Question 1(8 points)

Assuming U represents the universal set and S represents any set, which of the following is a correct set property?

Question 1 options:

	S U = S
	S =
	S U = U
	S =

Question 2(8 points)

Among the sets { {a, b, c, d} }, { {a, b}, {c, d} } and {a, b, c, d}, which has the highest cardinality?

Question 2 options:

	{ {a, b}, {c, d} }
	{a, b, c, d}
	All three have the same cardinality
	{ {a, b, c, d} }

Question 3(8 points)

Assume set A = {1, 3, 5, 9, 15} and B = {1, 5, 9, 11, 13}, which of the following represents the set A B?

Question 3 options:

	{1, 3, 5, 9, 11, 13, 15}
	{1, 5, 9, 13, 15}
	{3, 15}
	{11, 13}

Question 4(8 points)

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

Question 4 options:

	Step 5
	Step 4
	Step 3
	Steps 1 & 2

Question 5(8 points)

Which of the following best describes the necessary requirements for of every proof by induction?

Question 5 options:

	Some proofs do not require any base case or inductive case
	Every proof must have exactly one base case and an optional inductive case
	In every proof, both the base and inductive cases are optional
	Every proof must have at least one base case and one inductive case

Question 6(8 points)

Which of the following is the proper way to begin a proof by contradiction of the theorem "p q, p q p + q "?

Question 6 options:

	Suppose there exist two rational numbers whose sum is irrational
	Suppose there exist two irrational numbers whose sum is rational
	Suppose the sum of every two irrational numbers is rational
	Suppose the sum of every two rational numbers is irrational

Question 7(8 points)

Removing which of the following edges in the above graph will create a spanning tree?

Question 7 options:

	Edges ad and de
	Only edge ad
	Edges ad, df and de
	Edges ad and df

Question 8(8 points)

Which of the following is true about the above graph?

Question 8 options:

	The graph has two parallel edges
	The graph is connected
	The graph has one isolated vertex
	The graph has no circuits

Question 9(8 points)

Which of the following is true about spanning trees of the above graph?

Question 9 options:

	It has no possible spanning tree because it contains an isolated vertex
	It has many possible spanning trees
	It has only one possible spanning tree
	It has no possible spanning tree because it is disconnected

Question 10(8 points)

Which of the following is a postorder traversal of the above tree?

Question 10 options:

	a b d g e c f
	g d b e a c f
	g d e b f c a
	a b c d e f g

Question 11(8 points)

How many even integers are there between 300 and 500 inclusive?

Question 11 options:

	50
	101
	51
	100

Question 12(8 points)

In a Java programming class of 33 students, 18 are male. 10 students in that class received an A on the final project. 5 of those 10 were male. Choosing a student at random, what is the probability of choosing either a male student or a student who received an A on the final project?

Question 12 options:

	28/33
	1/3
	5/16
	23/33

Question 13(8 points)

Suppose a computer science major is required to complete 3 different programming projects and is offered 7 projects to choose from. How many choices are available?

Question 13 options:

	210
	35
	6
	22

Question 14(8 points)

Which of the following is true about the above arrow diagram?

Question 14 options:

	It does not define a relation because 3 is unrelated to any element of B
	It does not define a relation because A has more elements than B
	It properly defines a relation
	It does not define a relation because two of the arrows cross

Question 15(8 points)

Assume A is the set of positive integers less than 10 and B is the set of positive integers less than or equal to 20, and R is a relation from A to B defined as follows: R = {(a, b) | a A, b B, a is divisible by 4 b = 2a}. Which of the following ordered pairs belongs to that relation?

Question 15 options:

	(16, 32)
	(12, 6)
	(8, 16)
	(6, 12)

Question 16(8 points)

Given the relation R = {(n, m) | n, m , n = m}. Which of the following relations defines the inverse of R?

Question 16 options:

	R = {(n, m) | n, m , n
	R = {(n, m) | n, m , n m}
	R = {(n, m) | n, m , n m}
	R = {(n, m) | n, m , n = m}

Question 17(8 points)

Which of the following is the negation of the predicate calculus statement "x , x 0 1/x

Question 17 options:

	x , x 0 1/x
	x , x 0 1/x
	x , x
	x , x

Question 18(8 points)

Assuming P is the set of professors and S is the set of students, which of the following is the correct predicate calculus translation of the sentence "At least one professor teaches some student"?

Question 18 options:

	p P, s S, p teaches s
	p P, s S, p teaches s
	p P, s S, p teaches s
	p P, s S, p teaches s

Question 19(8 points)

Which of the following is the negation of the following sentence "No one loves spinach"?

Question 19 options:

	At least one person loves spinach
	Everyone doesn't love spinach
	Someone doesn't loves spinach
	Everyone loves spinach

Question 20(8 points)

Which of the following functions is a one-to-one correspondence?

Question 20 options:

	f: , where f(n) = n + 1
	f: , where f(n) = 5n + 2
	f: , where f(x) = 2x - 1
	f: , where f(n) = 6n

Question 21(8 points)

Which of the following is a synonym for a bijection?

Question 21 options:

	a one-to-one function
	an algebraic function
	a one-to-one correspondence
	an onto function

Question 22(8 points)

Given functions f: , where f(x) = x + 5x + 2, and g: , where g(x) = 6x + 1, which of the following functions is fg?

Question 22 options:

	fg: , where fg(x) = x + 11x + 2
	fg: , where fg(x) = 6x + 30x + 13
	fg: , where fg(x) = 36x + 42x + 8
	fg: , where fg(x) = 6x + 42x + 13

Question 23(8 points)

Which of the following is logically equivalent to p q?

Question 23 options:

	q p
	(q p)
	p q
	(p q)

Question 24(8 points)

Given the premises p q and r, which of the following is a correct inference?

Question 24 options:

	(p q) r)
	p (q r)
	p (q r)
	(p q) r

Question 25(8 points)

Which of the following describes the proposition (q (q (p p)))?

Question 25 options:

	It is a tautology
	It is both a tautology and a contradiction
	It is neither a tautology nor a contradiction
	It is a contradiction

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