WGU C960 – DISCRETE MATHEMATICS II OBJECTIVE
ASSESSMENT EXAM– QUESTIONS AND ANSWERS | VERIFIED
AND WELL DETAILED ANSWERS PLUS RATIONALES |
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1. Consider the proposition $p \land \neg p$. What is the truth value of this compound
proposition for all possible truth values of $p$?
A. True
B. False
C. Contingency
D. Undetermined
A conjunction of a proposition and its negation is a contradiction, which is always false
regardless of the truth value of the individual proposition.
2. Which of the following logical equivalences represents De Morgan's Law for the negation
of a disjunction?
A. $\neg(p \lor q) \equiv \neg p \land \neg q$
B. $\neg(p \land q) \equiv \neg p \lor \neg q$
C. $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$
D. $\neg(\neg p) \equiv p$
De Morgan's laws state that the negation of a disjunction is the conjunction of the negations,
making option A correct while option B represents the negation of a conjunction.
3. What is the contrapositive of the conditional statement "If a function is differentiable,
then it is continuous"?
A. If a function is continuous, then it is differentiable.
B. If a function is not differentiable, then it is not continuous.
C. If a function is not continuous, then it is not differentiable.
D. If a function is not differentiable, then it is continuous.
The contrapositive of $p \to q$ is given by $\neg q \to \neg p$, which reverses and negates both
components while preserving logical equivalence.
4. Let $P(x)$ denote the predicate "$x > 3$" over the domain of real numbers. Which of
the following expresses the proposition "There is a real number less than or equal to 3"?
A. $\forall x P(x)$
,B. $\exists x P(x)$
C. $\forall x \neg P(x)$
D. $\exists x \neg P(x)$
Saying that every real number is less than or equal to 3 is equivalent to stating that for all $x$,
the statement $x > 3$ is false, which is written as $\forall x \neg P(x)$.
5. Which rule of inference is applied in the following argument: "If it snows, school is
closed. It is snowing. Therefore, school is closed."?
A. Modus Tollens
B. Hypothetical Syllogism
C. Modus Ponens
D. Addition
Modus Ponens follows the valid argument form: $[p \land (p \to q)] \to q$, confirming the
consequent when the antecedent is affirmed.
6. What is the value of the sum $\sum_{i=1}^{4} (2i + 1)$?
A. 16
B. 20
C. 24
D. 28
Evaluating for each term: $(2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) = 3 + 5 + 7 + 9 = 24$.
7. Which proof technique assumes the negation of the conclusion to derive a contradiction?
A. Direct proof
B. Proof by contraposition
C. Proof by contradiction
D. Mathematical induction
Proof by contradiction involves assuming the negation of the statement to be proven and
showing that this assumption leads to a logical impossibility or contradiction.
8. In a proof by mathematical induction, what is the first step that must be established?
A. Inductive hypothesis
B. Base case
C. Inductive step
D. Conclusion
Mathematical induction requires establishing the base case (typically for $n = 1$ or $n = 0$)
to show the proposition holds for the initial value before proving the inductive step.
,9. What is the coefficient of $x^3 y^2$ in the expansion of $(x + y)^5$ according to the
Binomial Theorem?
A. 5
B. 10
C. 15
D. 20
The binomial coefficient $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{120}{6 \times 2} = 10$.
10. How many different 4-letter passwords can be formed from the English alphabet (26
letters) if repetition of letters is allowed?
A. 100
B. 17,576
C. 456,976
D. 358,800
Using the multiplication rule, since repetition is allowed, there are $26 \times 26 \times 26
\times 26 = 26^4 = 456,976$ possible passwords.
11. How many ways can a committee of 3 people be chosen from a group of 10 individuals?
A. 720
B. 210
C. 120
D. 504
Since order does not matter in a committee, combinations are used: $\binom{10}{3} = \frac{10
\times 9 \times 8}{3 \times 2 \times 1} = 120$.
12. What does the Pigeonhole Principle state about placing 13 pigeons into 12 pigeonholes?
A. At least one pigeonhole remains empty.
B. Exactly one pigeonhole contains two pigeons.
C. At least one pigeonhole must contain two or more pigeons.
D. No pigeonhole can contain more than one pigeon.
The Pigeonhole Principle guarantees that if $k+1$ or more objects are placed into $k$ boxes,
at least one box must contain multiple objects.
13. What is the coefficient of $x^2 y^3 z^2$ in the expansion of $(x + y + z)^7$?
A. 210
B. 420
C. 105
, D. 840
Using the multinomial coefficient formula $\frac{7!}{2!3!2!} = \frac{5040}{2 \times 6 \times 2}
= \frac{5040}{24} = 210$.
14. What is the solution to the linear homogeneous recurrence relation $a_n = 5a_{n-1} -
6a_{n-2}$ with initial conditions $a_0 = 1, a_1 = 3$?
A. $a_n = 2^n + 3^n$
B. $a_n = 3^n$
C. $a_n = 2^n$
D. $a_n = 2 \cdot 3^n - 2^n$
The characteristic equation is $r^2 - 5r + 6 = 0$, which factors into $(r-2)(r-3) = 0$, yielding
roots $r=2$ and $r=3$. Testing the initial conditions confirms $a_n = 3^n$.
15. What type of graph contains directed edges where each edge connects an ordered pair
of vertices?
A. Simple graph
B. Undirected graph
C. Directed graph (Digraph)
D. Complete graph
A directed graph consists of a set of vertices and a set of directed edges, where each edge has a
specific direction from a source to a target vertex.
16. In an undirected graph, what is the sum of the degrees of all vertices equal to?
A. The number of edges
B. Twice the number of edges
C. The number of vertices squared
D. Half the number of edges
The Handshaking Theorem states that the sum of the degrees of all vertices in an undirected
graph is always equal to twice the number of edges because each edge contributes to the
degree of two endpoints.
17. What is the maximum number of edges in a simple undirected graph with $n$ vertices?
A. $n(n-1)$
B. $\frac{n(n-1)}{2}$
C. $n^2$
D. $2n - 1$
ASSESSMENT EXAM– QUESTIONS AND ANSWERS | VERIFIED
AND WELL DETAILED ANSWERS PLUS RATIONALES |
GUARANTEED PASS | LATEST EXAM UPDATE | EXAM PREP |
STUDY GUIDE | PRACTICE TEST| DOWNLOAD INSTANT PDF
1. Consider the proposition $p \land \neg p$. What is the truth value of this compound
proposition for all possible truth values of $p$?
A. True
B. False
C. Contingency
D. Undetermined
A conjunction of a proposition and its negation is a contradiction, which is always false
regardless of the truth value of the individual proposition.
2. Which of the following logical equivalences represents De Morgan's Law for the negation
of a disjunction?
A. $\neg(p \lor q) \equiv \neg p \land \neg q$
B. $\neg(p \land q) \equiv \neg p \lor \neg q$
C. $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$
D. $\neg(\neg p) \equiv p$
De Morgan's laws state that the negation of a disjunction is the conjunction of the negations,
making option A correct while option B represents the negation of a conjunction.
3. What is the contrapositive of the conditional statement "If a function is differentiable,
then it is continuous"?
A. If a function is continuous, then it is differentiable.
B. If a function is not differentiable, then it is not continuous.
C. If a function is not continuous, then it is not differentiable.
D. If a function is not differentiable, then it is continuous.
The contrapositive of $p \to q$ is given by $\neg q \to \neg p$, which reverses and negates both
components while preserving logical equivalence.
4. Let $P(x)$ denote the predicate "$x > 3$" over the domain of real numbers. Which of
the following expresses the proposition "There is a real number less than or equal to 3"?
A. $\forall x P(x)$
,B. $\exists x P(x)$
C. $\forall x \neg P(x)$
D. $\exists x \neg P(x)$
Saying that every real number is less than or equal to 3 is equivalent to stating that for all $x$,
the statement $x > 3$ is false, which is written as $\forall x \neg P(x)$.
5. Which rule of inference is applied in the following argument: "If it snows, school is
closed. It is snowing. Therefore, school is closed."?
A. Modus Tollens
B. Hypothetical Syllogism
C. Modus Ponens
D. Addition
Modus Ponens follows the valid argument form: $[p \land (p \to q)] \to q$, confirming the
consequent when the antecedent is affirmed.
6. What is the value of the sum $\sum_{i=1}^{4} (2i + 1)$?
A. 16
B. 20
C. 24
D. 28
Evaluating for each term: $(2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) = 3 + 5 + 7 + 9 = 24$.
7. Which proof technique assumes the negation of the conclusion to derive a contradiction?
A. Direct proof
B. Proof by contraposition
C. Proof by contradiction
D. Mathematical induction
Proof by contradiction involves assuming the negation of the statement to be proven and
showing that this assumption leads to a logical impossibility or contradiction.
8. In a proof by mathematical induction, what is the first step that must be established?
A. Inductive hypothesis
B. Base case
C. Inductive step
D. Conclusion
Mathematical induction requires establishing the base case (typically for $n = 1$ or $n = 0$)
to show the proposition holds for the initial value before proving the inductive step.
,9. What is the coefficient of $x^3 y^2$ in the expansion of $(x + y)^5$ according to the
Binomial Theorem?
A. 5
B. 10
C. 15
D. 20
The binomial coefficient $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{120}{6 \times 2} = 10$.
10. How many different 4-letter passwords can be formed from the English alphabet (26
letters) if repetition of letters is allowed?
A. 100
B. 17,576
C. 456,976
D. 358,800
Using the multiplication rule, since repetition is allowed, there are $26 \times 26 \times 26
\times 26 = 26^4 = 456,976$ possible passwords.
11. How many ways can a committee of 3 people be chosen from a group of 10 individuals?
A. 720
B. 210
C. 120
D. 504
Since order does not matter in a committee, combinations are used: $\binom{10}{3} = \frac{10
\times 9 \times 8}{3 \times 2 \times 1} = 120$.
12. What does the Pigeonhole Principle state about placing 13 pigeons into 12 pigeonholes?
A. At least one pigeonhole remains empty.
B. Exactly one pigeonhole contains two pigeons.
C. At least one pigeonhole must contain two or more pigeons.
D. No pigeonhole can contain more than one pigeon.
The Pigeonhole Principle guarantees that if $k+1$ or more objects are placed into $k$ boxes,
at least one box must contain multiple objects.
13. What is the coefficient of $x^2 y^3 z^2$ in the expansion of $(x + y + z)^7$?
A. 210
B. 420
C. 105
, D. 840
Using the multinomial coefficient formula $\frac{7!}{2!3!2!} = \frac{5040}{2 \times 6 \times 2}
= \frac{5040}{24} = 210$.
14. What is the solution to the linear homogeneous recurrence relation $a_n = 5a_{n-1} -
6a_{n-2}$ with initial conditions $a_0 = 1, a_1 = 3$?
A. $a_n = 2^n + 3^n$
B. $a_n = 3^n$
C. $a_n = 2^n$
D. $a_n = 2 \cdot 3^n - 2^n$
The characteristic equation is $r^2 - 5r + 6 = 0$, which factors into $(r-2)(r-3) = 0$, yielding
roots $r=2$ and $r=3$. Testing the initial conditions confirms $a_n = 3^n$.
15. What type of graph contains directed edges where each edge connects an ordered pair
of vertices?
A. Simple graph
B. Undirected graph
C. Directed graph (Digraph)
D. Complete graph
A directed graph consists of a set of vertices and a set of directed edges, where each edge has a
specific direction from a source to a target vertex.
16. In an undirected graph, what is the sum of the degrees of all vertices equal to?
A. The number of edges
B. Twice the number of edges
C. The number of vertices squared
D. Half the number of edges
The Handshaking Theorem states that the sum of the degrees of all vertices in an undirected
graph is always equal to twice the number of edges because each edge contributes to the
degree of two endpoints.
17. What is the maximum number of edges in a simple undirected graph with $n$ vertices?
A. $n(n-1)$
B. $\frac{n(n-1)}{2}$
C. $n^2$
D. $2n - 1$
