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mathematics
contemporary mathematics
Contemporary Mathematics 1st Edition OpenStax - Solutions
I and \(J\)Use the figure to answer the following exercises. A pair of graphs is given. Identify three differences between them that demonstrate the graphs are not isomorphic. a 9 W b d g h k m n P t Graph G Graph H Graph I Graph J Graph K Graph L
\(I\) and \(K\)Use the figure to answer the following exercises. A pair of graphs is given. Identify three differences between them that demonstrate the graphs are not isomorphic. a 9 W b d g h k m n P t Graph G Graph H Graph I Graph J Graph K Graph L
\(I\) and \(L\)Use the figure to answer the following exercises. A pair of graphs is given. Identify three differences between them that demonstrate the graphs are not isomorphic. a 9 W b d g h k m n P t Graph G Graph H Graph I Graph J Graph K Graph L
\(J\) and \(K\)Use the figure to answer the following exercises. A pair of graphs is given. Identify three differences between them that demonstrate the graphs are not isomorphic. a 9 W b d g h k m n P t Graph G Graph H Graph I Graph J Graph K Graph L
\(K\) and \(L\)Use the figure to answer the following exercises. A pair of graphs is given. Identify three differences between them that demonstrate the graphs are not isomorphic. a 9 W b d g h k m n P t Graph G Graph H Graph I Graph J Graph K Graph L
\(\quad P\) and \(Q\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(P\) and \(R\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(P\) and \(U\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(Q\) and \(R\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(Q\) and \(S\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(Q\) and \(T\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(Q\) and \(V\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(S\) and \(V\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(T\) and \(U\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(U\) and \(V\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. W h n Z Graph P Graph Q Graph R Graph S Graph T Graph U Graph V
\(A\) and \(B\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. V W S H Graph A m n p b X y N g e f j k h Graph B Graph C
\(A\) and \(C\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. V W S H Graph A m n p b X y N g e f j k h Graph B Graph C
\(A\) and \(D\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. V W S H Graph A m n p b X y N g e f j k h Graph B Graph C
\(B\) and \(C\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. V W S H Graph A m n p b X y N g e f j k h Graph B Graph C
\(B\) and \(D\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. V W S H Graph A m n p b X y N g e f j k h Graph B Graph C
\(C\) and \(D\)Use the figure to answer the following exercises. In each exercise, a pair of graphs is given. Either give a reason that the graphs are not isomorphic, or show how one of the graphs can be transformed to look like the other. V W S H Graph A m n p b X y N g e f j k h Graph B Graph C
\(W\) and \(Z\)Use the figure to answer the following exercises. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. If they are isomorphic, name an isomorphism. If one is a subgraph of the other, indicate a correspondence between the vertices that
\(X\) and \(Y\)Use the figure to answer the following exercises. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. If they are isomorphic, name an isomorphism. If one is a subgraph of the other, indicate a correspondence between the vertices that
\(X\) and \(Z\)Use the figure to answer the following exercises. Determine if one graph is a subgraph of the other graph, or the two graphs are isomorphic. If they are isomorphic, name an isomorphism. If one is a subgraph of the other, indicate a correspondence between the vertices that
Find the complement of Graph \(W\).Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Find the complement of Graph \(Y\).Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Find the complement of Graph \(X\).Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Find an isomorphism between the complement of \(W\) and the complement of \(Y\) if one exists. If not, explain how you know.Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Are \(W\) and \(Y\) isomorphic? Explain how you know.Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Find an isomorphism between the complement of \(W\) and the Complement of \(X\) if one Exists. If not, explain how you know.Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Are \(W\) and \(X\) isomorphic? Explain how you know.Use the figure to answer the following exercises. b Graph W Graph X (m Graph Y Graph Z
Find the complement of the graph in the given figure representing direct flights between south Florida airports. Explain what the graph represents. Tampa Tampa TPA West Palm Beach PBI Fort Lauderdale FLL Miami Key West FYW
According to the Cook Political Report, a total of \(158,394,605\) votes were cast in the 2020 U.S. presidential election. Of those, 81,281,502 were cast for Joe Biden, 74,222,593 for Donald Trump, and 2,890,510 for other candidates. Although U.S. presidential elections are not determined by
Consider the results of the 2018 U.S. Senate primary for Maryland. Determine which candidate won the primary for the Republicans (R). Did the candidate win a majority or a plurality of Republican votes?
The student government bylaws of a particular college require that a new president is elected annually by plurality voting. In the event of a tie, the bylaws require the candidate(s) with the fewest votes to be removed from the ballot and a runoff election to be held with the remaining candidates.
There are six members on the board of a Parent Teacher Association (PTA) at a local elementary school: the president (P), the vice president (V), the recording secretary (R), the liaison to the administration (L), the treasurer (T), and the chief fundraiser (C). The board must decide which
A kindergarten class votes on their favorite colors using a ranked ballot. Use the results in the following table to answer the questions.1. How many students voted in total?2. How many students voted for yellow as their favorite color?3. How many students in the class indicated that blue was their
1. Suppose that 58 Star Wars fans were asked to vote for their favorite Star Wars character. They were given a ranked ballot, and the results are shown in the following table. Use ranked-choice voting to determine the winner. Number of Ballots 7 6 10 8 4 5 6 7 23 Han Solo 3 5 1 3 2 4 2 2 2 1 4
Consider the color preferences of the kindergarten class once more. Using the Borda count method, determine the total number of points the color blue received.
Answer the following questions using the table below, which summarizes Imaginarian voter preferences.1. Use the ranked-choice voting method to determine the winner of the election.2. Use the Borda count method to determine the winner of the election.3. Compare the results of the two methods. Did
According to Variety magazine, there were 8,469 eligible Oscar voters in 2020. To make our matrix easier to work with, we've rounded this number up to 8,700 . We'd never do this in a real voting situation. Suppose that the voter preferences for the ballot for Best Director had been as shown in the
A pairwise comparison matrix is given. Determine the winner by the pairwise comparison method. If there is not a winner, explain why. If there is a winner, tell whether the winner is a Condorcet candidate.
Suppose there is an election with 26 candidates, A through Z, and that Candidate C is a Condorcet candidate.1. How many points did Candidate \(\mathrm{C}\) win?2. What is the greatest number of points that any one of the other candidates could win?3. Is it possible for Candidate \(\mathrm{C}\) to
The Chionilis family is trying to decide on a restaurant again, but now they don't want to deal with multiple runoffs or even ranking. They will use the approval voting method shown in the following table. Each family member will approve their top two choices. Rainbow China won when multiple
What would be the outcome of the election if every member of the Chionilis family approved their top three choices from the table below? Options ARSTG Rainbow China 1 3 3 1 3 Dough Boys Pizza 2212 1 Taco City 3 4 2 4 2 Caribbean Flavor 414 3 4
Extrapolate the results of an approval method election using Table 11.3, the assumptions from Example 11.14, and the additional assumptions that the supporters of Al Gore would all approve Ralph Nader and that half of the supporters of George Bush would approve Pat Buchanan while half would approve
The juniors at a high school in Central Florida are voting for a theme park to visit for an end of the year field trip. The options are Disney's four theme parks: Animal Kingdom, Magic Kingdom, EPCOT Center, or Hollywood Studios.Determine the outcome of the vote by each of the given voting
Use the information in the table to answer the following questions.1. Which option is the winner when using the plurality voting method?2. Which option is the winner when using the ranked-choice voting method?3. Does the majority criterion apply? If so, for which voting method(s), if any, did the
Use the information in the following table to find the winner using each of the voting methods in parts 1 and 2 , then answer the question in part 3.1. Pairwise comparison 2. Borda count 3. Does the majority criterion apply? If so, for which voting method(s) did the majority criterion fail? Voters
Determine whether the Condorcet criterion applies based on the summary of ranked ballots given in the table below. Votes 3 2 Option A 13 Option B 2 1 Option C 3 2
Use the summary of ranked ballots below to find the winner and determine whether the Condorcet criterion is satisfied when each of the following voting methods are used. Recall that Option A is a Condorcet candidate.1. Plurality 2. Ranked-choice voting 3. Borda count Votes 3 2 Option A 13 Option B
Use the Favorite Large Dog Breed Ballot Summary to answer each question.1. Determine the winner of the election by Borda count.2. Suppose that the 61 voters in the third column increased their ranking of the winner by 1 . Determine the winner by Borda count with the new rankings.3. Does this
Use the Favorite Large Dog Breed Ballot Summary to answer each question.1. Determine the winner of the election by the ranked-choice method.2. Suppose that the 24 voters in the last column of the table increased their ranking of the winner by 1. Determine the winner by the ranked-choice method with
The local animal shelter is having another vote-by-donation charity event! This time, for a \(\$ 10\) donation, an individual can complete a ranked ballot indicating their favorite small dog breed, miniature, from the ones provided: miniature poodle, Yorkshire terrier, or Chihuahua. Use the summary
Use the initial ballot summary from Example 11.23 to answer the questions.1. Determine the winner of the election by the Borda count method.2. Determine the winner of the election by the Borda count method if Michael Scott is removed from the ballot.3. Does this election violate IIA?
The SAT is to be administered at a high school. In preparation, pencils have been distributed to each of the classrooms based on the room capacity. Use the information in the following table to answer each question.1. Find the part to part ratio of desks to pencils for each room. Represent it as
Refer again to the table providing information on classroom capacities and pencil distribution.1. Determine the number of pencils that would be allocated to a classroom \(F\) with 28 desks by multiplying the number of desks by the ratio of pencils to desks.2. Determine the number of pencils that
Table 11.9 contains a list of the five U.S. states ranked sixth through tenth in the number of representatives in the U.S. House of Representatives, along with the population of that state in 2021. Use the information in the table to answer the questions.1. What is the ratio of State Population to
By the end of the first U.S. Congress in 1791 , there were 13 states, 65 representative seats, and approximately \(3,929,214\) citizens. Find the standard divisor rounded to the nearest tenth.
The Hernandez family and the Higgins family went trick-or-treating together for Halloween last year. They returned with 313 pieces of candy, which they will now apportion to the families. The Hernandez family has three children and the Higgins family has four children.1. Identify the states, seats,
By the end of the first U.S. Congress in 1791 , there were 13 states, 65 representative seats, and approximately \(3,929,214\) citizens. In that year, the state of Delaware had a population of approximately 59,000 . people. Use this information and the standard divisor you found in Your Turn 11.27
This year the Hernandez family and the Higgins family were joined by the Ho family for Halloween trick-ortreating. The Hernandez family has three children, the Higgins family has four children, and the Ho family has two children. This time, they collected 527 pieces of candy, which they are going
The science department of a high school has received a grant for 34 laptops. They plan to apportion them among their six classrooms based on each classroom's student capacity. Use the standard quotas in the table below to answer each question.1. Use traditional rounding to determine the number of
The reading coach at an elementary school has 13 gift cards to distribute to their three students as a reward for time spent reading. When they calculated the standard quota for each student based on the number of minutes they student had read, the results were: 4.49 gift cards, 4.03 gift cards,
The apportionment of 70 new emergency blue lights in three parking lots is based on acreage. The standard quota for each lot is listed in the table below. Use this information to answer each question.1. Give the lower quota for each parking lot.2. Find the sum of the lower quotas.3. By how much
In the country of Imaginaria, there will be four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Suppose there will be 35 seats in the legislature of Imaginaria. Use Hamilton's method of apportionment to determine the number of seats in each state based on the populations in the
Apportionment Method \(\mathrm{V}\) has been used to allocate 125 seats to ten states as shown in the table below. Determine whether the apportionment satisfies the quota rule and justify your answer. State A State B State C State D State E State F State G Standard Quota41.26 10.70 16.00 13.11
Suppose the population of a state is 12 and the standard divisor is 0.225 .1. Find the state's standard quota.2. Decrease the standard divisor by 0.200 units and use the modified divisor to determine the modified quota for the state.3. Increase the modified divisor from part 2 by 0.100 units and
Let's return to the Imaginarian states of Fictionville, Pretendstead, Illusionham, and Mythbury. Suppose that there are going to be 35 seats in the legislature. This time use Jefferson's method of apportionment to determine the number of seats in each state based on the populations in the table
There are four states in Imaginaria: Fictionville, Pretendstead, Illusionham, and Mythbury. Assume there will be 35 seats in the legislature of Imaginaria. Use Adams's method of apportionment to determine the number of seats in each state based on the populations in the table below. How many seats
If you use Webster's method to apportion 35 legislative seats to the 4 states of Imaginaria, Fictionville, Pretendstead, Illusionham, and Mythbury, with the populations given in the table below, what is the resulting apportionment? Fictionville Pretendstead Illusionham Mythbury Total Population
You apportioned 35 legislative seats among the four states of Imaginaria using the Hamilton, Jefferson, Adams, and Webster methods of apportionment. To understand how the differences in the apportionments might affect Imaginarians, answer these questions.1. Which apportionment method would the
The following table displays the effect of increasing the size of the divisor. Observe the effect this has on the modified quotas of smaller states versus larger states and use the table to answer each question.1. When the divisor increases from 11,500 to 12,000 , how many representatives are lost
The 1900 census recorded the population of Colorado as 539,700 and that of the U.S. as \(76,212,168\).1. Calculate the standard divisor and standard quota for the State of Colorado based on a house size of 356.2. Calculate the standard divisor and standard quota for the State of Colorado based on a
Suppose that the founders of Imaginaria decide to have a parliament that apportions seats to four political parties based on the portion of the vote each party has earned. Also, suppose that Party A has 56.7 thousand votes, Party B has 38.5 thousand votes, Party C has 4.2 thousand votes, and Party
Suppose that 18 respirators are to be apportioned to three hospitals based on their capacities. The respirators are allocated based on the Hamilton method in 2020, then reallocated based on new capacities in 2021. The results are shown in the table below. How does this demonstrate the population
The country of Elbonia has three states: Mudston with a population of 866,000 ; WallaWalla with a population of 626,000 ; and Dilberta with a population of 256,000 . There are 38 seats in the Congress of Elbonia. The apportionment of the seats is determined by Hamilton's method to be: 19 for
The country of Narnia has grown from two states to three. The house size of the congress has been increased by five and the seats have been reapportioned to accommodate the new state of Chippingford. The constitution mandates the use of the Hamilton method of apportionment. Use this information and
The country of Neverland has two states: Neverwood with a population of 760,000 and Mermaids Lagoon with a population of 943,000 . The constitution of Neverland requires that the 84 congressional seats be apportioned by Hamilton's method. Neverwood has received 37 seats while Mermaids Lagoon has
New Mexico was admitted as the 47th state on January 6, 1912. Before New Mexico joined the union, the U.S. population was approximately \(76,000,000\) and the House of Representatives had 391 seats. The new state had a population of approximately 300,000 . Use this information to estimate the
Suppose that in 2016 , States \(A, B\), and \(C\) had populations of 13 million, 12 million, and 112 million, respectively. In 2020, State A has grown by 1 million residents, State B has lost 1 million residents, and State C has gained 2 million residents. Compare the apportionments in 2016 to 2020
Name three voting methods that use a ranked ballot.
Determine whether the following statement is true or false: The same ranked ballots may result in a different winner depending on which voting method is used.
Determine whether the following statement is true or false. A majority candidate is always a Condorcet candidate.
The \( \qquad \) method is a system of voting using ranked ballots in which each candidate is awarded points corresponding to the number of candidates ranked lower on each ballot.
The \( \qquad \) method is a system of voting using ranked ballots (or multiple elections) in which each candidate receives a point for each candidate they would beat in a one-on-one election and half a point for each candidate they would tie.
The \( \qquad \) method is a runoff voting system in which only the candidate(s) with the very least votes are eliminated.
Explain the differences between two-round voting and ranked-choice voting.
Which fairness criterion is violated by all four of the main ranked voting methods presented in this chapter?
Which of the four main ranked voting methods presented in this chapter satisfies the Condorcet criterion?
Which of the four main ranked voting methods presented in this section violates the majority criterion?
Which of the four main ranked voting methods presented in this section violates the monotonicity criterion?
According to Arrow's Impossibility Theorem, which of the four main ranked voting methods presented in this chapter violate at least one of the fairness criteria?
Determine whether the following statement is true or false and explain your reasoning: Any ranked election that violates the majority criterion also violates the Condorcet criterion.
Does Arrow's Impossibility Theorem apply to approval voting? Why or why not?
Arrow's Impossibility Theorem guarantees that ranked voting systems always lead to unfair elections.Determine whether each statement is true or false. Explain your reasoning.
Approval voting is in the class of voting systems called Cardinal Voting systems.Determine whether each statement is true or false. Explain your reasoning.
Student 1 says that the number of units of \(A\) is the product of the number of units of \(B\) times the ratio of \(A\) to \(B\). Student 2 says that the number of units of \(A\) is the quotient of the number of units of \(B\) divided by the ratio of \(A\) to \(B\).
Student 1 says that the number of units of \(A\) is the quotient of the number of units of \(B\) divided by the ratio of \(B\) to \(A\).Student 2 says that the number of units of \(B\) is the quotient of the number of units of \(A\) divided by the ratio of \(A\) to B.
Suppose there are 110 of item \(A\) and 55 of item \(B\).Student 1 says the ratio of \(A\) to \(B\) is \(\frac{1}{2}\) Student 2 says the ratio of \(B\) to \(A\) is \(\frac{1}{0.5}\).
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