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mathematics
contemporary mathematics

Contemporary Mathematics 1st Edition OpenStax - Solutions

\(A \rightarrow B \rightarrow G \rightarrow F \rightarrow E \rightarrow A\)Identify each sequence of vertices from Figure 12.134 as a closed walk, circuit (closed trail), and/or directed cycle (closed path). Select all that apply.
\(E \rightarrow F \rightarrow G \rightarrow K \rightarrow J \rightarrow F \rightarrow B \rightarrow A \rightarrow E\)Identify each sequence of vertices from Figure 12.134 as a closed walk, circuit (closed trail), and/or directed cycle (closed path). Select all that apply.
The colors do not follow the definition of a graph coloring.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The graph is a two-coloring.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The graph is a three-coloring.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The graph is a four-coloring.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The graph is planar.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The graph has a chromatic number of two.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
Fewer colors could be used to color the graph.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The chromatic number is more than four.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
The Four-Color Theorem applies to the graph.Use the graphs shown. Identify the graph or graphs with the given characteristics. Graph 1 R R R R P B Graph 3 Graph 4 Graph 2 P
Closed path: \(a \rightarrow b \rightarrow \square \rightarrow \square \rightarrow \square \rightarrow a\)Complete the sequence of vertices from the graph to create the indicated kind of closed walk. P h b C 9 f e
Closed path that visits edge ed twice: \(e \rightarrow d \rightarrow \square \rightarrow \square \rightarrow \square \rightarrow e\)Complete the sequence of vertices from the graph to create the indicated kind of closed walk. P h b C 9 f e
Directed cycle: \(e \rightarrow a \rightarrow \square \rightarrow \square \rightarrow \square \rightarrow e\)Complete the sequence of vertices from the graph to create the indicated kind of closed walk. P h b C 9 f e
Directed cycle: \(b \rightarrow d \rightarrow \square \rightarrow \square \rightarrow \square \rightarrow b\)Complete the sequence of vertices from the graph to create the indicated kind of closed walk. P h b C 9 f e
Circuit that visits vertex \(d\) twice: \(b \rightarrow c \rightarrow d \rightarrow \square \rightarrow \square \rightarrow \square \rightarrow b\)Complete the sequence of vertices from the graph to create the indicated kind of closed walk. P h b C 9 f e
Circuit that visits vertex \(f\) twice: \(g \rightarrow h \rightarrow f \rightarrow \square \rightarrow \square \rightarrow \square \rightarrow g\)Complete the sequence of vertices from the graph to create the indicated kind of closed walk. P h b C 9 f e
Find the chromatic number \(n\) for graph 1 and give an \(n\)-coloring.Use the graphs shown. a b P d e f Graph 1 a b g m Graph 2 Graph 3 Graph 4
Find the chromatic number \(n\) for graph 2 and give an \(n\)-coloring.Use the graphs shown. a b P d e f Graph 1 a b g m Graph 2 Graph 3 Graph 4
Find the chromatic number \(n\) for graph 3 and give an \(n\)-coloring.Use the graphs shown. a b P d e f Graph 1 a b g m Graph 2 Graph 3 Graph 4
Find the chromatic number \(n\) for graph 4 and give an \(n\)-coloring.Use the graphs shown. a b P d e f Graph 1 a b g m Graph 2 Graph 3 Graph 4
The graph has 15 vertices and contains a clique with 9 vertices.Indicate the smallest and largest possible chromatic number of the graph described.
A planar graph with 100 vertices and a clique with 3 vertices.Indicate the smallest and largest possible chromatic number of the graph described.
A planar graph with more than 2 vertices and no cliques.Indicate the smallest and largest possible chromatic number of the graph described.
A complete graph with 2,123 vertices.Indicate the smallest and largest possible chromatic number of the graph described.
In chess, a knight can move in any direction, but it must move two spaces then turn and move one more space. The 8 possible moves a knight can make from a space in the center of a five-by-five grid are shown in the first figure. A knight's tour is a sequence of moves by a knight on a chessboard (of
An open knight's tour on a five-by-five board is shown in the figure. Is it most accurately described as a walk, a trail, or a path?
A knight's tour is not possible on a four-by-four board like the one shown in the figure. Find an open tour of the board in which the knight is permitted to travel through a given space more than once to achieve the goal of visiting every space. Is your tour most accurately described as a walk,
Describe how the route can be represented as a graph. If there is no backtracking, in other words, the person never reverses direction, is the route followed by the postal delivery person best described as a walk, trail, path, closed walk, circuit, or directed cycle? Explain your reasoning.The
Draw the graph for the route described in the neighborhood. Find its chromatic number. Find a map coloring that shows the chromatic number.The neighborhood of Pines West has three cul-de-sacs that meet at an intersection as shown in the figure. A postal delivery person starts at the intersection
When radio towers are within range of each other, they must use different frequencies. In the graph, the vertices represent the towsers, and the edges indicate the towers are in range of each other. Use graph colorings to determine the number of frequencies needed to avoid two towers in the same
A Sudoku puzzle consists of a nine-by-nine grid which is subdivided into nine three-by-three smaller grids. Given an incomplete grid, the goal is to complete it in such a way that each row contains the numbers 1 through 9 , each column contains the numbers 1 through 9 , and each three by three grid
Draw a graph to show the relationships.At a wedding reception, the bride and groom would like their family and friends to get to know each other better. They have decided that if two people know each other, they will not seat them at the same table. The following is a list of guests and who they
Determine the minimum number of tables needed to seat the guests so that they will sit with people they don't know.At a wedding reception, the bride and groom would like their family and friends to get to know each other better. They have decided that if two people know each other, they will not
Give a coloring to support your conclusion.At a wedding reception, the bride and groom would like their family and friends to get to know each other better. They have decided that if two people know each other, they will not seat them at the same table. The following is a list of guests and who
Draw a graph to represent the map.The map of the states of Imaginaria is given. Use the figure to answer the following exercises. Fictionville Illusionham Mythbury Pretendstead
Determine the chromatic number of the graph you found.The map of the states of Imaginaria is given. Use the figure to answer the following exercises. Fictionville Illusionham Mythbury Pretendstead
Give a graph coloring that supports your answer in the previous question.The map of the states of Imaginaria is given. Use the figure to answer the following exercises. Fictionville Illusionham Mythbury Pretendstead
Color the provided map to correspond with the coloring you found in the previous question.The map of the states of Imaginaria is given. Use the figure to answer the following exercises. Fictionville Illusionham Mythbury Pretendstead
In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, and D 112.Which candidate has the plurality and wins the election?
Which candidate has a plurality?Use the table below. Options ABCDE Candidate 1 1 33 1 3 Candidate 2 2 1 1 2 4 Candidate 3 3 41 2 4 1 Candidate 4 4 2 4 3 2
Does the plurality candidate have a majority?Use the table below. Options ABCDE Candidate 1 1 33 1 3 Candidate 2 2 1 1 2 4 Candidate 3 3 41 2 4 1 Candidate 4 4 2 4 3 2
Determine the winner of the election by the Hare method based on the sample preference summary in the table.Use the table below. Options ABCDE Candidate 1 1 33 1 3 Candidate 2 2 1 1 2 4 Candidate 3 3 41 2 4 1 Candidate 4 4 2 4 3 2
Use ranked-choice voting to determine the two options in the final round and the number of votes they each receive in that round.Use the table below. Number of Ballots 10 20 15 5 Option A 1 4 3 4 Option B 2 3 4 2 Option C 4 2 1 3 Option D 3 1 2 1
Is there a winning option? If so, which option? Justify your answer.Use the table below. Number of Ballots 10 20 15 5 Option A 1 4 3 4 Option B 2 3 4 2 Option C 4 2 1 3 Option D 3 1 2 1
What are the Borda scores for each candidate?Use the table below. Number of Ballots 100 80 110 105 55 Candidate A 1 1 4 4 2 Candidate B 2 2 2 3 1 Candidate C 4 4. 1 1 4 Candidate D 3 3 3 2 3
Which candidate is the winner by the Borda count method?Use the table below. Number of Ballots 100 80 110 105 55 Candidate A 1 1 4 4 2 Candidate B 2 2 2 3 1 Candidate C 4 4. 1 1 4 Candidate D 3 3 3 2 3
Calculate the points received by each candidate in pairwise comparison matrix.Use Pairwise Comparison Matrix for Candidates U, V, W, X, and Y to answer Question.
Determine the winner of the pairwise comparison election represented by matrix. If there is a winner, determine whether the winner is a Condorcet candidate and explain your reasoning. If there is no winner, indicate this.Use Pairwise Comparison Matrix for Candidates U, V, W, X, and Y to answer
The ladies of The Big Bang Theory decide to hold their own approval voting election to determine the best option in Rock, Paper, Scissors, Lizard, Spock. Use the summary of their approval ballots in the table below to determine the number of votes for each candidate. Determine the winner, or state
Which candidate is the winner by the ranked-choice method?Use the table below. Percentage of Vote 40% 35% 25% Candidate A 1 3 2 Candidate B 2 1 3 Candidate C 3 m. 2 1
Suppose that you used the approval method and each voter approved their top two choices. Which candidate is the winner by the approval method?Use the table below. Percentage of Vote 40% 35% 25% Candidate A 1 3 2 Candidate B 2 1 3 Candidate C 3 m. 2 1
Which candidate is the winner by the Borda count method?Use the table below. Percentage of Vote 40% 35% 25% Candidate A 1 3 2 Candidate B 2 1 3 Candidate C 3 m. 2 1
In a Borda count election, the candidates have the following Borda scores: A 1245, B 1360, and C 787.Candidate A received \(55 \%\) of the first place rankings. Identify which fairness criteria, if any, are violated by characteristics of the described voter profile in this Borda count election.
Determine Borda score for each candidate, and the winner of the election using the Borda count method.Use the table below. Number of Ballots 8 10 12 4 Option A 1 3 2 1 Option B 3 14 4 Option C 4 2 1 2 Option D 2 4 3 3
Is there a majority candidate? If so, which candidate?Use the table below. Number of Ballots 8 10 12 4 Option A 1 3 2 1 Option B 3 14 4 Option C 4 2 1 2 Option D 2 4 3 3
Does the Borda method election violate the majority criterion? Justify your answer.Use the table below. Number of Ballots 8 10 12 4 Option A 1 3 2 1 Option B 3 14 4 Option C 4 2 1 2 Option D 2 4 3 3
In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, and D 16.The pairwise match up points for the same voter profiles would have been A 2, B 0, C 1, and D 3.Identify which fairness criteria, if any, are violated by characteristics of the described voter
Determine the winner of the election using the ranked-choice method.Use the table below. Number of Ballots 8 10 12 4 Option A 1 3 2 1 Option B 3 14 4 Option C 4 2 1 2 Option D 2 4 3 3
If the four voters in the last column rank \(\mathrm{C}\) ahead of \(\mathrm{A}\), which candidate wins by the ranked-choice method?Use the table below. Number of Ballots 8 10 12 4 Option A 1 3 2 1 Option B 3 14 4 Option C 4 2 1 2 Option D 2 4 3 3
Does this ranked-choice election violate the monotonicity criterion? Explain your reasoning.Use the table below. Number of Ballots 8 10 12 4 Option A 1 3 2 1 Option B 3 14 4 Option C 4 2 1 2 Option D 2 4 3 3
Determine the winner of the election by the Borda method.Use the table below. Number of Ballots 15 12 9 3 Option A 1 3 3 2 Option B 2 2 1 1 Option C 3 1 2 3
Does this Borda method election violate the IIA? Why or why not?Use the table below. Number of Ballots 15 12 9 3 Option A 1 3 3 2 Option B 2 2 1 1 Option C 3 1 2 3
Which of the ranked voting methods in this chapter, if any, meets the majority criterion, the head-to-head criterion, the monotonicity criterion, and the irrelevant alternatives criterion?Use the table below. Number of Ballots 15 12 9 3 Option A 1 3 3 2 Option B 2 2 1 1 Option C 3 1 2 3
Identify the states, the seats, and the state population (the basis for the apportionment) in the given scenario: The reading coach at an elementary school has 52 prizes to distribute to their students as a reward for time spent reading.
Use the given information to find the standard divisor to the nearest hundredth. Include the units. The total population is 2,235 automobiles, and the number of seats is 14 warehouses.
Use the given information to find the standard quota. Include the units. The state population is eight residents in a unit, and the standard divisor is 1.75 residents per parking space.
Determine the Hamilton apportionment for Scenario \(X\) in the table below. State A State B State C State D State E State F Total Seats Scenario X 17.63 26.62 10.81 16.01 13.69 15.24 100
Does the apportionment resulting from Method \(X\) in the table below satisfy the quota rule? Why or why not? Standard Quota Apportionment Method X State A State B State C State D State E 1.67 3.33 5.00 6.67 8.33 2 2 5 7 9
Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario?Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In Wakandan, the word "birnin" means "fortress
Find the modified upper quota for each state using a modified divisor of 2,250 . Is the sum of the modified quotas too high, too low, or equal to the house size?Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In
Use the Adams method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In
Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario?Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In Wakandan, the word "birnin" means "fortress city."
Use the Webster method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In
Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario?Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In Wakandan, the word "birnin" means "fortress city."
Which of the four methods of apportionment from this section are the residents of Birnin S'Yan likely to prefer? Justify your answer.Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In Wakandan, the word "birnin"
Does the change from a standard divisor to a modified divisor tend to change the number of seats for larger or smaller states more?Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In Wakandan, the word "birnin"
Which of the four apportionment methods-Jefferson, Adams, Hamilton, or Webster-satisfies the quota rule?Use the table below and the following information: In Wakanda, the domain of the Black Panther, King T'Challa, has six fortress cities. In Wakandan, the word "birnin" means "fortress city." King
A city purchased five new firetrucks and apportioned them among the existing fire stations. Although your neighborhood fire station has the same proportion of the city's firetrucks as before the new ones were purchased, it now has one fewer. Is this scenario an example of a quota rule violation,
The school resources officers in a county were reapportioned based on the most recent census. The number of students at Chapel Run Elementary went up while the number of students at Panther Trail Elementary went down, but Chapel Run now has 1 fewer resources officers while Panther Trail has one
How did the change in populations impact the apportionment?The house size is 24 seats. When the population of \(A\) increases by 28 percent, \(B\) increases by 26 percent, and C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to A 3.63, B 6.67 , and \(C 13.71\).
Is this apportionment an example of a paradox? Justify your answer.The house size is 24 seats. When the population of \(A\) increases by 28 percent, \(B\) increases by 26 percent, and C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to A 3.63, B 6.67 , and \(C
When the city of Cocoa annexed an adjacent unincorporated community, the number of seats on the city council was increased to maintain the standard ratio of citizens to seats, but one existing community of Cocoa still lost a seat on the city council to another existing community of Cocoa when the
How did the additional state impact the apportionment?The house size was 27.There were three states with standard quotas of A 6.39, B 11.40, and C 9.21. A fourth state was annexed, and the house size was increased to 35 . The new standard quotas are A \(6.38, B\) 11.37, C 9.19, and D 8.06.
Is this apportionment an example of a paradox? Justify your answer.The house size was 27.There were three states with standard quotas of A 6.39, B 11.40, and C 9.21. A fourth state was annexed, and the house size was increased to 35 . The new standard quotas are A \(6.38, B\) 11.37, C 9.19, and D
Demonstrate that the population paradox occurs when the Hamilton method is used.Suppose 11 seats are apportioned to States A, B, and C with populations of 50, 129, and 181 people, respectively. Then the populations of States \(A, B\), and \(C\) change to 57,151 , and 208 , respectively.
Demonstrate that the population paradox does not occur when the Jefferson method is used. Justify your answer.Suppose 11 seats are apportioned to States A, B, and C with populations of 50, 129, and 181 people, respectively. Then the populations of States \(A, B\), and \(C\) change to 57,151 , and
Demonstrate that the population paradox does not occur when the Adams method is used. Justify your answer.Suppose 11 seats are apportioned to States A, B, and C with populations of 50, 129, and 181 people, respectively. Then the populations of States \(A, B\), and \(C\) change to 57,151 , and 208 ,
Demonstrate that the population paradox does not occur when the Webster method is used. Justify your answer.Suppose 11 seats are apportioned to States A, B, and C with populations of 50, 129, and 181 people, respectively. Then the populations of States \(A, B\), and \(C\) change to 57,151 , and 208
Determine the winner of the election by plurality.Use the table below. Number of Ballots 12 17 15 13 Option A 1 24 3 Option B 3 1 3 2 Option C 4 3 1 4 Option D 2 4 2 1
Determine the Borda scores for each candidate to determine the winner by Borda count method.Use the table below. Number of Ballots 12 17 15 13 Option A 1 24 3 Option B 3 1 3 2 Option C 4 3 1 4 Option D 2 4 2 1
Create and analyze a pairwise comparison matrix based on the preference summary to determine the winner of the election by pairwise comparison.Use the table below. Number of Ballots 12 17 15 13 Option A 1 24 3 Option B 3 1 3 2 Option C 4 3 1 4 Option D 2 4 2 1
From the table below use ranked-choice voting to determine the winner of the election. Number of Ballots 28 5 30 5 16 16 Option L 3 2 1 1 2 3 Option R 1 1 3 2 3 2 Option E 2 3 2 3 1 1
In a Borda count election, the candidates have the following Borda scores: A 1345, B 1260, C 685.Candidate B received \(51 \%\) of the first-place rankings.Identify which fairness criteria, if any, are violated by characteristics of the described voter profile. Explain your reasoning.
In a plurality election, the candidates have the following percentages of first place votes: A 25, B 21, C 30, D 24.The pairwise matchup points for the same voter profiles would have been A 3, B 0, C 2, D 2.Identify which fairness criteria, if any, are violated by characteristics of the described
Determine the winner by ranked-choice voting if two of the voters in the second column up-rank the original winner. Refer to Question 4.Which fairness criterion, if any, is violated?Use the table below. Number of Ballots 13 14 11 Option A 12 2 1 3 3 Option B Option C Option D 3 2 4 1 4 - 4 1 2 3 3 4
Determine the winner by ranked-choice voting if candidate \(R\) is removed from the election. Refer to Question 4.Which fairness criterion, if any, is violated?Use the table below. Number of Ballots 13 14 11 Option A 12 2 1 3 3 Option B Option C Option D 3 2 4 1 4 - 4 1 2 3 3 4
Identify the states, the seats, and the state population (the basis for apportionment) in the given scenario.Use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members
Determine the standard divisor for the apportionment Use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations with
Determine each state's standard quota rounded to two decimal places.Use this information: The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents, and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations

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