8th Edition

ISBN: 9781260521337

Author: ROSEN

Publisher: MCG

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A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 2.4, Problem 35E

Show that ∑ j = 1 n ( a j − a j − 1 ) = a n − a 0 , where a 0 , a 1 , ... , a n is a sequence of real numbers. This type of sum is called telescoping.

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Chapter 2.4, Problem 34E

Chapter 2.4, Problem 36E

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Please fill in the rest of the steps of the proof of Thm 2.5. Show how "Repeating this step with n-1,n-2,...,2 in place of n" gives us the desired result.

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Chapter 2 Solutions

DISCRETE MATHEMATICS+ITS APPL. (LL)-W/A

Ch. 2.1 - List the members of these sets. { xx is a real...Ch. 2.1 - Use set builder notation to give a description of...Ch. 2.1 - Which of the intervals (0, 5), (0, 5], [0, 5), [0,...Ch. 2.1 - For each of these intervals, list all its elements...Ch. 2.1 - For each of these pairs of sets, determine whether...Ch. 2.1 - For each of these pairs of sets, determine whether...Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - For each of the following sets, determine whether...Ch. 2.1 - Prob. 10E

Ch. 2.1 - Determine whether each of these statements is true...Ch. 2.1 - Determine whether these statements are true or...Ch. 2.1 - Determine whether each of these statements is true...Ch. 2.1 - Prob. 14E Ch. 2.1 - Use a Venn diagram to illustrate the set of all...Ch. 2.1 - Prob. 16E Ch. 2.1 - Use a Venn diagram to illustrate the re1ationships...Ch. 2.1 - Use a Venn diagram to illustrate the relationships...Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - What is the cardinality of each of these sets? {a}...Ch. 2.1 - What is the cardinality of each of these sets? {}...Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - How many elements does each of these sets have...Ch. 2.1 - Determine whether each of these sets is the power...Ch. 2.1 - Prove that P(A)P(B) if and only if AB .Ch. 2.1 - Show that if AC and BD , then ABCD Ch. 2.1 - Let A={a,b,c,d} and B={y,z} . Find AB . BA .Ch. 2.1 - Prob. 30E Ch. 2.1 - That is the Cartesian product ABC , where A is the...Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Let A={a,b,c} , B={x,y} , and C={0,l} . Find ABC ....Ch. 2.1 - Find A2 if A={0,1,3} A={1,2,a,b}Ch. 2.1 - Find A3 if A={a} A={0,a}Ch. 2.1 - How many different elements does AB have if A has...Ch. 2.1 - How many different elements does ABC have if A has...Ch. 2.1 - How many different elements does An have when A...Ch. 2.1 - Show that ABBA , when A and B are nonempty, unless...Ch. 2.1 - Explain why ABC and (AB)C are not the same.Ch. 2.1 - Explain why (AB)(CD) and A(BC)D are not the same.Ch. 2.1 - Prove or disprove that if A and B are sets, then...Ch. 2.1 - Prove or disprove that if A, B, and C are nonempty...Ch. 2.1 - Translate each of these quantifications into...Ch. 2.1 - Translate each of these quantifications into...Ch. 2.1 - Find the truth set of each of these predicates...Ch. 2.1 - Find the truth set of each of these predicates...Ch. 2.1 - Prob. 49E Ch. 2.1 - Prob. 50E Ch. 2.1 - Prob. 51E Ch. 2.2 - Prob. 1E Ch. 2.2 - Suppose that A is the set of sophomores at your...Ch. 2.2 - Let A={1,2,3,4,5} and B={0,3,6} . Find AB . AB ....Ch. 2.2 - Let A={a,b,c,d,e} and B={a,b,c,d,e,f,g,h} . Find...Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - TABLE 1 Set Identities. Identity Name AU=AA=A...Ch. 2.2 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - Prob. 16E Ch. 2.2 - Show that if A and B are sets in a universe U then...Ch. 2.2 - Prob. 18E Ch. 2.2 - Prob. 19E Ch. 2.2 - Prob. 20E Ch. 2.2 - Prob. 21E Ch. 2.2 - Prob. 22E Ch. 2.2 - Prob. 23E Ch. 2.2 - Prob. 24E Ch. 2.2 - Prob. 25E Ch. 2.2 - Let A, B, and C be sets. Show that (AB)C=(AC)(BC)...Ch. 2.2 - Prob. 27E Ch. 2.2 - Prob. 28E Ch. 2.2 - Prob. 29E Ch. 2.2 - Prob. 30E Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 32E Ch. 2.2 - Let A and B be subsets of a universal set U. Show...Ch. 2.2 - Let A, B, and C be sets. Use the identity AB=AB ,...Ch. 2.2 - Prob. 35E Ch. 2.2 - Prob. 36E Ch. 2.2 - Prove or disprove that for all sets A, B, and C,...Ch. 2.2 - Prob. 38E Ch. 2.2 - Prob. 39E Ch. 2.2 - Prob. 40E Ch. 2.2 - Prob. 41E Ch. 2.2 - Prob. 42E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 44E Ch. 2.2 - Prob. 45E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 47E Ch. 2.2 - Prob. 48E Ch. 2.2 - Prob. 49E Ch. 2.2 - Prob. 50E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 52E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 54E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 58E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 64E Ch. 2.2 - Prob. 65E Ch. 2.2 - Prob. 66E Ch. 2.2 - Prob. 67E Ch. 2.2 - Prob. 68E Ch. 2.2 - Prob. 69E Ch. 2.2 - The successor of the set A is the set A{A} ....Ch. 2.2 - The Jaccard similarity J(A,B) of the finite sets A...Ch. 2.2 - Prob. 72E Ch. 2.2 - Prob. 73E Ch. 2.2 - Prob. 74E Ch. 2.2 - Prob. 75E Ch. 2.3 - Why is f not a function from R to R if f(x)=1/x?...Ch. 2.3 - Determine whether f is a function from Z to R if...Ch. 2.3 - Prob. 3E Ch. 2.3 - Find the domain and range of these functions. Note...Ch. 2.3 - Find the domain and range of these functions. Note...Ch. 2.3 - Find the domain and range of these functions. the...Ch. 2.3 - Find the domain and range of these functions. the...Ch. 2.3 - Find these values. 1.1 1.1 0.1 0.1 2.99 2.99 12+12...Ch. 2.3 - Find these values. 34 78 34 78 3 1 12+32 1252 Ch. 2.3 - Prob. 10E Ch. 2.3 - Which functions in Exercise 10 are onto? Determine...Ch. 2.3 - Determine whether each of these functions from Z...Ch. 2.3 - Prob. 13E Ch. 2.3 - Determine whether f:ZZZ is onto if f(m,n)=2mn ....Ch. 2.3 - Determine whether the function f:ZZZ is onto if...Ch. 2.3 - Consider these functions from the set of students...Ch. 2.3 - Consider these functions from the set of teachers...Ch. 2.3 - Specify a codomain for each of the functions in...Ch. 2.3 - Specify a codomain for each of the functions in...Ch. 2.3 - Prob. 20E Ch. 2.3 - Give an explicit formula for a function from the...Ch. 2.3 - Determine whether each of these functions is a...Ch. 2.3 - Determine whether each of these functions is a...Ch. 2.3 - Let f:RR and let f(x)0 for all xR . Show that f(x)...Ch. 2.3 - Let f:RR and 1et f(x)0 for all xR . Show that f(x)...Ch. 2.3 - Prove that a strictly increasing function from R...Ch. 2.3 - Prob. 27E Ch. 2.3 - Show that the function f(x)=ex from the set of...Ch. 2.3 - Prob. 29E Ch. 2.3 - Let S={1,0,2,4,7} . Find f(S) if f(x)=1 ....Ch. 2.3 - Let f(x)=x2/3 . Find f(S) if S={2,1,0,1,2,3}...Ch. 2.3 - Let f(x)=2x where the domain is the set of real...Ch. 2.3 - Prob. 33E Ch. 2.3 - Suppose that g is a function from A to B and f is...Ch. 2.3 - Prob. 35E Ch. 2.3 - If f and fog are one-to-one, does it follow that g...Ch. 2.3 - Prob. 37E Ch. 2.3 - Find fog and gof where f(x)=x2 and g(x)=x+2 , are...Ch. 2.3 - Prob. 39E Ch. 2.3 - Let f(x)ax+b and g(x)=cx+d , where a, b, c, and d...Ch. 2.3 - Show that the function f(x)ax+b from R to R, where...Ch. 2.3 - Prob. 42E Ch. 2.3 - Prob. 43E Ch. 2.3 - Let f be the function from R to R defined by...Ch. 2.3 - Let g(x)=|x| . Find g1({0}) . g1({1,0,1}) ....Ch. 2.3 - Prob. 46E Ch. 2.3 - Prob. 47E Ch. 2.3 - Show x+12 is the closest integer to the number x...Ch. 2.3 - Prob. 49E Ch. 2.3 - Show that if x is a real number, then xx=1 if x is...Ch. 2.3 - Prob. 51E Ch. 2.3 - Prob. 52E Ch. 2.3 - Prob. 53E Ch. 2.3 - Show that if x is a real number and n is an...Ch. 2.3 - Prob. 55E Ch. 2.3 - Prove that if x is a real number, then x=x and x=x...Ch. 2.3 - Prob. 57E Ch. 2.3 - Prob. 58E Ch. 2.3 - Prob. 59E Ch. 2.3 - How many bytes are required to encode n bits of...Ch. 2.3 - How many bytes are required to encode n bits of...Ch. 2.3 - How many ATM cells (described in Example 30) can...Ch. 2.3 - Data are transmitted over a particular Ethernet...Ch. 2.3 - Draw the graph of the function f(n)=1n2 from Z to...Ch. 2.3 - Draw the graph of the function f(x)=2x from R to...Ch. 2.3 - Draw the graph of the function f(x)=x/2 from R to...Ch. 2.3 - Prob. 67E Ch. 2.3 - Draw the graph of the function f(x)=x+x/2 from R...Ch. 2.3 - Draw graphs of each of these functions. f(x)=x+12...Ch. 2.3 - Prob. 70E Ch. 2.3 - Find the inverse function of f(x)=x3+1 .Ch. 2.3 - Suppose that f is an invertible function from Y to...Ch. 2.3 - Let S be a subset of a universal set U. The...Ch. 2.3 - Suppose that f is a function from A to B, where A...Ch. 2.3 - Prove or disprove each of these statements about...Ch. 2.3 - Prove or disprove each of these statements about...Ch. 2.3 - Prove that if x is a positive real number, then...Ch. 2.3 - Let x be a real number. Show that 3x=x+x+13+x+23 .Ch. 2.3 - For each of these partial functions, determine its...Ch. 2.3 - Prob. 80E Ch. 2.3 - Prob. 81E Ch. 2.3 - Show that a set S is infinite if and only if there...Ch. 2.4 - Find these terms of the sequence {an} , where...Ch. 2.4 - What is the term a8 of the sequence {an} if an ,...Ch. 2.4 - What are the terms a0,a1,a2 , and a3 of the...Ch. 2.4 - What are the terms a0,a1,a2 , and a3 of the...Ch. 2.4 - List the first 10 terms of each of these...Ch. 2.4 - List the first lo terms of each of these...Ch. 2.4 - Find at least three different sequences beginning...Ch. 2.4 - Find at least three different sequences beginning...Ch. 2.4 - Find the first five terms of the sequence defined...Ch. 2.4 - Find the first six terms of the sequence defined...Ch. 2.4 - Let an=2n+53n for n=0,1,2,,... Find a0,a1,a2,a3 ,...Ch. 2.4 - Show that the sequence {an} is a solution of the...Ch. 2.4 - Is the sequence {an} a solution of the recurrence...Ch. 2.4 - For each of these sequences find a recurrence...Ch. 2.4 - Show that the sequence {an} is a solution of the...Ch. 2.4 - Find the solution to each of these recurrence...Ch. 2.4 - Find the solution to each of these recurrence...Ch. 2.4 - A person deposits $1000 in an account that yields...Ch. 2.4 - Suppose that the number of bacteria in a colony...Ch. 2.4 - Assume that the population of the world in 2017...Ch. 2.4 - A factory makes custom sports cars at an...Ch. 2.4 - An employee joined a company in 2017 with a...Ch. 2.4 - Find a recurrence relation for the balance B(k)...Ch. 2.4 - Find a recurrence relation for the balance B(k)...Ch. 2.4 - For each of these lists of integers, provide a...Ch. 2.4 - For each of these lists of integers, provide a...Ch. 2.4 - *27. Show that if an denotes the nth positive...Ch. 2.4 - Let an , be the nth term of the sequence 1, 2, 2,...Ch. 2.4 - What are the values of these sums? k=15(k+1)...Ch. 2.4 - What are the values of these sums, where...Ch. 2.4 - What is the value of each of these sums of terms...Ch. 2.4 - Find the value of each of these sums. j=08(1+ ( 1...Ch. 2.4 - Compute each of these double sums. i=12j=13( i+j)...Ch. 2.4 - Compute each of these double sums. i=13j=12( i+j)...Ch. 2.4 - Show that j=1n(aja j1)=ana0 , where a0,a1,...,an...Ch. 2.4 - Use the identity 1/(k(k+1))=1/k1/(k+1) and...Ch. 2.4 - Sum both sides of the identity k2(k21)2=2k1 from...Ch. 2.4 - Use the technique given in Exercise 35, together...Ch. 2.4 - Find k=100200k . (Use Table 2.) TABLE 2 Some...Ch. 2.4 - Prob. 40E Ch. 2.4 - Find k=1020k2(k3) . (Use Table 2.) TABLE 2 Some...Ch. 2.4 - Find . k=1020(k1)(2k2+1) (Use Table 2.) TABLE 2...Ch. 2.4 - Find a formula for k=0mk , when m is a positive...Ch. 2.4 - Find a formula for k=0mk3 , when m is a positive...Ch. 2.4 - There is also a special notation for products. The...Ch. 2.4 - Express n! using product notation.Ch. 2.4 - Find j=04j! .Ch. 2.4 - Find j=04j! .Ch. 2.5 - Prob. 1E Ch. 2.5 - Determine whether each of these sets is finite,...Ch. 2.5 - Determine whether each of these sets is countable...Ch. 2.5 - Determine whether each of these sets is countable...Ch. 2.5 - Show that a finite group of guests arriving at...Ch. 2.5 - Suppose that Hilbert’s Grand Hotel is fully...Ch. 2.5 - Suppose that Hilbert’s Grand Hotel is fully...Ch. 2.5 - Show that a countably infinite number of guests...Ch. 2.5 - Suppose that a countably infinite number of buses,...Ch. 2.5 - Give an example of two uncountable sets A and B...Ch. 2.5 - Give an example of two uncountable sets A and B...Ch. 2.5 - Prob. 12E Ch. 2.5 - Prob. 13E Ch. 2.5 - Prob. 14E Ch. 2.5 - Prob. 15E Ch. 2.5 - Show that a subset of a countable set is also...Ch. 2.5 - Prob. 17E Ch. 2.5 - Prob. 18E Ch. 2.5 - Prob. 19E Ch. 2.5 - Show that if |A|=|B| and |B|=|C| , then |A|=|C| .Ch. 2.5 - Prob. 21E Ch. 2.5 - Suppose that A is a countable set. Show that the...Ch. 2.5 - Prob. 23E Ch. 2.5 - Prob. 24E Ch. 2.5 - Prob. 25E Ch. 2.5 - Prob. 26E Ch. 2.5 - Show that the union of a countable number of...Ch. 2.5 - Show that the set Z+Z+ is countable Ch. 2.5 - Prob. 29E Ch. 2.5 - Show that the set of real numbers that are...Ch. 2.5 - Show that Z+Z+ t is countable by showing that the...Ch. 2.5 - Show that when you substitute (3n+1)2 for each...Ch. 2.5 - Prob. 33E Ch. 2.5 - Show that (0, 1) and R have the same cardinality...Ch. 2.5 - Prob. 35E Ch. 2.5 - Prob. 36E Ch. 2.5 - Show that the set of all computer programs in a...Ch. 2.5 - Prob. 38E Ch. 2.5 - Prob. 39E Ch. 2.5 - Show that if S is a set, then there does not exist...Ch. 2.5 - In this exercise, we prove the Schröder-Bernstein...Ch. 2.6 - Let A=[111320461137] . What size is A? What is the...Ch. 2.6 - Find A + B, where A=[104122022],B=[135223230]...Ch. 2.6 - Find AB if A=[2132],B=[0413] A=[110123],B=[321102]...Ch. 2.6 - Find the product AB, where...Ch. 2.6 - Find a matrix A such that [2314]A=[3012] . [Hint:...Ch. 2.6 - Find a matric A such that [132211403]A=[713103137]Ch. 2.6 - Prob. 7E Ch. 2.6 - Prob. 8E Ch. 2.6 - Prob. 9E Ch. 2.6 - Prob. 10E Ch. 2.6 - Prob. 11E Ch. 2.6 - In this exercise we show that matrix...Ch. 2.6 - Prob. 13E Ch. 2.6 - The nn matrix A=[aij] is called a diagonal matrix...Ch. 2.6 - Let A=[1101] . Find a formula for An , whenever n...Ch. 2.6 - Show that (At)t=A .Ch. 2.6 - Prob. 17E Ch. 2.6 - Show that [231121113] Is the inverse of...Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Prob. 23E Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Let A=[1101] and B=[0110] Find AB . AB . AB .Ch. 2.6 - Prob. 27E Ch. 2.6 - Find the Boolean product of A and B, where...Ch. 2.6 - Prob. 29E Ch. 2.6 - Let A be a zeroone matrix. Show that AA=A . AA=A .Ch. 2.6 - Prob. 31E Ch. 2.6 - Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - In this exercise we will show that the Boolean...Ch. 2 - Prob. 1RQ Ch. 2 - What is the empty set? Show that the empty set is...Ch. 2 - Define |S|, the cardinality of the set S. Give a...Ch. 2 - Define the power set of a set S. When is the empty...Ch. 2 - Define the union. intersection, difference, and...Ch. 2 - Prob. 6RQ Ch. 2 - Explain the relationship between logical...Ch. 2 - Prob. 8RQ Ch. 2 - Prob. 9RQ Ch. 2 - Define the inverse of a function. When does a...Ch. 2 - Prob. 11RQ Ch. 2 - Conjecture a formula for the terms of the sequence...Ch. 2 - Prob. 13RQ Ch. 2 - What is the sum of the terms of the geometric...Ch. 2 - Show that the set of odd integers is countable.Ch. 2 - Prob. 16RQ Ch. 2 - Prob. 17RQ Ch. 2 - Prob. 18RQ Ch. 2 - Prob. 1SE Ch. 2 - Prob. 2SE Ch. 2 - Prob. 3SE Ch. 2 - Prob. 4SE Ch. 2 - Prob. 5SE Ch. 2 - Prob. 6SE Ch. 2 - Prob. 7SE Ch. 2 - Prob. 8SE Ch. 2 - Prob. 9SE Ch. 2 - Prob. 10SE Ch. 2 - Prob. 11SE Ch. 2 - Prob. 12SE Ch. 2 - Prob. 13SE Ch. 2 - Prob. 14SE Ch. 2 - Prob. 15SE Ch. 2 - *16. Suppose that f is a function from the set A...Ch. 2 - Prob. 17SE Ch. 2 - Prob. 18SE Ch. 2 - Prob. 19SE Ch. 2 - Prob. 20SE Ch. 2 - Prob. 21SE Ch. 2 - Prob. 22SE Ch. 2 - Prob. 23SE Ch. 2 - Prove that if x is a real number, then x/2/2=x/4 .Ch. 2 - Prob. 25SE Ch. 2 - Prob. 26SE Ch. 2 - Prove that if m is a positive integer and x is a...Ch. 2 - We define the Ulam numbers by setting u1=1 and...Ch. 2 - Prob. 29SE Ch. 2 - Determine a rule for generating the terms of the...Ch. 2 - Prob. 31SE Ch. 2 - Prob. 32SE Ch. 2 - Prob. 33SE Ch. 2 - Show that the set of all finite subsets of the set...Ch. 2 - Prob. 35SE Ch. 2 - Prob. 36SE Ch. 2 - Prob. 37SE Ch. 2 - Prob. 38SE Ch. 2 - Prob. 39SE Ch. 2 - Prob. 40SE Ch. 2 - Prob. 41SE Ch. 2 - Prob. 1CP Ch. 2 - Prob. 2CP Ch. 2 - Prob. 3CP Ch. 2 - Prob. 4CP Ch. 2 - Prob. 5CP Ch. 2 - Prob. 6CP Ch. 2 - Prob. 7CP Ch. 2 - Prob. 8CP Ch. 2 - Prob. 9CP Ch. 2 - Prob. 10CP Ch. 2 - Prob. 11CP Ch. 2 - Prob. 12CP Ch. 2 - Prob. 1CAE Ch. 2 - Prob. 2CAE Ch. 2 - Use a computational program or programs you have...Ch. 2 - Prob. 4CAE Ch. 2 - Prob. 5CAE Ch. 2 - Use a computational program or programs you have...Ch. 2 - Prob. 1WP Ch. 2 - Research where the concept of a function first...Ch. 2 - Explain the different ways in which the...Ch. 2 - Define the recently invented EKG sequence and...Ch. 2 - Prob. 5WP Ch. 2 - Expand the discussion of the continuum hypothesis...

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Show that ∑ j = 1 n ( a j − a j − 1 ) = a n − a 0 , where a 0 , a 1 , ... , a n is a sequence of real numbers. This type of sum is called telescoping .

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