas and concepts of program...
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