A

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let x be a real number. Prove the following statements are equivalent:

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let x be a real number. Prove the following statements are equivalent:

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let x be a real number. Prove the following statements are equivalent:

D

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let x be a real number. Prove the following statements are equivalent:

A

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Define f(x) and g(x) as follows for positive integers:

Which of the following are true statements? Justify your answers.

1. f(x) is O(g(x)).
2. g(x) is O(f(x)).
3. f(x) is Ω(g(x)).
4. g(x) is Ω(f(x)).

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Define f(x) and g(x) as follows for positive integers:

Which of the following are true statements? Justify your answers.

1. f(x) is O(g(x)).
2. g(x) is O(f(x)).
3. f(x) is Ω(g(x)).
4. g(x) is Ω(f(x)).

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Define f(x) and g(x) as follows for positive integers:

Which of the following are true statements? Justify your answers.

1. f(x) is O(g(x)).
2. g(x) is O(f(x)).
3. f(x) is Ω(g(x)).
4. g(x) is Ω(f(x)).

D

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Define f(x) and g(x) as follows for positive integers:

Which of the following are true statements? Justify your answers.

1. f(x) is O(g(x)).
2. g(x) is O(f(x)).
3. f(x) is Ω(g(x)).
4. g(x) is Ω(f(x)).

A

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Questions:

1. In a certain town of 1000 people, 600 people have private health insurance and 300 are eligible for a state health insurance plan. 120 people who have private health insurance are also eligible for the state plan. How many people who do not have private health insurance are also not eligible for the state plan?
2. Julio has a memory stick with 43MB of remaining space. He has four files he’d like to put on the stick, with sizes 9MB, 15MB, 19MB, and 23MB. How many different combinations of the files could he put on the stick? (Note: he doesn’t have to fill the stick. He can put none of the files on the stick if he wants.)

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Questions:

1. In a certain town of 1000 people, 700 people have private health insurance and 200 are eligible for a state health insurance plan. 140 people who have private health insurance are also eligible for the state plan. How many people who do not have private health insurance are also not eligible for the state plan?
2. Jane has a memory stick with 42MB of remaining space. She has four files she’d like to put on the stick, with sizes 9MB, 15MB, 19MB, and 23MB. How many different combinations of the files could she put on the stick? (Note: she doesn’t have to fill the stick. She can put none of the files on the stick if she wants.)

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Questions:

1. In a certain town of 1000 people, 800 people have private health insurance and 300 are eligible for a state health insurance plan. 240 people who have private health insurance are also eligible for the state plan. How many people who do not have private health insurance are also not eligible for the state plan?
2. Jing has a memory stick with 41MB of remaining space. She has four files she’d like to put on the stick, with sizes 3MB, 15MB, 19MB, and 23MB. How many different combinations of the files could she put on the stick? (Note: she doesn’t have to fill the stick. She can put none of the files on the stick if she wants.)

A

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Three divisions of five teams play a in league. At the end of the season, the top team in each division qualifies for the playoffs. In addition, the best of the remaining teams also qualifies for the playoffs. How many different sets of teams can make the playoffs? (Express your answer as a product and/or sum of integers.)

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Three divisions of six teams play a in league. At the end of the season, the top team in each division qualifies for the playoffs. In addition, the best of the remaining teams also qualifies for the playoffs. How many different sets of teams can make the playoffs? (Express your answer as a product and/or sum of integers.)

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Three divisions of four teams play a in league. At the end of the season, the top team in each division qualifies for the playoffs. In addition, the best of the remaining teams also qualifies for the playoffs. How many different sets of teams can make the playoffs? (Express your answer as a product and/or sum of integers.)

A

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Questions:

1. How many solutions are there to the equation

   (1)

   where each x_i is a nonnegative integer? (Express your answer as a product and/or sum of integers.)

2. How many nonnegative integer solutions are there to (3) with x₁ ≥ 1 and x₃ ≥ 3? (Express your answer as a product and/or sum of integers.)

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Questions:

1. How many solutions are there to the equation

   (2)

   where each x_i is a nonnegative integer? (Express your answer as a product and/or sum of integers.)

2. How many nonnegative integer solutions are there to (3) with x₂ ≥ 2 and x₃ ≥ 3? (Express your answer as a product and/or sum of integers.)

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Questions:

1. How many solutions are there to the equation

   (3)

   where each x_i is a nonnegative integer? (Express your answer as a product and/or sum of integers.)

2. How many nonnegative integer solutions are there to (3) with x₂ ≥ 2 and x₄ ≥ 4? (Express your answer as a product and/or sum of integers.)

B

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Questions:

Let a_n be the number of strings of length n containing only 0s, 1s, and 2s, with no consecutive 0s. So 012101 and 111111 are valid strings of length 6, but 100202 is not. Show that

for n ≥ 3. (Hint: Let a_n⁰ denote the number of strings of length n ending in a 0, and let a _n¹² denote the number of strings of length n ending in a 1 or a 2. We have a_n = a_n⁰ + a _n¹². Write a _n in terms of a_n-1⁰ and a _n-1¹², and work from there.)

A

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Question:
Find the solution to a_n = 5a_n-1 - 6a_n-2 for n ≥ 2 with a₀ = 4 and a₁ = 11.

B

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Question:
Find the solution to a_n = 5a_n-1 - 6a_n-2 for n ≥ 2 with a₀ = 4 and a₁ = 9.

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Find the solution to a_n = 5a_n-1 - 6a_n-2 for n ≥ 2 with a₀ = 5 and a₁ = 11.

B

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Question:
Draw two bipartite simple graphs with six vertices with degree sequence 3,3,2,2,1,1, which are not isomorphic. Why are they not isomorphic?

A

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Question:
Consider the following graph with 6 vertices:

1. Does this graph have a Euler path? (Just answer “Yes” or “No”, with justification. You do not need to draw the path, if it exists.)
2. Is this graph planar? (Justify your answer: if it is planar, draw it in the plane; if it is not planar use an appropriate theorem from class.)

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Consider the following graph with 6 vertices:

1. Does this graph have a Euler path? (Just answer “Yes” or “No”, with justification. You do not need to draw the path, if it exists.)
2. Is this graph planar? (Justify your answer: if it is planar, draw it in the plane; if it is not planar use an appropriate theorem from class.)

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Consider the following graph with 6 vertices:

1. Does this graph have a Euler path? (Just answer “Yes” or “No”, with justification. You do not need to draw the path, if it exists.)
2. Is this graph planar? (Justify your answer: if it is planar, draw it in the plane; if it is not planar use an appropriate theorem from class.)

A

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let R₁ and R₂ be two relations on a set A represented by the matrices

1. Is either relation transitive?
2. Find the matrix representing the relation R₁ ∘ R₂.

B

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let R₁ and R₂ be two relations on a set A represented by the matrices

1. Is either relation transitive?
2. Find the matrix representing the relation R₁ ∘ R₂.

C

No collaboration is permitted on the quizzes. Calculators are not permitted. Please ring your section below:

Question:
Let R₁ and R₂ be two relations on a set A represented by the matrices

1. Is either relation transitive?
2. Find the matrix representing the relation R₁ ∘ R₂.