Question

When we prove a statement about all natural numbers in two parts, first for all odd numbers, then for all even numbers, which of the following kinds of proof are we using? Proof by contraposition Proof by contradiction Direct proof Proof by cases 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? 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? 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 ? ?, ?m ? ?, n = 2m"? 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 ? ? ?, 1/q ? q"? q = 1 q = ½ q = -½ q = -1 Question 6 (10 points) Which of the following is true about the inductive hypothesis"? 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 7 (10 points) In the proof shown above, what is the correct justification for the seventh step? Modus tollens Hypothetical syllogism Modus ponens Contraposition 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? 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? 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 ? 1 ? 2 + 4 + ... + 2n = n² + n, which of the following steps requires the inductive hypothesis as its justification? 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)

          When we prove a statement about all natural numbers in two
parts, first for all odd numbers, then for all even numbers, which
of the following kinds of proof are we using?
 
 
Proof by contraposition
 
Proof by contradiction
 
Direct proof
 
Proof by cases
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?
 
 
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?
 
 
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 ? ?, ?m ? ?, n = 2m"?
 
 
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 ? ? ?, 1/q ? q"?
 
q = 1
 
q = ½
 
q = -½
 
q = -1
Question 6 (10 points)
 
Which of the following is true about the inductive
hypothesis"?
 
 
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 7 (10 points)
 
In the proof shown above, what is the correct justification for
the seventh step?
 
 
Modus tollens
 
Hypothetical syllogism
 
Modus ponens
 
Contraposition
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?
 
 
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?
 
 
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 ? 1 ? 2 + 4 +
... + 2n = n² + n, which of the following steps requires the
inductive hypothesis as its justification?
 
 
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 Diane A.

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