New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
statistics

Discrete Mathematics and Its Applications 7th edition Kenneth H. Rosen - Solutions

Suppose that the relation R on the finite set A is represented by the matrix MR. Show that the matrix that represents the symmetric closure of R is MR ∨ MtR.
When is it possible to define the "irreflexive closure" of a relation R, that is, a relation that contains R, is irreflexive, and is contained in every irreflexive relation that contains R?
Find all circuits of length three in the directed graph in Exercise 16.In problem
Let R be the relation on the set of all students containing the ordered pair (a, b) if a and b are in at least one common class and a ≠ b. When is (a, b) in a) R2? b) R3? c) R∗?
Suppose that the relation R is symmetric. Show that R∗ is symmetric.
Use Algorithm 1 to find the transitive closures of these relations on {1, 2, 3, 4}. a) {(1, 2), (2,1), (2,3), (3,4), (4,1)} b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)} c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)} d) {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)}
Use Warshall's algorithm to find the transitive closures of the relations in Exercise 25. a) {(1, 2), (2,1), (2,3), (3,4), (4,1)} b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)} c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)} d) {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)}
Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is a) Reflexive and transitive. b) Symmetric and transitive. c) Reflexive, symmetric, and transitive.
Let R be the relation {(a, b) | a divides b} on the set of integers. What is the symmetric closure of R?
Algorithms have been devised that use O(n2.8) bit operations to compute the Boolean product of two n × n zero- one matrices. Assuming that these algorithms can be used, give big-O estimates for the number of bit operations using Algorithm 1 and using Warshall's algorithm to find the transitive
Adapt Algorithm 1 to find the reflexive closure of the transitive closure of a relation on a set with n elements.
Show that the closure with respect to the property P of the relation R = {(0, 0), (0, 1), (1, 1), (2, 2)} on the set {0, 1, 2} does not exist if P is the property a) "Is not reflexive." b) "Has an odd number of elements."
Draw the directed graph of the reflexive closure of the relations with the directed graph shown.1.2.
Find the directed graphs of the symmetric closures of the relations with directed graphs shown in Exercises 5-7.In exercise1.2.
Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {(0, 0), (1, 1), (2, 2), (3, 3)} b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)} c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} d) {(0,
Show that the relation R consisting of all pairs (x, y) such that x and y are bit strings that agree in their first and third bits is an equivalence relation on the set of all bit strings of length three or more.
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if a + d = b + c. Show that R is an equivalence relation.
a) Show that the relation R on the set of all differentiable functions from R to R consisting of all pairs (f, g) such that f'(x) = g'(x) for all real numbers x is an equivalence relation. b) Which functions are in the same equivalence class as the function f (x) = x2?
Let R be the relation on the set of all URLs (or Web addresses) such that x R y if and only if the Web page at x is the same as the Web page at y. Show that R is an equivalence relation.
Determine whether the relation with the directed graph shown is an equivalence relation.
What are the equivalence classes of the equivalence relations in Exercise 2? a) {(a, b) | a and b are the same age} b) {(a, b) | a and b have the same parents}
What is the equivalence class of the bit string 011 for the equivalence relation in Exercise 25?
What are the equivalence classes of the bit strings in Exercise 30 for the equivalence relation from Exercise 12?a) 010b) 1011c) 11111d) 01010101
What are the equivalence classes of the bit strings in Exercise 30 for the equivalence relation R4 from Example 5 on the set of all bit strings? (Recall that bit strings s and t are equivalent under R4 if and only if they are equal or they are both at least four bits long and agree in their first
What is the congruence class [n]5 (that is, the equivalence class of n with respect to congruence modulo 5) when n is a) 2? b) 3? c) 6? d) −3?
Give a description of each of the congruence classes modulo 6.
a) What is the equivalence class of (1, 2) with respect to the equivalence relation in Exercise 15? b) Give an interpretation of the equivalence classes for the equivalence relation R in Exercise 15.
Which of these collections of subsets are partitions of {1, 2, 3, 4, 5, 6}? a) {1, 2}, {2, 3, 4}, {4, 5, 6} b) {1}, {2, 3, 6}, {4}, {5} c) {2, 4, 6}, {1, 3, 5} d) {1, 4, 5}, {2, 6}
Which of these collections of subsets are partitions of the set of bit strings of length 8? a) The set of bit strings that begin with 1, the set of bit strings that begin with 00, and the set of bit strings that begin with 01 b) The set of bit strings that contain the string 00, the set of bit
Which of these are partitions of the set Z × Z of ordered pairs of integers? a) The set of pairs (x, y), where x or y is odd; the set of pairs (x, y), where x is even; and the set of pairs (x, y), where y is even b) The set of pairs (x, y), where both x and y are odd; the set of pairs (x, y),
List the ordered pairs in the equivalence relations produced by these partitions of {0, 1, 2, 3, 4, 5}. a) {0}, {1, 2}, {3, 4, 5} b) {0, 1}, {2, 3}, {4, 5} c) {0, 1, 2}, {3, 4, 5} d) {0}, {1}, {2}, {3}, {4}, {5}
Show that the partition formed from congruence classes modulo 6 is a refinement of the partition formed from congruence classes modulo 3. A partition P1 is called a refinement of the partition P2 if every set in P1 is a subset of one of the sets in P2.
Define three equivalence relations on the set of buildings on a college campus. Determine the equivalence classes for each of these equivalence relations.
Show that the partition of the set of bit strings of length 16 formed by equivalence classes of bit strings that agree on the last eight bits is a refinement of the partition formed from the equivalence classes of bit strings that agree on the last four bits.
Show that the partition of the set of all identifiers in C formed by the equivalence classes of identifiers with respect to the equivalence relation R31 is a refinement of the partition formed by equivalence classes of identifiers with respect to the equivalence relation R8. (Compilers for "old"
Find the smallest equivalence relation on the set {a, b, c, d, e} containing the relation {(a, b), (a, c), (d, e)}.
Consider the equivalence relation from Example 2, namely, R = {(x, y) | x − y is an integer}. a) What is the equivalence class of 1 for this equivalence relation? b) What is the equivalence class of 1/2 for this equivalence relation?
Determine the number of different equivalence relations on a set with three elements by listing them.
Do we necessarily get an equivalence relation when we form the transitive closure of the symmetric closure of the reflexive closure of a relation?
Suppose we use Theorem 2 to form a partition P from an equivalence relation R. What is the equivalence relation R' that results if we use Theorem 2 again to form an equivalence relation from P?
Devise an algorithm to find the smallest equivalence relation containing a given relation.
Use Exercise 68 to find the number of different equivalence relations on a set with n elements, where n is a positive integer not exceeding 10.Let p(n) denote the number of different equivalence relations on a set with n elements (and by Theorem 2 the number of partitions of a set with n elements).
Show that the relation of logical equivalence on the set of all compound propositions is an equivalence relation. What are the equivalence classes of F and of T?
Suppose that A is a nonempty set, and f is a function that has A as its domain. Let R be the relation on A consisting of all ordered pairs (x, y) such that f (x) = f (y). a) Show that R is an equivalence relation on A. b) What are the equivalence classes of R?
Which of these relations on {0, 1, 2, 3} are partial orderings? Determine the properties of a partial ordering that the others lack. a) {(0, 0), (1, 1), (2, 2), (3, 3)} b) {(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)} c) {(0, 0), (1, 1), (1, 2), (2, 2), (3, 3)} d) {(0, 0), (1, 1), (1,
Find the duals of these posets. a) ({0, 1, 2}, ≤) b) (Z, ≥) c) (P (Z), ⊇) d) (Z+, |)
Find two incomparable elements in these posets. a) (P({0, 1, 2}),⊆) b) ({1, 2, 4, 6, 8}, |)
Find the lexicographic ordering of these n-tuples: a) (1, 1, 2), (1, 2, 1) b) (0, 1, 2, 3), (0, 1, 3, 2) c) (1, 0, 1, 0, 1), (0, 1, 1, 1, 0)
Find the lexicographic ordering of the bit strings 0, 01, 11, 001, 010, 011, 0001, and 0101 based on the ordering 0 < 1.
Draw the Hasse diagram for the "less than or equal to" relation on {0, 2, 5, 10, 11, 15}.
Draw the Hasse diagram for divisibility on the set a) {1, 2, 3, 4, 5, 6, 7, 8}. b) {1, 2, 3, 5, 7, 11, 13}. c) {1, 2, 3, 6, 12, 24, 36, 48}. d) {1, 2, 4, 8, 16, 32, 64}.
All ordered pairs in the partial ordering with the accompanying Hasse diagram.
What is the covering relation of the partial ordering {(A,B) | A ⊆ B} on the power set of S, where S = {a, b, c}?
Is (S,R) a poset if S is the set of all people in the world and (a, b) ∈ R, where a and b are people, if a) A is taller than b? b) A is not taller than b? c) A = b or a is an ancestor of b? d) A and b have a common friend?
Show that a finite poset can be reconstructed from its covering relation.
Answer these questions for the poset ({3, 5, 9, 15, 24, 45}, |). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of {3, 5}. f) Find the least upper bound of {3, 5}, if it exists. g) Find all lower
Answer these questions for the poset ({{1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}}, ⊆). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of {{2}, {4}}. f) Find the least upper
Which of these are posets? a) (Z, =) b) (Z, ≠) c) (Z, ≥) d) (Z, X)
Determine whether the relations represented by these zero-one matrices are partial orders.a)b) c)
Draw graph models, stating the type of graph (from Table 1) used, to represent airline routes where every day there are four flights from Boston to Newark, two flights from Newark to Boston, three flights from Newark to Miami, two flights from Miami to Newark, one flight from Newark to Detroit, two
Let G be a simple graph. Show that the relation R on the set of vertices of G such that uRv if and only if there is an edge associated to {u, v} is a symmetric, ir-reflexive relation on G.
The intersection graph of a collection of sets A1, A2, . . . , An is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph of these collections of sets.
Construct a niche overlap graph for six species of birds, where the hermit thrush competes with the robin and with the blue jay, the robin also competes with the mockingbird, the mockingbird also competes with the blue jay, and the nuthatch competes with the hairy woodpecker.
We can use a graph to represent whether two people were alive at the same time. Draw such a graph to represent whether each pair of the mathematicians and computer scientists with biographies in the first five chapters of this book who died before 1900 were contemporaneous. (Assume two people lived
Construct an influence graph for the board members of a company if the President can influence the Director of Research and Development, the Director of Marketing, and the Director of Operations; the Director of Research and Development can influence the Director of Operations; the Director of
In a round-robin tournament the Tigers beat the Blue Jays, the Tigers beat the Cardinals, the Tigers beat the Orioles, the Blue Jays beat the Cardinals, the Blue Jays beat the Orioles, and the Cardinals beat the Orioles. Model this outcome with a directed graph.
Explain how the two telephone call graphs for calls made during the month of January and calls made during the month of February can be used to determine the new telephone numbers of people who have changed their telephone numbers.
How can a graph that models e-mail messages sent in a network be used to find people who have recently changed their primary e-mail address?
Describe a graph model that represents whether each person at a party knows the name of each other person at the party. Should the edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed?
For each course at a university, there may be one or more other courses that are its prerequisites. How can a graph be used to model these courses and which courses are prerequisites for which courses? Should edges be directed or undirected? Looking at the graph model, how can we find courses that
Describe a graph model that represents traditional marriages between men and women. Does this graph have any special properties?
Construct a precedence graph for the following program: S1: x := 0 S2: x := x + 1 S3: y := 2 S4: z := y S5: x := x + 2 S6: y := x + z S7: z := 4
Describe a discrete structure based on a graph that can be used to model relationships between pairs of individuals in a group, where each individual may either like, dislike, or be neutral about another individual, and the reverse relationship may be different.
Determine whether the graph shown has directed or undirected edges, whether it has multiple edges, and whether it has one or more loops. Use your answers to determine the type of graph in Table 1 this graph is.
In Exercise find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. Identify all isolated and pendant vertices.
Construct the underlying undirected graph for the graph with directed edges in Figure 2.
What does the degree of a vertex represent in an academic collaboration graph? What does the neighborhood of a vertex represent? What do isolated and pendant vertices represent?
What do the in-degree and the out-degree of a vertex in a telephone call graph, as described in Example 4 of Section 10.1, represent? What does the degree of a vertex in the undirected version of this graph represent?
What do the in-degree and the out-degree of a vertex in a directed graph modeling a round-robin tournament represent?
Use Exercise 18 to show that in a group of people, there must be two people who are friends with the same number of other people in the group.
Determine whether the graph is bipartite. You may find it useful to apply Theorem 4 and answer the question by determining whether it is possible to assign either red or blue to each vertex so that no two adjacent vertices are assigned the same color.
Suppose that there are four employees in the computer support group of the School of Engineering of a large university. Each employee will be assigned to support one of four different areas: hardware, software, networking, and wireless. Suppose that Ping is qualified to support hardware,
Suppose that there are five young women and five young men on an island. Each man is willing to marry some of the women on the island and each woman is willing to marry any man who is willing to marry her. Suppose that Sandeep is willing to marry Tina and Vandana; Barry is willing to marry Tina,
In Exercise find the number of vertices, the number of edges, and the degree of each vertex in the given undirected graph. Identify all isolated and pendant vertices.
Suppose there is an integer k such that every man on a desert island is willing to marry exactly k of the women on the island and every woman on the island is willing to marry exactly k of the men. Also, suppose that a man is willing to marry a woman if and only if she is willing to marry him. Show
For the graph G in Exercise 1 find a) The sub-graph induced by the vertices a, b, c, and f. b) The new graph G1 obtained from G by contracting the edge connecting b and f .
How many vertices and how many edges do these graphs have? a) Kn b) Cn c) Wn d) Km,n e) Qn
Find the degree sequence of each of the following graphs. a) K4 b) C4 c) W4 d) K2,3 e) Q3
What is the degree sequence of Kn, where n is a positive integer? Explain your answer.
How many edges does a graph have if its degree sequence is 5, 2, 2, 2, 2, 1? Draw such a graph.
Determine whether each of these sequences is graphic. For those that are, draw a graph having the given degree sequence. a) 3, 3, 3, 3, 2 b) 5, 4, 3, 2, 1 c) 4, 4, 3, 2, 1 d) 4, 4, 3, 3, 3 e) 3, 2, 2, 1, 0 f) 1, 1, 1, 1, 1
Show that a sequence d1, d2, . . . , dn of nonnegative integers in non increasing order is a graphic sequence if and only if the sequence obtained by reordering the terms of the sequence d2 − 1, . . . , dd1+1 − 1, dd1+2, . . . , dn so that the terms are in non increasing order is a graphic
Show that every non increasing sequence of nonnegative integers with an even sum of its terms is the degree sequence of a pseudo graph, that is, an undirected graph where loops are allowed.
How many sub graphs with at least one vertex does K3 have?
Can a simple graph exist with 15 vertices each of degree five?
Draw all sub graphs of this graph.
For which values of n are these graphs regular? a) Kn b) Cn c) Wn d) Qn
How many vertices does a regular graph of degree four with 10 edges have?
In Exercise find the union of the given pair of simple graphs. (Assume edges with the same endpoints are the same.)
The complementary graph G of a simple graph G has the same vertices as G. Two vertices are adjacent in G if and only if they are not adjacent in G. Describe each of these graphs. (a) Kn (b) Km.n (c) Cn (d) Qn

Showing 69400 - 69500 of 88243

First
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved