uestion 1(10 points)

Saved

Which of the following best describe the contradiction in the above proof by contradiction?

Question 1 options:

	Line 4 contradicts line 1
	Line 4 contradicts line 2
	Line 3 contradicts line 1
	Line 3 contains the entire contradiction

Question 2(10 points)

Saved

In the proof shown above, which step requires algebraic substition as its justification?

Question 2 options:

	Step 5
	Step 3
	Step 2
	Step 4

Question 3(10 points)

Saved

In the proof by induction of the theorem n , n 1 2 + 4 + ... + 2n = n + n, which of the following would be the proper way to state the inductive hypothesis?

Question 3 options:

	Assume for n = k, 2 + 4 + ... + 2k = k + k
	Assume for n = k, 2 + 4 + ... + 2(k + 1) = (k + 1) + k
	Assume for n = k + 1, 2 + 4 + ... + 2k = k + k
	Assume for n = k, 2 + 4 + ... + 2(k + 1) = k + k

Question 4(10 points)

Suppose in proving the proposition p q, we show that p is false. What type of proof have we used?

Question 4 options:

	Vacuous proof
	Trivial proof
	Direct proof
	Proof by contradiction

Question 5(10 points)

Which of the following real number counterexamples provides a disproof of the universal statement "x , 10x x"?

Question 5 options:

	x = -10
	x = 5
	x = 10
	x = 0

Question 6(10 points)

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?

Question 6 options:

	Proof by cases
	Direct proof
	Proof by contradiction
	Proof by contraposition

Question 7(10 points)

In the proof shown above, what is the correct justification for the seventh step?

Question 7 options:

	Modus tollens
	Contraposition
	Modus ponens
	Hypothetical syllogism

Question 8(10 points)

Which of the following best describes the necessary requirements for the base case(s) of every proof by induction?

Question 8 options:

	Every proof must have exactly one base case
	Some proofs do not require any base case
	No proofs require any base case
	Every proof must have at least one base case

Question 9(10 points)

In the proof shown above, what is the correct justification for the third step?

Question 9 options:

	Definition of odd integers
	Definition of even integers
	Algebra
	Integers are closed under addition

Question 10(10 points)

Assuming the predicate E(n) means n is even, which of the following is the proper way to begin a proof by contradiction of the theorem "n , m , E(n) E(m) E(n + m)"?

Question 10 options:

	Suppose there exist two even integers whose sum is odd
	Suppose that the sum of every two odd integers is even
	Suppose that the sum of every two even integers is odd
	Suppose there exist two odd integers whose sum is even