Proof and logic form the cornerstone of rigorous reasoning in fields like mathematics, philosophy, and computer science. Logic is the systematic study of valid inference, encompassing rules and structures—such as propositions, syllogisms, and truth tables—that ensure arguments are sound and free from fallacy. A proof, in turn, is a formal demonstration that establishes the truth of a statement, relying on logical principles to derive conclusions from axioms, definitions, and previously verified theorems. Through deductive reasoning, proofs eliminate doubt, providing an unbreakable chain of evidence that upholds the integrity of knowledge.
Table of contents
- Part 1: OnlineExamMaker AI quiz maker – Make a free quiz in minutes
- Part 2: 20 proof & logic quiz questions & answers
- Part 3: Automatically generate quiz questions using AI Question Generator
Part 1: OnlineExamMaker AI quiz maker – Make a free quiz in minutes
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Part 2: 20 proof & logic quiz questions & answers
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1. Which of the following is a valid proof by contradiction for the statement “√2 is irrational”?
A. Assume √2 is rational, then it can be expressed as a fraction, leading to a contradiction.
B. Assume √2 is irrational, then derive a rational form.
C. Directly show √2 as a fraction.
D. Use induction to prove it.
Answer: A
Explanation: Assuming √2 is rational leads to expressing it as a/b where a and b are integers with no common factors, but this results in an even/odd contradiction, proving it irrational.
2. In propositional logic, what is the truth value of (P ∧ Q) → R if P is true, Q is false, and R is true?
A. True
B. False
C. Undefined
D. Cannot be determined
Answer: A
Explanation: (P ∧ Q) is false (since Q is false), and a false premise implies the implication is true.
3. Which logical equivalence is correctly stated?
A. P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)
B. P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
C. ¬(P → Q) ≡ P ∧ Q
D. P ↔ Q ≡ (P → Q) ∧ (Q → P)
Answer: D
Explanation: The biconditional P ↔ Q is equivalent to both implications (P → Q) and (Q → P) being true.
4. For the statement “If n is even, then n^2 is even,” what type of proof would verify this?
A. Proof by induction
B. Direct proof
C. Proof by contradiction
D. Counterexample
Answer: B
Explanation: A direct proof shows that if n = 2k, then n^2 = 4k^2, which is even.
5. In a truth table for P XOR Q, when is the result true?
A. When both P and Q are true
B. When both P and Q are false
C. When exactly one of P or Q is true
D. When P is true and Q is false only
Answer: C
Explanation: XOR (exclusive or) is true only when the inputs differ, meaning one is true and the other is false.
6. Which of the following arguments is valid?
A. All A are B. C is A. Therefore, C is not B.
B. If P, then Q. Q is true. Therefore, P is true.
C. If P, then Q. P is true. Therefore, Q is true.
D. No A are B. C is A. Therefore, C is B.
Answer: C
Explanation: This is modus ponens, a valid form where the premise implies the conclusion.
7. What is the contrapositive of the statement “If it is raining, then the ground is wet”?
A. If the ground is not wet, then it is not raining.
B. If it is not raining, then the ground is not wet.
C. If the ground is wet, then it is raining.
D. The ground is wet only if it is raining.
Answer: A
Explanation: The contrapositive swaps and negates the parts: ¬Q → ¬P.
8. In mathematical induction, what must be proven after the base case?
A. The statement for n = 1 only
B. The inductive step for all n
C. The statement is true for n + 1 assuming n
D. A counterexample exists
Answer: C
Explanation: The inductive step assumes the statement is true for k and proves it for k + 1.
9. Which fallacy is present in the argument: “All dogs have fur. Fido has fur. Therefore, Fido is a dog”?
A. Affirming the consequent
B. Denying the antecedent
C. Affirming a disjunct
D. No fallacy
Answer: D
Explanation: The argument is valid; it’s an example of a correct syllogism.
10. For the set of premises: P → Q, Q → R, and P, what can be concluded?
A. R
B. ¬Q
C. P ∧ Q
D. Q ∨ R
Answer: A
Explanation: From P → Q and P, we get Q; from Q → R and Q, we get R (hypothetical syllogism).
11. What is the negation of “For all x, P(x)”?
A. For all x, ¬P(x)
B. There exists x such that ¬P(x)
C. There exists x such that P(x)
D. For all x, P(x) is false
Answer: B
Explanation: The negation of a universal quantifier is an existential quantifier with the negated predicate.
12. In a direct proof of “The sum of two even numbers is even,” how would you proceed?
A. Assume the sum is odd and find a contradiction.
B. Let the numbers be 2a and 2b, and show the sum is 2(a + b).
C. Use induction on the numbers.
D. Find a counterexample.
Answer: B
Explanation: Representing even numbers as 2a and 2b directly shows their sum is even.
13. Which of the following is a tautology?
A. P ∧ ¬P
B. P ∨ ¬P
C. P → ¬P
D. P ↔ ¬P
Answer: B
Explanation: The law of excluded middle states that P or not P is always true.
14. For the statement “n^3 + 5 is odd if n is even,” what proof method could disprove it?
A. Direct proof
B. Counterexample
C. Induction
D. Contradiction
Answer: B
Explanation: A counterexample, like n=2 (even), shows 8 + 5 = 13 (odd), so it’s true, but this tests validity.
15. In logic, what does “P ≡ Q” mean?
A. P implies Q only
B. P and Q are equivalent
C. P or Q
D. Not P and not Q
Answer: B
Explanation: ≡ denotes logical equivalence, meaning P and Q have the same truth value.
16. Which is an example of a valid modus tollens argument?
A. If P, then Q. Q is true. So P is true.
B. If P, then Q. P is true. So Q is true.
C. If P, then Q. ¬Q is true. So ¬P is true.
D. If P, then Q. ¬P is true. So ¬Q is true.
Answer: C
Explanation: Modus tollens states that if P implies Q and Q is false, then P is false.
17. What is required to prove “Every prime number greater than 2 is odd” by contradiction?
A. Assume a prime greater than 2 is even and show it leads to 2 being not prime.
B. Directly list primes.
C. Use induction up to infinity.
D. Assume it’s even and derive a contradiction with primality.
Answer: D
Explanation: Assuming an even prime greater than 2 (like 4) leads to it being divisible by 2 and not prime.
18. In predicate logic, “∃x ∀y (P(x, y))” means:
A. For all y, there exists x such that P(x, y)
B. There exists x such that for all y, P(x, y)
C. For all x and y, P(x, y)
D. There exists y such that for all x, P(x, y)
Answer: B
Explanation: The quantifiers are read in order: first ∃x, then ∀y.
19. For the inequality proof: “Prove that for all integers n > 1, n^2 > n.”
A. Use direct proof by subtracting n from n^2.
B. Use induction starting from n=2.
C. Assume n^2 ≤ n and contradict.
D. Find a counterexample for n=1.
Answer: B
Explanation: Induction works: base case n=2 (4>2), and assume for k, prove for k+1.
20. Which statement about logical fallacies is accurate?
A. Ad hominem attacks the argument, not the person.
B. Slippery slope assumes one event leads to a chain of events without proof.
C. Appeal to authority is always valid.
D. Circular reasoning provides new evidence.
Answer: B
Explanation: Slippery slope fallacy exaggerates a chain of consequences without sufficient evidence.
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