Quiz3_Answers
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Subject
Mathematics
Date
Feb 20, 2024
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Which of the following best describes the necessary requirements for the base case(s) of every proof by induction?
Some proofs do not require any base case
No proofs require any base case
Every proof must have exactly one base case
X
Every proof must have at least one base case
Question 2
10 / 10
points
When prove that all positive integers less than 4 are prime numbers by showing that 1, 2 and 3 are prime numbers, which
of the following kinds of proof are we using?
X
Exhaustive proof
Proof by induction
Trivial proof
Proof by contradiction
Question 3
0 / 10
points
In the proof by induction of the theorem ∀
n ∈
ℕ
, n ≥ 1 → 2 + 4 + ... + 2n = n² + n, which of the following steps would consitute the correct final step of the proof?
Stating that (k² + 2k + 1) + (k + 1) = k² + (k + 1)
X
Stating that (k² + 2k + 1) + (k + 1) = (k + 1)² + (k + 1)
Stating that (k² + 2k + 1) + (k + 1) = (k + 1)² + (k² + 1)
Stating that (k² + 2k + 1) + (k + 1) = (k + 1)² + k
Question 4
10 / 10
points
In the proof shown above, which steps require the definition of
rational numbers as their justification?
X
Steps 1, 2 & 5
Steps 1 & 3
Steps 1 & 2
Steps 3 & 5
Question 5
10 / 10
points
Suppose that in proving that the sum of two odd numbers is even, we begin by saying "Assume that the sum of two odd numbers is odd". Which of the following types of proof are we using?
X
Proof by contradiction
Proof by contraposition
Direct proof
Proof by cases
Question 6
10 / 10
points
Which of the following is the proper way to begin a proof by contradiction of the theorem "
∀
p ∀
q, p ∈
ℚ
∧
q ∈
ℚ
→ pq ∈
ℚ
"?
Suppose the product of every two irrational numbers is rational
Suppose the product of every two rational numbers is irrational
Suppose there exist two irrational numbers whose product is rational
X
Suppose there exist two rational numbers whose product is irrational
Question 7
0 / 10
points
Which of the following best describe the contradiction in the above proof by contradiction?
Line 3 contradicts line 1
Line 4 contradicts line 2
Line 3 contains the entire contradiction
X
Line 4 contradicts line 1
Question 8
10 / 10
points
Which of the following integer counterexamples provides a disproof of the universal statement "
∀
n ∈
ℤ
, n ≥ -n"?
n = 0
n = 10
X
n = -1
n = 1
Question 9
10 / 10
points
In the proof shown above, what is the correct justification in the eight step?
Modus ponens
Modus tollens
X
Conjunctive addition
Contraposition
Question 10
0 / 10
points
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In the proof shown above, what is the correct justification for the third step?
Definition of odd integers
Integers are closed under addition
X
Algebra
Definition of even integers