WGU C960 Discrete Mathematics II Exam |
Objective Assessment Revision 2026 | All
Questions And Solutions (Rationales) | Exam
Prep Q&A 2026/2027
1. Which statement is logically equivalent to the negation of the proposition “If a graph is
connected, then every vertex is reachable from every other vertex”?
A. The graph is connected, and every vertex is reachable from every other vertex.
B. The graph is not connected, or some vertex is reachable from every other vertex.
C. The graph is connected, and there exist two vertices such that one is not reachable from
the other.
D. The graph is not connected, and no vertex is reachable from any other vertex.
Rationale: The negation of an implication p → q is p ∧ ¬q. Therefore, the graph is connected
while at least one pair of vertices fails the reachability condition.
2. A software engineer models a system using a relation R on the set of all employees. The
relation holds when two employees belong to the same department. Which property must R
satisfy?
A. Reflexivity, symmetry, and transitivity.
B. Antisymmetry and irreflexivity.
C. Symmetry but not transitivity.
D. Transitivity but not reflexivity.
Rationale: Every employee belongs to the same department as themselves, so the relation is
reflexive. If employee a shares a department with b, then b shares it with a, establishing
symmetry. If a shares a department with b and b shares one with c, all three belong to the
same department, establishing transitivity.
3. A database contains 80 records. Exactly 35 records satisfy condition A, 30 satisfy
condition B, and 12 satisfy both conditions. How many records satisfy at least one
condition?
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,A. 53
B. 65
C. 77
D. 47
Rationale: By inclusion-exclusion, |A ∪ B| = |A| + |B| − |A ∩ B| = 35 + 30 − 12 = 53.
Therefore, the correct answer is A.
4. A security system generates a four-character access code using digits 0 through 9.
Repetition is permitted, and leading zeros are valid. How many distinct access codes are
possible?
A. 1,000
B. 10,000
C. 5,040
D. 40,000
Rationale: Each of the four positions has 10 independent choices. The multiplication principle
gives 10 × 10 × 10 × 10 = 10,000 possible codes.
5. A project manager must assign four distinct employees to four distinct roles. There are
nine eligible employees, and each employee can hold only one role. How many assignments
are possible?
A. 126
B. 3,024
C. 6,561
D. 3024 divided by 4
Rationale: Because the roles are distinct and employees cannot be reused, order matters. The
number of assignments is the permutation P(9,4) = 9 × 8 × 7 × 6 = 3,024.
6. A student argues that the proposition p → q is equivalent to q → p. Which statement
correctly evaluates this claim?
A. The claim is correct whenever p and q are both true.
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,B. The claim is correct because implications are symmetric.
C. The claim is false in general; the equivalent form of p → q is ¬q → ¬p.
D. The claim is false because p → q is always equivalent to ¬p → q.
Rationale: An implication is logically equivalent to its contrapositive, ¬q → ¬p, not generally
to its converse, q → p. For example, a true implication can have a false converse.
7. A computing system represents a set of integers using the property P(x): x is even. Which
expression correctly represents the set of integers that are not even?
A. {x ∈ ℤ : P(x)}
B. {x ∈ ℤ : ¬P(x)}
C. {x ∈ ℤ : P(x) ∧ ¬P(x)}
D. {x ∈ ℤ : P(x) → ¬P(x)}
Rationale: The complement of the set of even integers within the integers consists of all
integers for which the predicate “x is even” is false. These are precisely the odd integers.
8. A graph has six vertices, and every vertex has degree three. How many edges does the
graph contain?
A. 18
B. 12
C. 9
D. 6
Rationale: The handshaking lemma states that the sum of vertex degrees equals twice the
number of edges. Thus, 6 × 3 = 2|E|, giving |E| = 9.
9. A developer wants to prove that the sum of the first n positive integers is n(n + 1)/2.
Which approach is most directly suited to proving the formula for every positive integer n?
A. Mathematical induction.
B. Proof by contradiction using only n = 1.
C. Exhaustive enumeration of all positive integers.
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, D. A truth table with n rows.
Rationale: Mathematical induction establishes a base case and proves that truth for an
arbitrary positive integer k implies truth for k + 1. This demonstrates the formula for all
positive integers.
10. A relation R on the integers is defined by aRb if and only if a − b is divisible by 5.
Which statement is correct?
A. R is symmetric but not transitive.
B. R is reflexive and symmetric but not transitive.
C. R is transitive but not reflexive.
D. R is an equivalence relation.
Rationale: Since a − a = 0 is divisible by 5, R is reflexive. If 5 divides a − b, it also divides b − a,
so R is symmetric. If 5 divides a − b and b − c, it divides their sum a − c, so R is transitive.
11. A network administrator uses a simple undirected graph to represent direct
communication links. The graph contains seven vertices, and every vertex is connected to
every other vertex. How many edges are present?
A. 14
B. 21
C. 42
D. 49
Rationale: A complete simple undirected graph on n vertices has n(n − 1)/2 edges. For seven
vertices, the number is 7 × = 21.
12. An application checks whether a number is divisible by 2 or divisible by 3. Which
logical expression represents the condition that the number is divisible by exactly one of
these two divisors?
A. P ∧ Q
B. P ∨ Q
C. (P ∨ Q) ∧ ¬(P ∧ Q)
D. ¬P ∧ ¬Q
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Objective Assessment Revision 2026 | All
Questions And Solutions (Rationales) | Exam
Prep Q&A 2026/2027
1. Which statement is logically equivalent to the negation of the proposition “If a graph is
connected, then every vertex is reachable from every other vertex”?
A. The graph is connected, and every vertex is reachable from every other vertex.
B. The graph is not connected, or some vertex is reachable from every other vertex.
C. The graph is connected, and there exist two vertices such that one is not reachable from
the other.
D. The graph is not connected, and no vertex is reachable from any other vertex.
Rationale: The negation of an implication p → q is p ∧ ¬q. Therefore, the graph is connected
while at least one pair of vertices fails the reachability condition.
2. A software engineer models a system using a relation R on the set of all employees. The
relation holds when two employees belong to the same department. Which property must R
satisfy?
A. Reflexivity, symmetry, and transitivity.
B. Antisymmetry and irreflexivity.
C. Symmetry but not transitivity.
D. Transitivity but not reflexivity.
Rationale: Every employee belongs to the same department as themselves, so the relation is
reflexive. If employee a shares a department with b, then b shares it with a, establishing
symmetry. If a shares a department with b and b shares one with c, all three belong to the
same department, establishing transitivity.
3. A database contains 80 records. Exactly 35 records satisfy condition A, 30 satisfy
condition B, and 12 satisfy both conditions. How many records satisfy at least one
condition?
Page 1 of 105
,A. 53
B. 65
C. 77
D. 47
Rationale: By inclusion-exclusion, |A ∪ B| = |A| + |B| − |A ∩ B| = 35 + 30 − 12 = 53.
Therefore, the correct answer is A.
4. A security system generates a four-character access code using digits 0 through 9.
Repetition is permitted, and leading zeros are valid. How many distinct access codes are
possible?
A. 1,000
B. 10,000
C. 5,040
D. 40,000
Rationale: Each of the four positions has 10 independent choices. The multiplication principle
gives 10 × 10 × 10 × 10 = 10,000 possible codes.
5. A project manager must assign four distinct employees to four distinct roles. There are
nine eligible employees, and each employee can hold only one role. How many assignments
are possible?
A. 126
B. 3,024
C. 6,561
D. 3024 divided by 4
Rationale: Because the roles are distinct and employees cannot be reused, order matters. The
number of assignments is the permutation P(9,4) = 9 × 8 × 7 × 6 = 3,024.
6. A student argues that the proposition p → q is equivalent to q → p. Which statement
correctly evaluates this claim?
A. The claim is correct whenever p and q are both true.
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,B. The claim is correct because implications are symmetric.
C. The claim is false in general; the equivalent form of p → q is ¬q → ¬p.
D. The claim is false because p → q is always equivalent to ¬p → q.
Rationale: An implication is logically equivalent to its contrapositive, ¬q → ¬p, not generally
to its converse, q → p. For example, a true implication can have a false converse.
7. A computing system represents a set of integers using the property P(x): x is even. Which
expression correctly represents the set of integers that are not even?
A. {x ∈ ℤ : P(x)}
B. {x ∈ ℤ : ¬P(x)}
C. {x ∈ ℤ : P(x) ∧ ¬P(x)}
D. {x ∈ ℤ : P(x) → ¬P(x)}
Rationale: The complement of the set of even integers within the integers consists of all
integers for which the predicate “x is even” is false. These are precisely the odd integers.
8. A graph has six vertices, and every vertex has degree three. How many edges does the
graph contain?
A. 18
B. 12
C. 9
D. 6
Rationale: The handshaking lemma states that the sum of vertex degrees equals twice the
number of edges. Thus, 6 × 3 = 2|E|, giving |E| = 9.
9. A developer wants to prove that the sum of the first n positive integers is n(n + 1)/2.
Which approach is most directly suited to proving the formula for every positive integer n?
A. Mathematical induction.
B. Proof by contradiction using only n = 1.
C. Exhaustive enumeration of all positive integers.
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, D. A truth table with n rows.
Rationale: Mathematical induction establishes a base case and proves that truth for an
arbitrary positive integer k implies truth for k + 1. This demonstrates the formula for all
positive integers.
10. A relation R on the integers is defined by aRb if and only if a − b is divisible by 5.
Which statement is correct?
A. R is symmetric but not transitive.
B. R is reflexive and symmetric but not transitive.
C. R is transitive but not reflexive.
D. R is an equivalence relation.
Rationale: Since a − a = 0 is divisible by 5, R is reflexive. If 5 divides a − b, it also divides b − a,
so R is symmetric. If 5 divides a − b and b − c, it divides their sum a − c, so R is transitive.
11. A network administrator uses a simple undirected graph to represent direct
communication links. The graph contains seven vertices, and every vertex is connected to
every other vertex. How many edges are present?
A. 14
B. 21
C. 42
D. 49
Rationale: A complete simple undirected graph on n vertices has n(n − 1)/2 edges. For seven
vertices, the number is 7 × = 21.
12. An application checks whether a number is divisible by 2 or divisible by 3. Which
logical expression represents the condition that the number is divisible by exactly one of
these two divisors?
A. P ∧ Q
B. P ∨ Q
C. (P ∨ Q) ∧ ¬(P ∧ Q)
D. ¬P ∧ ¬Q
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