ESTUDYR
D420 DISCRETE MATH 1 WGU QUESTIONS AND
CORRECT DETAILED ANSWERS (VERIFIED ANSWERS) |
ASSURED SUCCESS
1. What does exclusive or (⊕) mean?
A. Both options are true
B. Neither option is true
C. One or the other, but not both (✔ )
D. Both options must be false
Rationale: Exclusive or means one of the statements is true, but not both.
2. What type of disjunction is represented by inclusive or?
A. A logical "or" that is true if at least one statement is true (✔ )
B. A logical "or" that is true only if both statements are true
C. A negation of "and"
D. A conditional statement
Rationale: Inclusive or is true when either or both conditions hold.
3. What is the correct order of operations in logic if there are no parentheses?
A. ∧, ∨, ¬
B. ¬ (not), ∧ (and), ∨ (or) (✔ )
C. ∨, ∧, ¬
D. ∧, ¬, ∨
Rationale: Negation is evaluated first, followed by conjunction and disjunction.
4. What is the truth table row count for three variables?
A. 6 rows
B. 8 rows (✔ )
C. 4 rows
D. 16 rows
,ESTUDYR
Rationale: For nnn variables, the truth table has 2n2^n2n rows. 23=82^3 = 823=8.
5. What is the converse of p→qp \to qp→q?
A. ¬q→¬p\neg q \to \neg p¬q→¬p
B. p→¬qp \to \neg qp→¬q
C. q→pq \to pq→p (✔ )
D. ¬p→q\neg p \to q¬p→q
Rationale: The converse swaps the hypothesis and conclusion.
6. What is the contrapositive of p→qp \to qp→q?
A. p→qp \to qp→q
B. q→pq \to pq→p
C. ¬q→p\neg q \to p¬q→p
D. ¬q→¬p\neg q \to \neg p¬q→¬p (✔ )
Rationale: Contrapositive negates and swaps the hypothesis and conclusion.
7. What is a biconditional statement?
A. p∨qp \lor qp∨q
B. p→qp \to qp→q
C. p↔qp \leftrightarrow qp↔q (✔ )
D. ¬p∧q\neg p \land q¬p∧q
Rationale: p↔qp \leftrightarrow qp↔q is true when ppp and qqq have the same truth value.
8. Which variable type is bound by a quantifier?
A. Bound variable (✔ )
B. Free variable
C. Dependent variable
D. Independent variable
Rationale: Bound variables are limited to the scope of quantifiers like ∀\forall∀ or ∃\exists∃.
,ESTUDYR
9. In the statement (∀xP(x))∧Q(x)(\forall x P(x)) \land Q(x)(∀xP(x))∧Q(x), which variable is free?
A. All variables are bound
B. xxx in Q(x)Q(x)Q(x) (✔ )
C. xxx in P(x)P(x)P(x)
D. No variables are free
Rationale: xxx in P(x)P(x)P(x) is bound by ∀\forall∀, but xxx in Q(x)Q(x)Q(x) is not.
10. According to De Morgan’s laws, ¬(∀xP(x))\neg(\forall x P(x))¬(∀xP(x)) is equivalent to:
A. ∀x¬P(x)\forall x \neg P(x)∀x¬P(x)
B. ∃x¬P(x)\exists x \neg P(x)∃x¬P(x) (✔ )
C. ¬∃xP(x)\neg \exists x P(x)¬∃xP(x)
D. ∀xP(x)\forall x P(x)∀xP(x)
Rationale: De Morgan’s laws allow negation to distribute through quantifiers, flipping them.
11. What makes an argument valid?
A. Hypotheses are always true
B. The conclusion is false
C. The conclusion is true whenever all hypotheses are true (✔ )
D. All hypotheses must be false
Rationale: Validity guarantees the conclusion when the hypotheses are true.
12. What is a theorem?
A. A self-evident truth
B. A statement that can be proven (✔ )
C. A hypothesis
D. An assumption
Rationale: Theorems require logical proof.
13. What is a proof by contrapositive?
, ESTUDYR
A. Assuming ppp to prove qqq
B. Testing all possible cases
C. Proving ¬q→¬p\neg q \to \neg p¬q→¬p to establish p→qp \to qp→q (✔ )
D. Using axioms to confirm p→qp \to qp→q
Rationale: The contrapositive is logically equivalent to the original statement.
14. Which is NOT true about a counterexample?
A. It disproves a universal statement
B. It contradicts the conclusion of a hypothesis
C. It proves the hypothesis is true (✔ )
D. It assigns values to variables
Rationale: A counterexample falsifies, not proves, universal claims.
15. What type of reasoning does a direct proof use?
A. Testing multiple cases
B. Contradiction
C. Assuming ppp to directly prove qqq (✔ )
D. Negating hypotheses
Rationale: Direct proofs logically derive qqq from ppp.
16. What does the symbol ∨\lor∨ represent in logic?
A. Negation
B. Conjunction
C. Disjunction (or) (✔ )
D. Implication
Rationale: ∨\lor∨ means "or," true if at least one operand is true.
17. Which truth table row results in p∨qp \lor qp∨q being false?
A. p=T,q=Fp = T, q = Fp=T,q=F
B. p=F,q=Fp = F, q = Fp=F,q=F (✔ )
D420 DISCRETE MATH 1 WGU QUESTIONS AND
CORRECT DETAILED ANSWERS (VERIFIED ANSWERS) |
ASSURED SUCCESS
1. What does exclusive or (⊕) mean?
A. Both options are true
B. Neither option is true
C. One or the other, but not both (✔ )
D. Both options must be false
Rationale: Exclusive or means one of the statements is true, but not both.
2. What type of disjunction is represented by inclusive or?
A. A logical "or" that is true if at least one statement is true (✔ )
B. A logical "or" that is true only if both statements are true
C. A negation of "and"
D. A conditional statement
Rationale: Inclusive or is true when either or both conditions hold.
3. What is the correct order of operations in logic if there are no parentheses?
A. ∧, ∨, ¬
B. ¬ (not), ∧ (and), ∨ (or) (✔ )
C. ∨, ∧, ¬
D. ∧, ¬, ∨
Rationale: Negation is evaluated first, followed by conjunction and disjunction.
4. What is the truth table row count for three variables?
A. 6 rows
B. 8 rows (✔ )
C. 4 rows
D. 16 rows
,ESTUDYR
Rationale: For nnn variables, the truth table has 2n2^n2n rows. 23=82^3 = 823=8.
5. What is the converse of p→qp \to qp→q?
A. ¬q→¬p\neg q \to \neg p¬q→¬p
B. p→¬qp \to \neg qp→¬q
C. q→pq \to pq→p (✔ )
D. ¬p→q\neg p \to q¬p→q
Rationale: The converse swaps the hypothesis and conclusion.
6. What is the contrapositive of p→qp \to qp→q?
A. p→qp \to qp→q
B. q→pq \to pq→p
C. ¬q→p\neg q \to p¬q→p
D. ¬q→¬p\neg q \to \neg p¬q→¬p (✔ )
Rationale: Contrapositive negates and swaps the hypothesis and conclusion.
7. What is a biconditional statement?
A. p∨qp \lor qp∨q
B. p→qp \to qp→q
C. p↔qp \leftrightarrow qp↔q (✔ )
D. ¬p∧q\neg p \land q¬p∧q
Rationale: p↔qp \leftrightarrow qp↔q is true when ppp and qqq have the same truth value.
8. Which variable type is bound by a quantifier?
A. Bound variable (✔ )
B. Free variable
C. Dependent variable
D. Independent variable
Rationale: Bound variables are limited to the scope of quantifiers like ∀\forall∀ or ∃\exists∃.
,ESTUDYR
9. In the statement (∀xP(x))∧Q(x)(\forall x P(x)) \land Q(x)(∀xP(x))∧Q(x), which variable is free?
A. All variables are bound
B. xxx in Q(x)Q(x)Q(x) (✔ )
C. xxx in P(x)P(x)P(x)
D. No variables are free
Rationale: xxx in P(x)P(x)P(x) is bound by ∀\forall∀, but xxx in Q(x)Q(x)Q(x) is not.
10. According to De Morgan’s laws, ¬(∀xP(x))\neg(\forall x P(x))¬(∀xP(x)) is equivalent to:
A. ∀x¬P(x)\forall x \neg P(x)∀x¬P(x)
B. ∃x¬P(x)\exists x \neg P(x)∃x¬P(x) (✔ )
C. ¬∃xP(x)\neg \exists x P(x)¬∃xP(x)
D. ∀xP(x)\forall x P(x)∀xP(x)
Rationale: De Morgan’s laws allow negation to distribute through quantifiers, flipping them.
11. What makes an argument valid?
A. Hypotheses are always true
B. The conclusion is false
C. The conclusion is true whenever all hypotheses are true (✔ )
D. All hypotheses must be false
Rationale: Validity guarantees the conclusion when the hypotheses are true.
12. What is a theorem?
A. A self-evident truth
B. A statement that can be proven (✔ )
C. A hypothesis
D. An assumption
Rationale: Theorems require logical proof.
13. What is a proof by contrapositive?
, ESTUDYR
A. Assuming ppp to prove qqq
B. Testing all possible cases
C. Proving ¬q→¬p\neg q \to \neg p¬q→¬p to establish p→qp \to qp→q (✔ )
D. Using axioms to confirm p→qp \to qp→q
Rationale: The contrapositive is logically equivalent to the original statement.
14. Which is NOT true about a counterexample?
A. It disproves a universal statement
B. It contradicts the conclusion of a hypothesis
C. It proves the hypothesis is true (✔ )
D. It assigns values to variables
Rationale: A counterexample falsifies, not proves, universal claims.
15. What type of reasoning does a direct proof use?
A. Testing multiple cases
B. Contradiction
C. Assuming ppp to directly prove qqq (✔ )
D. Negating hypotheses
Rationale: Direct proofs logically derive qqq from ppp.
16. What does the symbol ∨\lor∨ represent in logic?
A. Negation
B. Conjunction
C. Disjunction (or) (✔ )
D. Implication
Rationale: ∨\lor∨ means "or," true if at least one operand is true.
17. Which truth table row results in p∨qp \lor qp∨q being false?
A. p=T,q=Fp = T, q = Fp=T,q=F
B. p=F,q=Fp = F, q = Fp=F,q=F (✔ )
