Question

Question 2 (10 points) Suppose we prove that the proposition p → q is true by showing instead that ¬q → ¬p is true. Which of the following types of proofs have we used? Question 2 options: Direct proof Trivial proof Proof by contraposition Proof by contradiction Question 3 (10 points) In the proof shown above, which step(s) require(s) Lemma 1 as part of its justification? Question 3 options: Step 4 Step 5 Step 3 Steps 1 & 2 Question 4 (10 points) Which of the following is the proper way to begin a proof by contradiction of the theorem "∀n ∈ ℕ, ∃m ∈ ℕ, n = 2m"? Question 4 options: Suppose there is some natural number that is not equal to twice any natural number Suppose there is some natural number that is equal to twice every natural number Suppose there is some natural number that is not equal to twice some natural number Suppose every natural number is not equal to twice any natural number Question 5 (10 points) Which of the following rational number examples provides a disproof of the universal statement "∀q ∈ ℚ, 1/q ≤ q"? Question 5 options: q = 1 q = ½ q = -½ q = -1 Question 6 (10 points) Which of the following is true about the inductive hypothesis? Question 6 options: It is stated at the conclusion of the proof and used as a justification for one of the previous steps It is stated at the beginning of the proof and used as a justification for one of the subsequent steps It is stated at the conclusion of the proof but never as a justification for any of the previous steps It is stated at the beginning of the proof but never as a justification for any of the subsequent steps Question 8 (10 points) Which line in the above proof by contradiction relies on closure of the integers under negation as part of its justification? Question 8 options: Line 4 Line 2 Line 1 Line 3 Question 9 (10 points) In the proof shown above, what is the correct justification for the first step? Question 9 options: Integers are closed under addition Algebra Definition of odd integers Definition of even integers Question 10 (10 points) In the proof by induction of the theorem ∀n ∈ ℕ, n ≥ 1 → 2 + 4 + ... + 2n = n² + n, which of the following steps requires the inductive hypothesis as its justification? Question 10 options: Stating that (k + 1)² + (k + 1) = (k² + 2k + 1) + (k + 1) Stating that (2 + 4 + ... + k) + 2(k + 1) = (k² + k) + 2(k + 1) Stating that (k² + k) + 2(k + 1) = k² + 3k + 2 Stating that (k² + 2k + 1) + (k + 1) = (k + 1)² + (k + 1)

          Question 2 (10 points)
Suppose we prove that the proposition p → q is true by showing instead that ¬q → ¬p is true. Which of the following types of proofs have we used?

Question 2 options:
Direct proof
Trivial proof
Proof by contraposition
Proof by contradiction

Question 3 (10 points)
In the proof shown above, which step(s) require(s) Lemma 1 as part of its justification?

Question 3 options:
Step 4
Step 5
Step 3
Steps 1 & 2

Question 4 (10 points)
Which of the following is the proper way to begin a proof by contradiction of the theorem "∀n ∈ ℕ, ∃m ∈ ℕ, n = 2m"?

Question 4 options:
Suppose there is some natural number that is not equal to twice any natural number
Suppose there is some natural number that is equal to twice every natural number
Suppose there is some natural number that is not equal to twice some natural number
Suppose every natural number is not equal to twice any natural number

Question 5 (10 points)
Which of the following rational number examples provides a disproof of the universal statement "∀q ∈ ℚ, 1/q ≤ q"?

Question 5 options:
q = 1
q = ½
q = -½
q = -1

Question 6 (10 points)
Which of the following is true about the inductive hypothesis?

Question 6 options:
It is stated at the conclusion of the proof and used as a justification for one of the previous steps
It is stated at the beginning of the proof and used as a justification for one of the subsequent steps
It is stated at the conclusion of the proof but never as a justification for any of the previous steps
It is stated at the beginning of the proof but never as a justification for any of the subsequent steps

Question 8 (10 points)
Which line in the above proof by contradiction relies on closure of the integers under negation as part of its justification?

Question 8 options:
Line 4
Line 2
Line 1
Line 3

Question 9 (10 points)
In the proof shown above, what is the correct justification for the first step?

Question 9 options:
Integers are closed under addition
Algebra
Definition of odd integers
Definition of even integers

Question 10 (10 points)
In the proof by induction of the theorem ∀n ∈ ℕ, n ≥ 1 → 2 + 4 + ... + 2n = n² + n, which of the following steps requires the inductive hypothesis as its justification?

Question 10 options:
Stating that (k + 1)² + (k + 1) = (k² + 2k + 1) + (k + 1)
Stating that (2 + 4 + ... + k) + 2(k + 1) = (k² + k) + 2(k + 1)
Stating that (k² + k) + 2(k + 1) = k² + 3k + 2
Stating that (k² + 2k + 1) + (k + 1) = (k + 1)² + (k + 1)

Added by Brian T.

Question

Please give Ace some feedback