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Questions and Answers

What is the time complexity of the function has_dominator in the worst-case scenario?

O(n log n)

O(1)

O(n^2) (correct)

O(n)

Which inequality best describes the behavior of the function 𝑓(𝑛) = 𝑛^4 with respect to 𝑔(𝑛) = 2^𝑛 in big-omega notation?

𝑓 does not belong to either Ω or O of 𝑔 (correct)

𝑓 ∈ Ω(𝑔)

𝑓 ∈ O(𝑔)

𝑓 is both Ω and O of 𝑔

In the function has_dominator, what condition is being checked within the nested loops?

If the list contains duplicate elements

If the elements are in ascending order

If any element is equal to the maximum element

If one element is less than another element (correct)

For a natural number 𝑚 > 1, what can be concluded about the prime number 𝑝 in relation to the expression (𝑚! + 2) ≤ 𝑝 ≤ (𝑚! + 𝑚)?

No prime number 𝑝 can exist within that range Signup and view all the answers

Which of the following statements correctly describes the summation formula provided in the content?

It computes the total number of comparisons in has_dominator Signup and view all the answers

Given that (𝑛 + 1)^4 = 𝑛^4 + 4𝑛^3 + 6𝑛^2 + 4𝑛 + 1, what mathematical property does this equation illustrate?

It represents the binomial theorem Signup and view all the answers

In the context of algorithm analysis, what does big-oh notation represent?

An upper limit on the time complexity Signup and view all the answers

Which of the following statements correctly reflects the natural number properties used in the context provided?

Factorial growth outpaces linear growth of integers Signup and view all the answers

What is the binary representation of the decimal number 61?

111101 Signup and view all the answers

What is the consequence of the inductive step being improperly handled in the proof?

The assumption does not hold for larger n. Signup and view all the answers

Which pair of predicates can correctly demonstrate the condition given in the statement? (∀𝑛 ∈ ℕ, 𝑃(𝑛) ⇒ 𝑄(𝑛)) ⇒ (∃𝑚 ∈ ℕ, 𝑃(𝑚) ∧ 𝑄(𝑚))

𝑃(𝑛): 𝑛 is a multiple of 2, 𝑄(𝑛): 𝑛 is a multiple of 4 Signup and view all the answers

In the proof by induction, how is the base case verified for P(1)?

By proving a graph with one vertex has no edges. Signup and view all the answers

For a complete bipartite graph G = (V, E), what can be inferred if |V1| = ⌊|V|/2⌋?

Every vertex in V1 connects to every vertex in V2. Signup and view all the answers

Why is P stated as connected in the context of graph theory?

Because all vertices can be reached from any starting vertex. Signup and view all the answers

What is a necessary condition for a graph G to be considered connected when |E| = |V| - 1?

There must be no isolated vertices. Signup and view all the answers

What will be the formula for 𝑖(𝑠), representing the value of 𝑖 after 𝑠 iterations in the doubler function?

𝑖(𝑠) = 2^𝑠 Signup and view all the answers

What role does the assumption P(n) play during the proof process?

It acts as the induction hypothesis. Signup and view all the answers

What is the correct general formula for the number of iterations of the loop in the doubler function for an input 𝑛?

2^k < n, where k is the number of iterations Signup and view all the answers

Which of the following statements correctly expresses that there is exactly one dog that has fleas and lives in the arctic using the defined predicates?

∃𝑑 ∈ 𝐷 (𝐴(𝑑) ∧ 𝐹(𝑑) ∧ ∀𝑑' (𝐹(𝑑') ∧ 𝐴(𝑑') ⇒ 𝑑 = 𝑑')) Signup and view all the answers

In proving that path P is connected, what must be shown explicitly?

That there is a route between every pair of vertices in P. Signup and view all the answers

In the context of the proof, what does it imply when a complete bipartite graph contains two partitions?

Every vertex in one partition connects to all vertices in the other partition. Signup and view all the answers

Study Notes

Question 1: Short Answer

Part (a): Binary Representation of 61: The binary representation of 61 is 111101.
Part (b): Predicates P(n) and Q(n): Define predicates to make (∀𝑛 ∈ ℕ, 𝑃(𝑛) ⇒ 𝑄(𝑛)) ⇒ (∃𝑚 ∈ ℕ, 𝑃(𝑚) ∧ 𝑄(𝑚)) true and (∃𝑛 ∈ ℕ, 𝑃(𝑛) ⇒ 𝑄(𝑛)) ⇒ (∀𝑚 ∈ ℕ, 𝑃(𝑚) ∧ 𝑄(𝑚)) false. One possible solution involves creating conditions where the implication is always true in the first statement but only true for some n in the second.
Part (c): Moduli: The number 𝑖 that satisfies 𝑖 ≡ 2 (mod 3) and 𝑖 ≡ 4 (mod 5) is 14.
Part (d): doubler(17) Loop Iterations: The loop in doubler(17) iterates 3 times.
Part (e): Formula for i(s): The formula for 𝑖(𝑠), the value of 𝑖 after 𝑠 iterations of the doubler loop, is 𝑖(𝑠) = 2^s.
Part (f): Iterations for doubler(n): The number of iterations for doubler(n) is ⌈log₂(√𝑛)⌉.
Part (g): Predicate Logic Statement for Dogs: The statement "There is exactly one dog that has fleas and lives in the arctic" can be expressed as: ∃𝑑 ∈ 𝐷, (𝐴(𝑑) ∧ 𝐹(𝑑)) ∧ (∀𝑑′ ∈ 𝐷, (𝐴(𝑑′) ∧ 𝐹(𝑑′)) ⇒ 𝑑 = 𝑑′).
Part (h): Units Digit of 3⁹⁸³¹²: The units digit of 3⁹⁸³¹² is 1.

Question 2: Number Theory

Part (a): Moduli Proof: To prove that if 𝑎 ≡ 𝑏 (mod 17) and 𝑎 ≡ 𝑏 (mod 19), then 𝑎 ≡ 𝑏 (mod 17 × 19), utilize the given theorem about prime factorization and the properties of modular arithmetic. Since 17 and 19 are prime, if they both divide (a-b), then their product (17*19) must also divide (a-b).
Part (b): Prime-Free Range: Prove that there is no prime number 𝑝 such that (𝑚!+ 2) ≤ 𝑝 ≤ (𝑚!+ 𝑚) for natural number 𝑚 > 1. All integers within that range are composite because they are divisible by integers from 2 to m..
Part (c): Faulty Proof Analysis: The inductive proof for P(n): "∀𝐺 = (𝑉 , 𝐸), |𝐸| = |𝑉 | − 1 ⇒ 𝐺 is connected" breaks down at n=3. A graph with 3 vertices and 2 edges might not be connected, showing the inductive step is flawed for this specific case.

Question 3: Big-Omega/Big-Oh

Part (a): Proof that f ∉ Ω(g): Prove that 𝑛⁴ ∉ Ω(2𝑛) by showing that no constants c_Ω and n_Ω exist to satisfy the definition of big-Omega for all n ≥ n_Ω.
Part (b): Proof that f ∈ 𝑂(g): Prove that 𝑛⁴ ∈ 𝑂(2^𝑛) by finding constants c_𝑂 and n_𝑂 that satisfy the Big-Oh definition for all n ≥ n_𝑂.

Question 4: Runtime Analysis

Part (a): Upper Bound for has_dominator: The worst-case runtime for has_dominator is 𝑂(𝑛²). This is because of its nested loops, leading to roughly n(n-1)/2 comparisons in the worst case, which is of the order of n². A formal proof would involve showing that the number of operations is bounded above by a quadratic function of n for all sufficiently large n.

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Questions and Answers

What is the time complexity of the function has_dominator in the worst-case scenario?

Which inequality best describes the behavior of the function 𝑓(𝑛) = 𝑛^4 with respect to 𝑔(𝑛) = 2^𝑛 in big-omega notation?

In the function has_dominator, what condition is being checked within the nested loops?

For a natural number 𝑚 > 1, what can be concluded about the prime number 𝑝 in relation to the expression (𝑚! + 2) ≤ 𝑝 ≤ (𝑚! + 𝑚)?

Which of the following statements correctly describes the summation formula provided in the content?

Given that (𝑛 + 1)^4 = 𝑛^4 + 4𝑛^3 + 6𝑛^2 + 4𝑛 + 1, what mathematical property does this equation illustrate?

In the context of algorithm analysis, what does big-oh notation represent?

Which of the following statements correctly reflects the natural number properties used in the context provided?

What is the binary representation of the decimal number 61?

What is the consequence of the inductive step being improperly handled in the proof?

Which pair of predicates can correctly demonstrate the condition given in the statement? (∀𝑛 ∈ ℕ, 𝑃(𝑛) ⇒ 𝑄(𝑛)) ⇒ (∃𝑚 ∈ ℕ, 𝑃(𝑚) ∧ 𝑄(𝑚))

In the proof by induction, how is the base case verified for P(1)?

For a complete bipartite graph G = (V, E), what can be inferred if |V1| = ⌊|V|/2⌋?

Why is P stated as connected in the context of graph theory?

What is a necessary condition for a graph G to be considered connected when |E| = |V| - 1?

What will be the formula for 𝑖(𝑠), representing the value of 𝑖 after 𝑠 iterations in the doubler function?

What role does the assumption P(n) play during the proof process?

What is the correct general formula for the number of iterations of the loop in the doubler function for an input 𝑛?

Which of the following statements correctly expresses that there is exactly one dog that has fleas and lives in the arctic using the defined predicates?

In proving that path P is connected, what must be shown explicitly?

In the context of the proof, what does it imply when a complete bipartite graph contains two partitions?

Study Notes

Question 1: Short Answer

Question 2: Number Theory

Question 3: Big-Omega/Big-Oh

Question 4: Runtime Analysis

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