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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

Prove parts (a) and (c) of Theorem 5.10. If f: A → B and B1,B2 ⊂ B, then (a) f-1{B1 ∩ B2) = f-l(B1) ∩ (b) f-1 (B1, ∪ B2) = f-1'(B1,) ∪ f-1'(B2); and (c) f-1 (B1)- f-1 (B1).
(a) For A = (-2, 7] c R define the functions /, g: A -* R byf(x) = 2x - 4 and g(x) = 2x2 - 8/x + 2verify that f = g.(b) Is the result in part (a) affected if we change A to [-7, 2)?
Let f: Z †’ N be defined by(a) Prove that f is one-to-one and onto.(b) Determine f-l.
Let f, g, h, k: N→N where f(n) = 3n, g(n) = [n/3], h(n) = [(n + 1)/3], and k(n) = (n + 2)/3], for each n ∈ N. (a) For each n ∈ N what are (g o f)(h o f) and {k o f){n?)l (b) Do the results in part (a) contradict Theorem 5.7?
Let f, g: R → R, where g(x) = 1 - x + x2 and f(x) = ax + b. If (g o f)(x) = 9x2 - 9x + 3, determine a, b.
If °U is a given universe with (fixed) S, T ⊂ °U, define g: P(U) SP(U) by g(A) = T D (5 U A) for A c °ll. Prove that g2 = g.
Let f, g,h: Z ^ Z be defined by /(jc) = x - 1, gW = 3*, Determine (a) f o g, g o f, g o h, h o g, f o (g o A), (f o g) o h; (b) f2, f3, g2, g3, h2, h3, h500.
Let f : A → B, g. B → C. Prove that (a) if g o f: A → C is onto, then g is onto; and (b) if g o f: A → C is one-to-one, then f is one-to-one.
(a) Find the inverse of the function f: R → R+ defined by f(x) = e2x+5. (b) Show that f o f-l = 1R+ and f-1 o f = 1R.
Use the results of Table 5.11 to determine the best "big-Oh" form for each of the following functions f: Z+ → R. (a) f(n) = 3n + 7 (b) f(n) = 3 + sin(l/n) (c) f(n) = n3 - 5n2 + 25n - 165 (d) f(n) = 5n2 + 3n log2n (e) f(n) = n2 + (n - l)3 (f) f(n) = n(n + 1)(n+2)/(n + 3) (g) f(n) = 2 + 4 + 6
(a) Prove that f ∈ O(f) for all f: Z+ R.(b) Let f, g: Z+ → R. If f ∈ O (g) and g ∈ 0(f), prove that 0(f) = 0(g). That is, prove that for all h: Z+ R, if h is dominated by f, then h is dominated by g, and conversely.(c) If f, g: Z+ → R, prove that if 0(f) = 0(g), then f ∈ 0(g) and g ∈
The following is analogous to the "big-Oh" notation introduced in conjunction with Definition 5.23.For f, g: Z+ → R we say that / is of order at least g if there exist constants M ∈ R+ and f: ∈ Z+ such that |f(n)| > M\g(n) for all n ∈ Z+, where n > k. In this case we write f ∈ Ω
Let f, g: Z+ → R. Prove that f ∈ Ω (g) if and only if g ∈ 0(f).
a) Let f: Z+ †’ R where f(n) = ˆ‘ni=1 1. When n = 4, for example, we have f(n) = f(4) = 1 + 2 + 3 + 4 > 2 + 3 + 4>2 + 2 + 2 = 3- 2 = [(4+ l)/2]2 = 6 > (4/2)2 = (n/2)1. For n = 5, we find f(n) = /(5) = l+2 + 3 + 4 + 5>3 + 4 + 5 > 3 + 3 + 3 = 3ˆ™ 3 = [(5 + 1)(n/2)|3 = 9 >
For f, g: Z+ → R, we say that f is "big Theta of g," and write f ∈ ⊙(g), when there exist constants m1, m2 ∈ R+ and k ∈ Z+ such that m1|g(n)| < |f(n)| < m2|g(n)|, for all n ∈ Z+, where n > Prove that f ∈ ⊖(g) if and only if f e ^ (g) and f ∈ O(g).
Let f, g: Z+ → R. Prove that f ∈ ⊖(g) if and only if g ∈ ⊖(f).
Let f, g: Z+ R, where f(n) = n and g(n) = n + (1/n), for n ∈ Z+. Use Definition 5.23 to show that f e 0(g) and g ∈ O(f).
In each of the following, f g: Z+ → R. Use Definition 5.23 to show that g dominates f.a) f(n) = 100 log2n, g(n) = (1/2)nb) f(n) = 2n, g(n) = 22n - 1000c) f(n) - 3n2, g(n) = 2n + 2n
Let f, g: Z+ R be defined by f(n) = n + 100, g(n) = n2. Use Definition 5.23 to show that f ∈ O (g) but g ∉ O(f).
Let f, g: Z+ → R, where f(n) = n2 + and g(n) = (1/2) n3, for n ∈ Z+. Use Definition 5.23 to show that f ∈ O(g) but g ∉ O(f).
Let f, g: Z+ †’ R be defined as followsVerify that f ˆ‰ 0(g) and g ˆ‰ O(f).
Let f, g: Z+ †’ R where f(n) = n and g(n) = log2 n, for n ˆˆ Z+. Show that g ˆˆ O(f) but ˆ‰ 0(g).This requires the use of calculus
Let f, g, h: Z+ R where f ∈ O(g) and g ∈ 0 ().Prove that f ∈ 0(h)
If g: Z+ → R and c ∈ R, we define the function eg: Z+ → R by (c(g)(n) = c(g(n)), for each n ∈ Z+. Prove that if f, g: Z+ R with f ∈ O(g), then f ∈ O(cg) for all c ∈ R, c ≠ 0.
In each of the following pseudocode program segments, the integer variables i, j,n, and sum are declared earlier in the program. The value of n (a positive integer) is supplied by the user prior to execution of the segment. In each case we define the time-complexity function f(n) to be the number
When the linear search algorithm is applied to the array a1, a2, a3, . .. , an (of n distinct integers) for the integer key, suppose the probability that key has the value at is i /[n (n + 1)], for 1 < i < n. Under these circumstances, what is the average number of array elements examined?
(a) Write a computer program (or develop an algorithm) to determine the location of the first entry in an array a1, a2, a3, . .. , an of integers that repeats a previous entry in the array.(b) Determine the worst-case complexity for the implementation developed in part (a).
(a) Write a computer program (or develop an algorithm) to determine the location of the first entry at in an array a1, a2, a3, . . . , an of integers, where at (b) Determine the worst-case complexity for the implementation developed in part (a).
(a) Write a computer program (or develop an algorithm) to locate the first occurrence of the maximum value in an array a1, a2, a3, ... , an of integers. (Here n ∈ Z+ and the entries in the array need not be distinct.)b) Determine the worst-case complexity function for the implementation developed
(a) Write a computer program (or develop an algorithm) to determine the minimum and maximum values in an array a1, a2, a3, ... , an of integers. (Here n ∈ Z+ with n > 2, and the entries in the array need not be distinct.)b) Determine the worst-case complexity function for the implementation
We first note how the polynomial in the previous exercise can be written in the nested multiplication method:8 + x(-10 + x{l + x(-2 + x(3 + 12x)))).Using this representation, the following pseudocode procedure (implementing Horner's method) can be used to evaluate the given polynomial.Answer the
Let a1, a2, a3, . .. be the integer sequence defined recursively by(1) a1{ = 0; and(2) For n > 1, an = 1 + a[n/2].Prove that an = [log2n] for all n ∈ Z+.
Let a1, a2, a3, ... be the integer sequence defined recursively by1) a1 = 0; and2) For n > 1, an = 1 + a(n/2).Find an explicit formula for an and prove that your formula is correct.
Suppose the probability that the integer key is in the array a1, a2, a3, ..., an (of ft distinct integers) is 3/4 and that each array element has the same probability of containing this value. If the linear search algorithm of Example 5.70 is applied to this array and value of key, what is the
Let A, B ⊂ °U. Prove that(a) (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B); and(b) (A × B) ∪ (B ∪ A) ⊂ (A ∪ B) × (A ∪ B).
Let A1, A and B be sets with {1, 2, 3, 4, 5} = A1 ⊂ A, B = {s, t u, v, w, x}, and f: A1 → B. If f can be extended to A in 216 ways, what is |A|?
Let A = {1, 2, 3, 4, 5} and B = {t, u, v, w, x, y, z}. (a) If a function f: A → B is randomly generated, what is the probability that it is one-to-one? (b) Write a computer program (or develop an algorithm) to generate random functions f. A → B and have the program print out how many functions
Let S be a set of seven positive integers the maximum of which is at most 24. Prove that the sums of the elements in all the nonempty subsets of S cannot be distinct.
In a ten-day period Ms. Rosatone typed 84 letters to different clients. She typed 12 of these letters on the first day, seven on the second day, and three on the ninth day, and she finished the last eight on the tenth day. Show that for a period of three consecutive days Ms. Rosatone typed at least
If {x1, x2, .... x7} ⊂ Z+, show that for some i ≠ j, either x1 + xj or xt - xj is divisible by 10.
Let n ∈ Z+, n odd. If i1i2, . . . , in is a permutation of the integers 1, 2, ..., n, prove that (1 - i 1)(2 - i2) ∙ ∙ ∙ ∙ ∙ ∙ ∙ (n - in) is an even integer. (Which counting principle is at work here?)
With both of their parents working, Thomas, Stuart, and Craig must handle ten weekly chores among themselves, (a) In how many ways can they divide up the work so that everyone is responsible for at least one chore? (b) In how many ways can the chores be assigned if Thomas, as the eldest, must mow
Let n ∈ N, n > 2. Show that S(n, 2) = 2n-1 x - 1.
Mrs. Blasi has five sons (Michael, Rick, David, Kenneth, and Donald) who enjoy reading books about sports. With Christmas approaching, she visits a bookstore where she finds 12 different books on sports. a) In how many ways can she select nine of these books? b) Having made her purchase, in how
Determine whether each of the following statements is true or false. For each false statement give a counterexample. a) If f:A →B and (a, b), (a, c) ∈ f, then b = c. b) If f: A B is a one-to-one correspondence and A, B are finite, then A = B. c) If f: A → B is one-to-one, then f is
If n ∈ Z+ with n > 4, verify that 5(n, n - 2) = (n3) + 3(n4).
Let f: R → R where f(ab) = af(b) + bf(a), for all a, b ∈ R. (a) What is f(l)? (b) What is f(0)? (c) If n ∈ Z+, a ∈ R, prove that f(an) = nan~l f (a).
Let A, B c N with 1 < |A| < |B|. If there are 262,144 relations from A to B, determine all possibilities for |A| and |B|.
If °U1, °U2 are universal sets with A, B ⊂ °U1, and C, D ⊂. °U2, prove that (A ∩ 5) × (C ∩ D) = (A × C) ∩ (B × D).
Determine all real numbers x for which x2 - [x] = 1/2.
Let R ⊂ Z+ × Z+ be the relation given by the following recursive definition.1) (1, 1) ∈ R; and2) For all (a, b) ∈ R, the three ordered pairs (a + 1, b), (a + 1, b + 1), and (a + 1, b + 2) are also in RProve that 2a > b for all (a, b) ∈ R.
Let a, b denote fixed real numbers and suppose that R→ R is defined by f(x) = a (x +b) - b, x ∈ R. (a) Determine f2(x) and f3(x). (b) Conjecture a formula for fn(x), where n ∈ Z+. Now establish the validity of your conjecture.
For ∑ = {x, y, z}, let A, B ⊂ ∑* be given by A = {xy} and B = {λ, x}. Determine (a) AB; (b) BA; (c) B3; (d) B+; (e) A*.
Given an alphabet E, is there a language A ⊂ ∑* where A* = A?
For E = {0, 1} determine whether the string 00010 is in each of the following languages (taken from E*). (a) {0, 1}* (b) {000, 101}{10, 11} (c) {00}{0}{10} (d) {000}*{1}{0} (e) {00}*{10} (f) {0}*{1}*{0}*
For E = {0, 1} describe the strings in A* for each of the following languages A⊂E*.a) {01} b) {000}c) {0, 010} d) {1, 10}
For ∑ = {0, 1} determine all possible languages A, B ⊂ E* where AB = {01, 000, 0101, 0111, 01000, 010111}.
Given a nonempty language A⊂∑*, prove that if A2 = A, then λ ∈ A.
For a given alphabet E, let a ∈ ∑ - with a fixed. Define the functions pa, sa, r:∑ -> ∑* and the function d: E+ → E* as follows: (i) The prefix (by a) function: pa(x) = ax, x ∈ ∑*. (ii) The suffix (by a) function: sa(x) = xa, x ∈ ∑*. (iii) The reversal function: r(λ) = λ for x
If A(≠ 0) is a language and A2 = A, prove that A = A*.
Provide the proofs for the remaining parts of Theorems 6.1 and 6.2.
Prove that for all finite languages A, B ⊂ ∑*, |AB| < |A||B|.
For ∑ [w,x,y,:) determine the number string in of length 5 (a) that start with r; (b) with precisely iwo w's; (C) with no w's; (d) with an even number of ws.
For ∈ = {x, y}, use finite languages from ∑* (as in Example 6.12), together with set operations, to describe the set of strings in ∑* that (a) Contain exactly one occurrence of x; (b) Contain exactly two occurrences of x; (c) Begin with x; (d) End in yxy; (e) Begin with x or end in yxy
For ∑ = {0, 1}, let A ⊂ ∑* be the language defined recursively as follows:(1) The symbols 0,1 are both in A - this is the base for our definition; and(2) For each word x in A, the word 0 × 1 is also in A - this constitutes the recursive process.(a) Find four different words - two of length 3
Provide a recursive definition for each of the following languages A⊂∑* where E = {0, 1}. (a) x ∈ A if (and only if) the number of 0's in x is even. (b) x ∈ A if (and only if) all of the l's in x precede all of the 0's.
Use the recursive definition given in Example 6.15 to verify that each of the following strings is in the language A of that example.(a) (())() (b) (())()() (c) ()(()())
For an alphabet ∈ a string x in ∑* is called a palindrome if x = xR - that is, x is equal to its reversal. If A ⊂ ∑* where A = {x ∈ ∑* E*|x = xR}, how can we define the language A recursively?
For ∑ = {0, 1}, let A ⊂ ∑*, where A = {00, 1}. How many strings in A* have length 3? length 4? length 5? length 6?
For E = {0, 1}, let A ∈ ∑*, where A = {00, 111}. How many strings in A* have length 19?
Let ∑ = {a, b, c}. Determine the smallest number of words one must select from ∑4 to guarantee that at least two of the words start and end with the same letter.
Let E = {ß, x, y, z} where P denotes a blank, so xß x, ß ß ≠ P, and xß ≠ xy but xλy = xy. Compute each of the following: (a) ||λ|| (b) || λ λ || (c) ||ß|| (d) ||ß ß || (e) || ß3 || (f) ||xßßy|| (g) ||ßλ|| (h) ||λ10||
Let ∑ be an alphabet. Let xi g ∈ for ∑ 1 < i < 100 (where xi ≠ xj for all 1 < i < j < 100). How many nonempty substrings are there for the string s = x1x2 ∙∙∙∙∙∙ x100?
For the alphabet E = {0, 1}, let A, B, C Š‚ ˆ‘* be the following languages:languages:a = {0, 1, 00, 11, 000, 111, 0000, 1111},B = {w ˆˆ ˆ‘*|2 C = {wˆˆ ˆ‘* 2 > ||w||}.Determine the following subsets (languages) of ˆ‘*.(a) A
Let A = {10, 11}, B = {00, 1} be languages for the alphabet E = {0, 1}. Determine each of the following: (a) AB (b) BA (c) A3; (d) B2.
If A, B, C, and D are languages over ∑, prove that (a) (A ⊂ B ^ C ⊂ D) 4 AC ⊂ BD and (b) Aϕ = ϕA = ϕ.
Using the finite state machine of Example 6.17, find the output for each of the following input stringsand determine the last internal state in the transition process. (Assume that we always start at s0. (a) x = 1010101 (b) x = 1001001 (c) x = 101
For the finite state machine of Example 6.17, an input string x, starting at state s0, produces the output string 00101. Determine
Let Mbe a finite state machine where S = {s0, s1, s2, s3}, and v, w are determined by Table 6.7. (a) Starting at s0, what is the output for the input string abbcccl (b) Draw the state diagram for this finite state machine.
Give the state table and the state diagram for the vending machine of Example 6.18 if the cost of a package of chewing gum (peppermint or spearmint) is increased to 25e/.
A finite state machine M ={0, 1} and is determined by the state diagram shown in Fig. 6.5. (a) Determine the output string for the input string 110111, starting at s0. What is the last transition state? (b) Answer part (a) for the same string but with s as the starting state. What about si and s3
Machine M hasand is determined by the state diagram shown in Fig. 6.6.(a) Describe in words what this finite state machine does.(b) What must state s1 remember?(c) Find two languagessuch that for every x e AB, w(s0, x) has 1 as a suffix.
Let M =determined by the state diagram shown in Fig. 6.7.(a) Find the output for the input string x = 0110111011.(b) Give the transition table for this finite state machine.(c) Starting in state s0, if the output for an input string x is 0000001, determine all possibilities for x.(d) Describe in
(a) Find the state table for the finite state machine in Fig. 6.8, where(b) Let with ||x|| = 4. If 1 is a suffix of w(s0, x), what are the possibilities for the string x ? (c) Let A Š‚ {0, 1}* be the language where w(s0, x) has 1 as a suffix for all x in A. Determine A. (d) Find the
Let f = 0 = {0, 1}. (a) Construct a state diagram for a finite state machine that recognizes each occurrence of 0000 in a string x ∈ f. (Here overlapping is allowed.) (b) Construct a state diagram for a finite state machine that recognizes each string x ∈ f* that ends in 0000 and has length 4k,
Answer Exercise 1 for each of the sequences 0110 and 1010.
Construct a state diagram for a finite state machine withthat recognizes all strings in the language {0, 1}*{00}ˆª{0, 1}*{11}.
Fora string x ˆˆ f* is said to have even parity if it contains an even number of l's. Construct a state diagram for a finite state machine that recognizes all nonempty strings of even parity.
Table 6.12 defines v and co for a finite state machine M where(a) Draw the state diagram for M.(b) Determine the output for the following input sequences, starting at s0 in each case: (i) x = 111; (ii) x = 1010; (iii) x = 00011.(c) Describe in words what machine M does.(d) How is this machine
Show that it is not possible to construct a finite state machine that recognizes precisely those sequences in the language A = {0.1 | j ∈ +, i > j}. (Here the alphabet for A is ∑ = {0, 1}.)
For each of the machines in Table 6.13, determine the transient states, sink states, submachines (where f1 = {0, 1}), and strongly connected submachines (where f1 = {0, 1}).
Let ∑ = {w, x, y] and ∑2 = {x, y, z} be alphabets. If A1 = {xtyj|i, j ∈ Z+, j > i > 1}, A2 = {wtyJi, j ∈ Z+, i > j > 1}, A3 = {wlxJylzJ\i, j ∈ Z+, j > i > 1}, and A4 = {zJ (wz)lwJ\i, j ∈ Z+, i > 1, j > 2}, determine whether each of the following statements is true or false۔ (a) Ai is a
Withlet M be the finite state machine given in Table 6.15. Here s0 is the starting state. Let A Š‚ f + where x Š‚ A if and only if the last symbol in w(s0, x) is 1. [There may be more than one 1 in the output string w(s0, x).] Construct a finite state machine wherein the last
Let $ - f = {0, 1} for the two finite state machines M1 and M2, given in Tables 6.16 and 6.17, respectively. The starting state for M1 is 50, whereas S3 is the starting state for M2.We connect these machines as shown in Fig. 6.19. Here each output symbol from M1 becomes an input symbol for M2. For
Although the state diagram seems more convenient than the state table when we are dealing with a finite state machineas the input strings get longer and the sizes of S,F and O increase, the state table proves useful when simulating the machine on a computer. The block form of the table suggests
For languages A, B ⊂ ∑*, does A* ⊂ B* => A ⊂ B?
Give an example of a language A over an alphabet E, where (A2) * ≠ (A*)2.
For ∑ = {0, 1} consider the languages A, B, C ⊂ ∑* where A = {01, 11}, B = {01, 11, 111}, and C = {01, 11, 1111}. (a) How are A* and B* related? (b) How about A* and C*?
Let M be the finite state machine shown in Fig. 6.17. For states s1, sj, where 0
Let M be the finite state machine in Fig. 6.18.(a) Find the state table for this machine.(b) Explain what this machine does.(c) How many distinct input strings x are there such that ||x|| = 8 and v(s0, x) = s0? Ho many are there with M = 12?

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