Discrete Mathematics And Its Applications
8th Edition
ISBN: 9781260091991
Author: NA
Publisher: Mc graw hill
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Chapter 8, Problem 7WP
To determine
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The level curves of the given function at the given value at
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Find all solutions of the polynomial congruence
x²+4x+1 = 0 (mod 143).
(The solutions of the congruence x² + 4x+1=0 (mod 11) are x = 3,4 (mod 11) and the
solutions of the congruence x² +4x+1 = 0 (mod 13) are x = 2,7 (mod 13).)
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Determine whether each function is an injection and determine whether each is a surjection.The notation Z_(n) refers to the set {0,1,2,...,n-1}. For example, Z_(4)={0,1,2,3}. f: Z_(6) -> Z_(6) defined by f(x)=x^(2)+4(mod6). g: Z_(5) -> Z_(5) defined by g(x)=x^(2)-11(mod5). h: Z*Z -> Z defined by h(x,y)=x+2y. j: R-{3} -> R defined by j(x)=(4x)/(x-3).
Chapter 8 Solutions
Discrete Mathematics And Its Applications
Ch. 8.1 - Use mathematical induction to verify the formula...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - A vending machine dispensing books of stamps...Ch. 8.1 - A country uses as currency coins with values of 1...Ch. 8.1 - How many was are there to pay a bill of 17 pesos...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...
Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - Messages are transmitted over a communications...Ch. 8.1 - A bus driver pays all tolls, using only nickels...Ch. 8.1 - a) Find the recurrence relation satisfied by Rn,...Ch. 8.1 - a) Find the recurrence relation satisfied by Rn,...Ch. 8.1 - a) Find the recurrence relation satisfied by Sn,...Ch. 8.1 - Find a recurrence relation for the number of bit...Ch. 8.1 - How many bit sequences of length seven contain an...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - Show that the Fibonacci numbers satisfy the...Ch. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - a) Use the recurrence relation developed in...Ch. 8.1 - In the Tower of Hanoi puzzle, suppose our goal is...Ch. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Prob. 35ECh. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Prob. 37ECh. 8.1 - Prob. 38ECh. 8.1 - Show that the Reve’s puzzle with four disks can be...Ch. 8.1 - Prob. 40ECh. 8.1 - Show that if R(n) is the number of moves used by...Ch. 8.1 - Prob. 42ECh. 8.1 - Prob. 43ECh. 8.1 - Prob. 44ECh. 8.1 - Prob. 45ECh. 8.1 - Prob. 46ECh. 8.1 - Prob. 47ECh. 8.1 - Prob. 48ECh. 8.1 - Show that an2=an2an+2an .Ch. 8.1 - Prob. 50ECh. 8.1 - Prob. 51ECh. 8.1 - Prob. 52ECh. 8.1 - Construct the algorithm described in the text...Ch. 8.1 - Use Algorithm 1 to determine the maximum number of...Ch. 8.1 - For each part of Exercise 54, use your algorithm...Ch. 8.1 - In this exercise we will develop a dynamic...Ch. 8.1 - Dynamic programming can be used to develop an...Ch. 8.2 - Determine which of these are linear homogeneous...Ch. 8.2 - Determine which of these are linear homogeneous...Ch. 8.2 - Solve these recurrence relations together with the...Ch. 8.2 - Solve these recurrence relations together with the...Ch. 8.2 - Prob. 5ECh. 8.2 - Prob. 6ECh. 8.2 - Prob. 7ECh. 8.2 - A model for the number of lobsters caught per year...Ch. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 8.2 - The Lucas numbers satisfy the recurrence relation...Ch. 8.2 - Find the solution to an=2an1+an2+2an3 for n = 3,4,...Ch. 8.2 - Find the solution to an=7an2+6an3 with a0=9,a1=10...Ch. 8.2 - Find the solution to an=5an24an4 with...Ch. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prove this identity relating the Fibonacci numbers...Ch. 8.2 - Solve the recurrence relation an=6an112an2+8an3...Ch. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - What is the general form of the solutions of a...Ch. 8.2 - Consider the nonhomogeneous linear recurrence...Ch. 8.2 - Consider the nonhomogeneous linear recurrence...Ch. 8.2 - a) Determine values of the constants A and B such...Ch. 8.2 - What is the general form of the particular...Ch. 8.2 - What is the general form of the particular...Ch. 8.2 - a) Find all solutions of the recurrence relation...Ch. 8.2 - Prob. 29ECh. 8.2 - Prob. 30ECh. 8.2 - Find all solutions of the recurrence relation...Ch. 8.2 - Find the solution of the recurrence relation...Ch. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Find the solution of the recurrence relation...Ch. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Solve the simultaneous recurrence relations...Ch. 8.2 - Prob. 41ECh. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - Prob. 45ECh. 8.2 - Suppose that there are two goats on an island...Ch. 8.2 - Prob. 47ECh. 8.2 - Prob. 48ECh. 8.2 - Use Exercise 48 to solve the recurrence relation...Ch. 8.2 - It can be shown that Cn, the average number of...Ch. 8.2 - Prob. 51ECh. 8.2 - Prob. 52ECh. 8.2 - Prob. 53ECh. 8.3 - How many comparisons are needed for a binary...Ch. 8.3 - Prob. 2ECh. 8.3 - Multiply (1110)2 and (1010)2 using the fast...Ch. 8.3 - Express the fast multiplication algorithm in...Ch. 8.3 - Determine a value for the constant C in Example...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Suppose that f(n)=2f(n/2)+3 when is an even...Ch. 8.3 - Prob. 9ECh. 8.3 - Find f(n) when n=2k , where f satisfies the...Ch. 8.3 - Give a big-O estimate for the function f in...Ch. 8.3 - Find f(n) when n=3k , where f satisfies the...Ch. 8.3 - Give a big-O estimate for the function f in...Ch. 8.3 - Suppose that there are n=2k terms in an...Ch. 8.3 - How many rounds are in the elimination tournament...Ch. 8.3 - Prob. 16ECh. 8.3 - Suppose that the votes of n people for different...Ch. 8.3 - Suppose that each person in a group of n people...Ch. 8.3 - a) Set up a divide-and-conquer recurrence relation...Ch. 8.3 - a) Set up a divide-and-conquer recurrence relation...Ch. 8.3 - Suppose that the function f satisfies the...Ch. 8.3 - Suppose that the function f satisfies the...Ch. 8.3 - This exercise deals with the problem of finding...Ch. 8.3 - Apply the algorithm described in Example 12 for...Ch. 8.3 - Apply the algorithm described in Example 12 for...Ch. 8.3 - Use pseudocode to describe the recursive algorithm...Ch. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - Prob. 30ECh. 8.3 - Prob. 31ECh. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.4 - Find the generating function for the finite...Ch. 8.4 - Find the generating function for the finite...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - Prob. 5ECh. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - Find the coefficient of x10in the power series of...Ch. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to find the number of...Ch. 8.4 - In how many ways can 25 identical donuts be...Ch. 8.4 - Use generating functions to find the number of...Ch. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Prob. 21ECh. 8.4 - Prob. 22ECh. 8.4 - Prob. 23ECh. 8.4 - Prob. 24ECh. 8.4 - Explain how generating functions can be used to...Ch. 8.4 - Explain how generating functions can be used to...Ch. 8.4 - Prob. 27ECh. 8.4 - Prob. 28ECh. 8.4 - Use generating functions (and a computer algebra...Ch. 8.4 - Use generating functions (and a computer algebra...Ch. 8.4 - Prob. 31ECh. 8.4 - If G(x) is the generating function for the...Ch. 8.4 - Prob. 33ECh. 8.4 - Prob. 34ECh. 8.4 - Prob. 35ECh. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Prob. 37ECh. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Prob. 40ECh. 8.4 - Prob. 41ECh. 8.4 - Prob. 42ECh. 8.4 - (Calculus required) Let {Cn}be the sequence of...Ch. 8.4 - Use generating functions to prove Pascal’s...Ch. 8.4 - Use generating functions to prove Vandermonde’s...Ch. 8.4 - Prob. 46ECh. 8.4 - Prob. 47ECh. 8.4 - Prob. 48ECh. 8.4 - Find the sequence with each of these functions as...Ch. 8.4 - Find the sequence with each of these functions as...Ch. 8.4 - A coding system encodes messages using strings of...Ch. 8.4 - A coding system encodes messages using strings of...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Prob. 55ECh. 8.4 - Prob. 56ECh. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Suppose that X is a random variable on a sample...Ch. 8.4 - Prob. 60ECh. 8.4 - Prob. 61ECh. 8.4 - Show that if X and Y are independent random...Ch. 8.5 - How many elements are in A1A2 if there are 12...Ch. 8.5 - There are 345 students at a college who have taken...Ch. 8.5 - A survey of households in the United States...Ch. 8.5 - A marketing report concerning personal computers...Ch. 8.5 - Find the number of elements A1A2A3 if there are...Ch. 8.5 - Prob. 6ECh. 8.5 - There are 2504 computer science students at a...Ch. 8.5 - In a survey of 270 college students, it is found...Ch. 8.5 - How many students are enrolled in a course either...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - How many swings of length eight do not contain six...Ch. 8.5 - How many permutations of the 26 letters of the...Ch. 8.5 - How many permutations of the 10 digits either...Ch. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - How many terms are there in the formula for the...Ch. 8.5 - Prob. 21ECh. 8.5 - Prob. 22ECh. 8.5 - Prob. 23ECh. 8.5 - Prob. 24ECh. 8.5 - Let E1, E2 ,and E3 be three events from a sample...Ch. 8.5 - Prob. 26ECh. 8.5 - Find the probability that when four numbers from 1...Ch. 8.5 - Prob. 28ECh. 8.5 - Prob. 29ECh. 8.5 - Prob. 30ECh. 8.5 - Prob. 31ECh. 8.6 - Suppose that in a bushel of 100 apples there are...Ch. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - Prob. 4ECh. 8.6 - Find the number of primes less than 200 using the...Ch. 8.6 - Prob. 6ECh. 8.6 - How many positive integers less than 10,000 are...Ch. 8.6 - Prob. 8ECh. 8.6 - How many ways are there to distribute six...Ch. 8.6 - In how many ways can eight distinct balls be...Ch. 8.6 - In how many ways can seven different jobs be...Ch. 8.6 - List all the derangements of {1, 2,3, 4}.Ch. 8.6 - Prob. 13ECh. 8.6 - Prob. 14ECh. 8.6 - A machine that inserts letters into envelopes goes...Ch. 8.6 - A group of n students is assigned seats for each...Ch. 8.6 - Prob. 17ECh. 8.6 - Prob. 18ECh. 8.6 - Prob. 19ECh. 8.6 - Prob. 20ECh. 8.6 - For which positive integers n is Dn, the number of...Ch. 8.6 - Prob. 22ECh. 8.6 - Prob. 23ECh. 8.6 - Prob. 24ECh. 8.6 - Prob. 25ECh. 8.6 - How many derangements of {1, 2, 3, 4, 5, 6} end...Ch. 8.6 - Prove Theorem 1.Ch. 8 - a) What is a recurrence re1aon? b) Find a...Ch. 8 - Explain how the Fibonacci numbers are used to...Ch. 8 - a) Find a recurrence relation for the number of...Ch. 8 - Prob. 6RQCh. 8 - a) Explain how to solve linear homogeneous...Ch. 8 - Prob. 8RQCh. 8 - Prob. 9RQCh. 8 - a) Give a formula for the number of elements in...Ch. 8 - a) Give a formula for the number of elements in...Ch. 8 - Prob. 12RQCh. 8 - Explain how the principle of inclusion-exclusion...Ch. 8 - Prob. 14RQCh. 8 - Prob. 15RQCh. 8 - a) Define a derangement. b) Why is counting the...Ch. 8 - A group of 10 people begin a chain letter, with...Ch. 8 - A nuclear reactor has created 18 grams of a...Ch. 8 - Every hour the U.S. government prints 10,000 more...Ch. 8 - Suppose that every hour there are two new bacteria...Ch. 8 - Messages are sent over a communications channel...Ch. 8 - Prob. 6SECh. 8 - How many ways are there to form these postages...Ch. 8 - Prob. 8SECh. 8 - Solve the recurrence relation an=a2n1/bn2 if a0=1...Ch. 8 - Prob. 10SECh. 8 - Find the solution of the recurrence relation...Ch. 8 - Prob. 12SECh. 8 - Prob. 13SECh. 8 - Prob. 14SECh. 8 - Prob. 15SECh. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - Find the solution to the recurrence relation...Ch. 8 - Find the solution to the recurrence relation...Ch. 8 - Give a big-O estimate for the size of f in...Ch. 8 - Find a recurrence relation that describes the...Ch. 8 - Prob. 23SECh. 8 - Prob. 24SECh. 8 - Prob. 25SECh. 8 - Find an where a) an=3 . b) an=4n+7 . c) an=n2+n+1Ch. 8 - Prob. 27SECh. 8 - Prob. 28SECh. 8 - Prob. 29SECh. 8 - Prob. 30SECh. 8 - Prob. 31SECh. 8 - Prob. 32SECh. 8 - Prob. 33SECh. 8 - Prob. 34SECh. 8 - Prob. 35SECh. 8 - How many terms are needed when the...Ch. 8 - How many solutions in positive integers are there...Ch. 8 - How many positive integers less than 1,000,000 are...Ch. 8 - How many positive integers less than 200 are a)...Ch. 8 - How many ways are there to assign six different...Ch. 8 - What is the probability that exactly one person is...Ch. 8 - How many bit stings of length six do not contain...Ch. 8 - What is the probability that a bit string of...Ch. 8 - Prob. 1CPCh. 8 - Prob. 2CPCh. 8 - Prob. 3CPCh. 8 - Prob. 4CPCh. 8 - Prob. 5CPCh. 8 - Prob. 6CPCh. 8 - Prob. 7CPCh. 8 - Prob. 8CPCh. 8 - Prob. 9CPCh. 8 - Prob. 10CPCh. 8 - Prob. 11CPCh. 8 - Prob. 12CPCh. 8 - Given a positive integer n, list all the...Ch. 8 - Prob. 1CAECh. 8 - Prob. 2CAECh. 8 - Find as many prime Fibonacci numbers as you can....Ch. 8 - Prob. 4CAECh. 8 - Prob. 5CAECh. 8 - Prob. 6CAECh. 8 - Prob. 7CAECh. 8 - Prob. 8CAECh. 8 - Prob. 9CAECh. 8 - List all the derangements of 1,2,3,4,5,6,7,8 .Ch. 8 - Prob. 11CAECh. 8 - Find the original source where Fibonacci presented...Ch. 8 - Explain how the Fibonacci numbers arise in a...Ch. 8 - Prob. 3WPCh. 8 - Discuss as mans different problems as possible...Ch. 8 - Prob. 5WPCh. 8 - Prob. 6WPCh. 8 - Prob. 7WPCh. 8 - Prob. 8WPCh. 8 - Describe the solution of Ulam’s problem (see...Ch. 8 - Discuss variations of Ulam’s problem (see Exercise...Ch. 8 - Prob. 11WPCh. 8 - Describe how sieve methods are used in number...Ch. 8 - Look up the rules of the old French card game of...Ch. 8 - Prob. 14WPCh. 8 - Describe the Polyá theory of counting and the kind...Ch. 8 - The problème des ménages (the problem of the...Ch. 8 - Explain how rook polynomials can be used to solve...
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- Determine whether each function is an injection and determine whether each is a surjection.arrow_forwardLet A = {a, b, c, d}, B = {a,b,c}, and C = {s, t, u,v}. Draw an arrow diagram of a function for each of the following descriptions. If no such function exists, briefly explain why. (a) A function f : AC whose range is the set C. (b) A function g: BC whose range is the set C. (c) A function g: BC that is injective. (d) A function j : A → C that is not bijective.arrow_forwardLet f:R->R be defined by f(x)=x^(3)+5.(a) Determine if f is injective. why?(b) Determine if f is surjective. why?(c) Based upon (a) and (b), is f bijective? why?arrow_forward
- 25-30. Normal and tangential components For the vector field F and curve C, complete the following: a. Determine the points (if any) along the curve C at which the vector field F is tangent to C. b. Determine the points (if any) along the curve C at which the vector field F is normal to C. c. Sketch C and a few representative vectors of F on C. 25. F = (2½³, 0); c = {(x, y); y − x² = 1} 26. F = x (23 - 212) ; C = {(x, y); y = x² = 1}) , 2 27. F(x, y); C = {(x, y): x² + y² = 4} 28. F = (y, x); C = {(x, y): x² + y² = 1} 29. F = (x, y); C = 30. F = (y, x); C = {(x, y): x = 1} {(x, y): x² + y² = 1}arrow_forward٣/١ B msl kd 180 Ka, Sin (1) I sin () sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed, 120 x 50 G 5005 1000 s = 1000-950 Copper bosses 5kW Rotor input 5 0.05 : loo kw 6) 1 /0001 ined sove in peaper I need a detailed solution on paper please وه اذا ميريد شرح الكتب فقط ١٥٠ DC 7) rotor a ' (y+xlny + xe*)dx + (xsiny + xlnx + dy = 0. Q1// Find the solution of: ( 357arrow_forward۳/۱ R₂ = X2 2) slots per pole per phase 3/31 B. 180 msl Kas Sin (I) 1sin() sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30): 0.866 4) Rotating 5) Synchronous speeds 120×50 looo G 1000-950 1000 Copper losses 5kw Rotor input 5 loo kw 0.05 6) 1 اذا ميريد شرح الكتب فقط look 7) rotor DC ined sove in peaper I need a detailed solution on paper please 0 64 Find the general solution of the following equations: QI//y(4)-16y= 0. Find the general solution of the following equations: Q2ll yll-4y/ +13y=esinx.arrow_forward
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