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Unit 1 - Practice Quiz

MTH401

1 Which of the following is the definition of a proposition?

A. A declarative sentence that is either true or false, but not both.
B. A declarative sentence that is both true and false simultaneously.
C. A command or a question.
D. A variable that can take any value.

2 Let be a proposition. The statement "It is not the case that " is denoted by which logical operator?

A. Conjunction ()
B. Disjunction ()
C. Negation ()
D. Implication ()

3 The conjunction of propositions and () is true if and only if:

A. is true and is false.
B. is false and is true.
C. Both and are true.
D. At least one of or is true.

4 Under what condition is the conditional statement false?

A. is true and is true.
B. is true and is false.
C. is false and is true.
D. is false and is false.

5 The statement is called a biconditional. It is true when:

A. is true and is false.
B. and have different truth values.
C. and have the same truth values.
D. is false and is true.

6 Which of the following is the contrapositive of the implication ?

A.
B.
C.
D.

7 What is the converse of the implication ?

A.
B.
C.
D.

8 What is the inverse of the implication ?

A.
B.
C.
D.

9 A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a:

A. Contradiction
B. Tautology
C. Contingency
D. Fallacy

10 Which of the following represents one of De Morgan's Laws?

A.
B.
C.
D.

11 The logical equivalence is known as:

A. Logical Implication Rule (Material Implication)
B. Modus Ponens
C. De Morgan's Law
D. Double Negation

12 If a compound proposition is always false, it is called a:

A. Tautology
B. Contradiction
C. Implication
D. Satisfiable proposition

13 Which law states that ?

A. Associative Law
B. Commutative Law
C. Distributive Law
D. Identity Law

14 In the statement , the symbol is called the:

A. Existential quantifier
B. Universal quantifier
C. Uniqueness quantifier
D. Negation operator

15 The statement is true if and only if:

A. is true for every in the domain.
B. is false for every in the domain.
C. There is at least one in the domain for which is true.
D. There is exactly one in the domain for which is true.

16 What is the negation of the statement ?

A.
B.
C.
D.

17 What is the negation of the statement ?

A.
B.
C.
D.

18 If the domain is integers, which statement is true?

A.
B.
C.
D.

19 When quantifiers are nested, does the order matter? i.e., Is always equivalent to ?

A. Yes, they are always equivalent.
B. No, they are never equivalent.
C. No, the order generally changes the meaning.
D. Yes, provided the domain is infinite.

20 The notation denotes:

A. There exists at least one such that .
B. There exists a unique such that .
C. There does not exist any such that .
D. For all unique , is true.

21 A valid logical argument consisting of a set of premises and a conclusion is called a:

A. Proof
B. Fallacy
C. Paradox
D. Hypothesis

22 In a Direct Proof of the theorem , what is the first step?

A. Assume is true.
B. Assume is false.
C. Assume is true.
D. Assume is true.

23 An integer is even if there exists an integer such that:

A.
B.
C.
D.

24 An integer is odd if there exists an integer such that:

A.
B.
C.
D.

25 To prove by Contraposition, what do you assume and what do you deduce?

A. Assume , deduce .
B. Assume , deduce .
C. Assume , deduce .
D. Assume , deduce .

26 Identify the type of proof used here: To prove "If is even, then is even", we assume is odd and show that is odd.

A. Direct Proof
B. Proof by Contraposition
C. Vacuous Proof
D. Proof by Cases

27 In a Proof by Contradiction to prove proposition , what is the starting assumption?

A. Assume is true.
B. Assume is false (i.e., assume ).
C. Assume nothing.
D. Assume a related tautology.

28 Which proof method relies on the logical equivalence to show that if is false, the implication is automatically true?

A. Direct Proof
B. Trivial Proof
C. Vacuous Proof
D. Proof by Contradiction

29 A Trivial Proof of is constructed by showing that:

A. is false.
B. is true.
C. and are both false.
D. implies .

30 Consider the statement: "All integers in the empty set are even." This is true. What type of proof justifies this?

A. Direct Proof
B. Vacuous Proof
C. Proof by Contradiction
D. Constructive Existence Proof

31 To prove a biconditional statement , one usually proves:

A. only.
B. only.
C. Both and .
D. Neither, one proves .

32 What is a Counterexample?

A. A specific case that proves a universal statement is false.
B. A proof that shows an existential statement is false.
C. An example that supports a theorem.
D. A step in a direct proof.

33 Which of the following serves as a counterexample to the statement: "For every integer , "?

A.
B.
C.
D. None of the above (the statement is true for integers).

34 Which of the following is a counterexample to "Every prime number is odd"?

A. 3
B. 5
C. 2
D. 9

35 The proof method that divides the domain into distinct classes and proves the theorem for each class is called:

A. Proof by Contraposition
B. Proof by Exhaustion (or Proof by Cases)
C. Vacuous Proof
D. Trivial Proof

36 The phrase "Without Loss of Generality" (WLOG) is often used in proofs when:

A. We are skipping steps to save time.
B. We make an arbitrary choice that does not affect the validity of the proof because other cases are symmetric.
C. The proof is too difficult to complete.
D. We are assuming the conclusion is true.

37 What common mistake is made in the following argument? "If it rains, the ground is wet. The ground is wet. Therefore, it rained."

A. Denying the antecedent
B. Affirming the consequent
C. Begging the question
D. Circular reasoning

38 What common mistake is made in the following argument? "If it rains, the ground is wet. It did not rain. Therefore, the ground is not wet."

A. Denying the antecedent
B. Modus Ponens
C. Modus Tollens
D. Affirming the consequent

39 Begging the question (or circular reasoning) occurs when:

A. The conclusion is proven using logical steps.
B. One or more steps of the proof rely on the truth of the conclusion itself.
C. The proof is very short.
D. A counterexample is found.

40 Proving that is irrational is classically done using:

A. Direct Proof
B. Proof by Contradiction
C. Trivial Proof
D. Vacuous Proof

41 A proof that shows an element exists such that is true, by explicitly finding such an , is called a:

A. Constructive existence proof
B. Non-constructive existence proof
C. Uniqueness proof
D. Universal generalization

42 If a proof demonstrates that there must be an such that is true, but does not tell us explicitly which it is, it is called a:

A. Constructive existence proof
B. Non-constructive existence proof
C. Direct proof
D. Trivial proof

43 The rule of inference called Modus Ponens states:

A. From and , derive .
B. From and , derive .
C. From and , derive .
D. From and , derive .

44 The rule of inference called Modus Tollens states:

A. From and , derive .
B. From and , derive .
C. From , derive .
D. From , derive .

45 The logical operator Exclusive OR (XOR), denoted , is true when:

A. Both and are true.
B. Both and are false.
C. Exactly one of or is true.
D. At least one of or is true.

46 Which of the following expresses " is a sufficient condition for "?

A.
B.
C.
D.

47 Which of the following expresses " is a necessary condition for "?

A.
B.
C.
D.

48 The statement " unless " translates logically to:

A.
B.
C.
D.

49 In a truth table for , how many rows (combinations of truth values) are required?

A. 4
B. 6
C. 8
D. 9

50 If we want to prove that "Given real number , if , then is not an integer", which proof strategy is most promising?

A. Direct Proof (solving the cubic equation directly).
B. Proof by Contradiction (Assume is an integer root).
C. Trivial Proof.
D. Mathematical Induction.