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WGU C959 Discrete Mathematics I (Math
2800) Comprehensive OA 2024 – Complete
Exam Prep & Study Guide
Unit 1: Logic and Proofs
1. Multiple Choice
What is the truth value of the compound proposition 𝑝 ∧ ¬𝑝?
A) True
B) False
C) Cannot be determined
D) Sometimes true
ANSWER>>: B) False
Rationale: This is a classic logical contradiction. A proposition and its negation
cannot both be true at the same time, so their conjunction is always false.
Key Tip: Memorize the basic truth tables for negation, conjunction, disjunction,
and implication—they are the foundation for everything else.
2. Multiple Choice
Which of the following is logically equivalent to 𝑝 → 𝑞?
A) ¬𝑝 → ¬𝑞
B) ¬𝑞 → ¬𝑝
C) 𝑞 → 𝑝
D) ¬𝑝 ∨ ¬𝑞
ANSWER>>: B) ¬𝑞 → ¬𝑝
Rationale: This is the contrapositive. A conditional statement is always logically
equivalent to its contrapositive. For example, "If it is raining, then the ground is
wet" is equivalent to "If the ground is not wet, then it is not raining."
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Key Tip: Remember that the converse (𝑞 → 𝑝) and inverse (¬𝑝 → ¬𝑞)
are not equivalent to the original conditional.
3. Fill in the Blank
A statement that is always true, regardless of the truth values of its variables, is
called a __________.
ANSWER>>: tautology
Rationale: Tautologies are propositions that are true under every possible
interpretation. Examples include 𝑝 ∨ ¬𝑝 (law of excluded middle).
Key Tip: Use truth tables to verify whether a complex proposition is a tautology,
contradiction, or contingency.
4. Multiple Choice
Consider the argument:
• If it snows, then school is canceled.
• It snows.
• Therefore, school is canceled.
This argument form is called:
A) Denying the hypothesis
B) Affirming the conclusion
C) Modus ponens
D) Modus tollens
ANSWER>>: C) Modus ponens
Rationale: Modus ponens (Latin for "mode that affirms") has the form: If p then
q; p; therefore q.
Key Tip: Modus ponens affirms the antecedent, while modus tollens denies the
consequent (¬𝑞, therefore ¬𝑝).
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5. Multiple Choice
What is the negation of the statement "All students in this class have taken
calculus"?
A) No students in this class have taken calculus
B) Some students in this class have not taken calculus
C) All students in this class have not taken calculus
D) Some students in this class have taken calculus
ANSWER>>: B) Some students in this class have not taken calculus
Rationale: The negation of a universal statement (∀𝑥𝑃(𝑥)) is an existential
statement with a negated predicate (∃𝑥¬𝑃(𝑥)). To disprove that "all have taken
calculus," you only need to find one who hasn't.
Key Tip: When negating quantified statements, ∀ becomes ∃ and vice versa, and
the predicate is negated.
6. Multiple Select
Which of the following are valid proof techniques? (Select all that apply)
A) Direct proof
B) Proof by contradiction
C) Proof by intimidation
D) Proof by induction
ANSWER>>: A) Direct proof, B) Proof by contradiction, D) Proof by induction
Rationale: Direct proof, proof by contradiction, and proof by induction are all
legitimate mathematical proof techniques. "Proof by intimidation" is not a valid
method—it's a joke among mathematicians.
Key Tip: The main proof types you'll encounter are direct, contrapositive,
contradiction, induction, and exhaustion (cases).
7. Multiple Choice
To prove a statement of the form 𝑝 → 𝑞 by contrapositive, you would assume:
A) p is true and prove q is true
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B) q is false and prove p is false
C) p is false and prove q is false
D) q is true and prove p is true
ANSWER>>: B) q is false and prove p is false
Rationale: The contrapositive of 𝑝 → 𝑞 is ¬𝑞 → ¬𝑝. To prove it, you assume the
conclusion is false (¬𝑞) and show the hypothesis must be false (¬𝑝).
Key Tip: Proof by contrapositive is especially useful when the conclusion being
false gives you useful information to work with.
8. Fill in the Blank
A proof that shows a statement is true by demonstrating that assuming it is false
leads to a logical inconsistency is called a proof by __________.
ANSWER>>: contradiction
Rationale: In a proof by contradiction (also called indirect proof), you assume the
negation of what you want to prove and derive a contradiction, showing the
original statement must be true.
Key Tip: Proof by contradiction is powerful for proving "there is no" or "there
exists exactly one" statements.
9. Multiple Choice
Let P(x) be "x is a mammal" and Q(x) be "x has fur." What does ∀𝑥(𝑃(𝑥) →
𝑄(𝑥)) mean?
A) All mammals have fur
B) All furry things are mammals
C) There exists a mammal with fur
D) If something is furry, then it is a mammal
ANSWER>>: A) All mammals have fur
Rationale: The statement reads: "For all x, if x is a mammal, then x has fur." This
means every mammal has the property of having fur.
WGU C959 Discrete Mathematics I (Math
2800) Comprehensive OA 2024 – Complete
Exam Prep & Study Guide
Unit 1: Logic and Proofs
1. Multiple Choice
What is the truth value of the compound proposition 𝑝 ∧ ¬𝑝?
A) True
B) False
C) Cannot be determined
D) Sometimes true
ANSWER>>: B) False
Rationale: This is a classic logical contradiction. A proposition and its negation
cannot both be true at the same time, so their conjunction is always false.
Key Tip: Memorize the basic truth tables for negation, conjunction, disjunction,
and implication—they are the foundation for everything else.
2. Multiple Choice
Which of the following is logically equivalent to 𝑝 → 𝑞?
A) ¬𝑝 → ¬𝑞
B) ¬𝑞 → ¬𝑝
C) 𝑞 → 𝑝
D) ¬𝑝 ∨ ¬𝑞
ANSWER>>: B) ¬𝑞 → ¬𝑝
Rationale: This is the contrapositive. A conditional statement is always logically
equivalent to its contrapositive. For example, "If it is raining, then the ground is
wet" is equivalent to "If the ground is not wet, then it is not raining."
,2
Key Tip: Remember that the converse (𝑞 → 𝑝) and inverse (¬𝑝 → ¬𝑞)
are not equivalent to the original conditional.
3. Fill in the Blank
A statement that is always true, regardless of the truth values of its variables, is
called a __________.
ANSWER>>: tautology
Rationale: Tautologies are propositions that are true under every possible
interpretation. Examples include 𝑝 ∨ ¬𝑝 (law of excluded middle).
Key Tip: Use truth tables to verify whether a complex proposition is a tautology,
contradiction, or contingency.
4. Multiple Choice
Consider the argument:
• If it snows, then school is canceled.
• It snows.
• Therefore, school is canceled.
This argument form is called:
A) Denying the hypothesis
B) Affirming the conclusion
C) Modus ponens
D) Modus tollens
ANSWER>>: C) Modus ponens
Rationale: Modus ponens (Latin for "mode that affirms") has the form: If p then
q; p; therefore q.
Key Tip: Modus ponens affirms the antecedent, while modus tollens denies the
consequent (¬𝑞, therefore ¬𝑝).
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5. Multiple Choice
What is the negation of the statement "All students in this class have taken
calculus"?
A) No students in this class have taken calculus
B) Some students in this class have not taken calculus
C) All students in this class have not taken calculus
D) Some students in this class have taken calculus
ANSWER>>: B) Some students in this class have not taken calculus
Rationale: The negation of a universal statement (∀𝑥𝑃(𝑥)) is an existential
statement with a negated predicate (∃𝑥¬𝑃(𝑥)). To disprove that "all have taken
calculus," you only need to find one who hasn't.
Key Tip: When negating quantified statements, ∀ becomes ∃ and vice versa, and
the predicate is negated.
6. Multiple Select
Which of the following are valid proof techniques? (Select all that apply)
A) Direct proof
B) Proof by contradiction
C) Proof by intimidation
D) Proof by induction
ANSWER>>: A) Direct proof, B) Proof by contradiction, D) Proof by induction
Rationale: Direct proof, proof by contradiction, and proof by induction are all
legitimate mathematical proof techniques. "Proof by intimidation" is not a valid
method—it's a joke among mathematicians.
Key Tip: The main proof types you'll encounter are direct, contrapositive,
contradiction, induction, and exhaustion (cases).
7. Multiple Choice
To prove a statement of the form 𝑝 → 𝑞 by contrapositive, you would assume:
A) p is true and prove q is true
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B) q is false and prove p is false
C) p is false and prove q is false
D) q is true and prove p is true
ANSWER>>: B) q is false and prove p is false
Rationale: The contrapositive of 𝑝 → 𝑞 is ¬𝑞 → ¬𝑝. To prove it, you assume the
conclusion is false (¬𝑞) and show the hypothesis must be false (¬𝑝).
Key Tip: Proof by contrapositive is especially useful when the conclusion being
false gives you useful information to work with.
8. Fill in the Blank
A proof that shows a statement is true by demonstrating that assuming it is false
leads to a logical inconsistency is called a proof by __________.
ANSWER>>: contradiction
Rationale: In a proof by contradiction (also called indirect proof), you assume the
negation of what you want to prove and derive a contradiction, showing the
original statement must be true.
Key Tip: Proof by contradiction is powerful for proving "there is no" or "there
exists exactly one" statements.
9. Multiple Choice
Let P(x) be "x is a mammal" and Q(x) be "x has fur." What does ∀𝑥(𝑃(𝑥) →
𝑄(𝑥)) mean?
A) All mammals have fur
B) All furry things are mammals
C) There exists a mammal with fur
D) If something is furry, then it is a mammal
ANSWER>>: A) All mammals have fur
Rationale: The statement reads: "For all x, if x is a mammal, then x has fur." This
means every mammal has the property of having fur.
