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Unit 1 - Practice Quiz

MTH401 50 Questions
0 Correct 0 Wrong 50 Left
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1 Which of the following sentences is a proposition?

A. What time is it?
B. Read this sentence carefully.
C.
D. Toronto is the capital of Canada.

2 Let be true and be false. What is the truth value of ?

A. True
B. False
C. Cannot be determined
D. Both True and False

3 What is the contrapositive of the conditional statement ?

A.
B.
C.
D.

4 Which of the following is logically equivalent to according to De Morgan's Laws?

A.
B.
C.
D.

5 A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called a(n):

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

6 What is the converse of the statement: "If it is raining, then the ground is wet"?

A. If the ground is wet, then it is raining.
B. If it is not raining, then the ground is not wet.
C. If the ground is not wet, then it is not raining.
D. It is raining if and only if the ground is wet.

7 The statement is true if and only if:

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

8 Which logical equivalence represents the definition of the conditional statement using disjunction?

A.
B.
C.
D.

9 Let denote the statement "". What is the truth value of the quantification where the domain consists of all real numbers?

A. True
B. False
C. Undefined
D. Contingent

10 How is the statement "There exists an such that is true" denoted?

A.
B.
C.
D.

11 What is the negation of the statement ?

A.
B.
C.
D.

12 In the statement , what does the order of quantifiers imply if the domain is integers?

A. There is one specific that works for all .
B. For every integer , there is a corresponding integer (additive inverse).
C. For every , there exists an .
D. The variables and are independent.

13 When proving a statement of the form is false, what is the most common method?

A. Direct Proof
B. Proof by Contraposition
C. Finding a Counterexample
D. Proof by Cases

14 Which rule of inference is the basis of Direct Proof?

A. Modus Ponens
B. Modus Tollens
C. Hypothetical Syllogism
D. Simplification

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

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

16 To prove by Contraposition, what do we assume?

A. is true
B. is true
C. is true
D. is true

17 A proof where we assume and derive a contradiction (like ) to prove is true is called:

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

18 If the hypothesis of a conditional statement is known to be false, the implication is always true. This is the basis of:

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

19 If the conclusion of a conditional statement is known to be true, the implication is always true. This is the basis of:

A. Direct Proof
B. Trivial Proof
C. Vacuous Proof
D. Indirect Proof

20 Which of the following implies that an argument is valid?

A. The conclusion is true.
B. The premises are true.
C. If the premises are all true, the conclusion must be true.
D. The premises are false.

21 Which logical fallacy is committed in the following argument?
"If it rains, the grass is wet. The grass is wet. Therefore, it rained."

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

22 What is the inverse of ?

A.
B.
C.
D.

23 Which law states ?

A. Associative Law
B. Distributive Law
C. De Morgan's Law
D. Absorption Law

24 To prove a biconditional statement , one usually proves:

A. only
B. only
C. and
D.

25 When proving by cases, what must be true about the cases?

A. They must be infinite.
B. They must overlap.
C. They must be exhaustive (cover all possibilities).
D. They must be contradictory.

26 The statement " is irrational" is typically proved using which method?

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

27 What is the logical equivalent of "Unless , "?

A.
B.
C.
D.

28 Which quantifier is used to express "There is a unique such that "?

A.
B.
C.
D.

29 Let be "". If the domain is real numbers, what is the truth value of ?

A. True
B. False
C. Cannot be determined
D. Depends on x

30 A logical error where the step to be proved is assumed implicitly in the premises is called:

A. Modus Ponens
B. Begging the Question (Circular Reasoning)
C. Non Sequitur
D. Reductio ad Absurdum

31 Which of the following is an example of a Constructive Existence Proof for ?

A. Showing leads to a contradiction.
B. Finding a specific element such that is true.
C. Showing is always true.
D. Assume is false.

32 In logic, what is a Lemma?

A. The main theorem of a paper.
B. A minor theorem used as a stepping stone to prove a major theorem.
C. A direct consequence of a theorem.
D. A statement accepted without proof.

33 What is the negation of the proposition: "Every student in the class has taken a course in calculus"?

A. No student in the class has taken a course in calculus.
B. Some student in the class has taken a course in calculus.
C. There exists a student in the class who has not taken a course in calculus.
D. Every student has not taken calculus.

34 What is the truth value of (Exclusive OR)?

A. True
B. False
C. Depends on p
D. Undefined

35 The phrase "without loss of generality" (WLOG) is used in proofs to:

A. Skip difficult steps.
B. Assert that by proving one case, similar cases are also proved due to symmetry.
C. Introduce a general variable.
D. State the conclusion.

36 If we want to prove that the set of primes is infinite, which proof strategy is historically used?

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

37 Which fallacy is committed here? ". is true. Therefore ."

A. Valid Argument (Modus Ponens)
B. Denying the Antecedent
C. Affirming the Consequent
D. Valid Argument (Modus Tollens)

38 The statement "For every real number , if , then " can be proved directly by:

A. Assuming and showing .
B. Assuming , multiplying inequality by (since ), getting .
C. Finding an example where .
D. Showing that implies .

39 What is the logical operator for the phrase "necessary and sufficient condition"?

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

40 In the domain of integers , let be " is even". The statement " is odd" is equivalent to:

A.
B.
C.
D. Both B and C

41 Which of the following implies is False?

A.
B.
C.
D.

42 A Non-constructive Existence Proof establishes is true by:

A. Producing an .
B. Showing that the non-existence of such an leads to a contradiction.
C. Checking all possible .
D. Using an algorithm.

43 Consider the argument: "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." This is an example of:

A. Universal Instantiation
B. Universal Generalization
C. Existential Instantiation
D. Existential Generalization

44 If is true, does it logically follow that is true?

A. Yes, always.
B. No, not necessarily.
C. Only if the domain is finite.
D. Only if P and Q are the same.

45 What is the Scope of a quantifier?

A. The size of the domain.
B. The part of the logical expression to which the quantifier applies.
C. The truth value of the quantifier.
D. The number of variables quantified.

46 A Corollary is best defined as:

A. An axiom.
B. A statement that follows readily from a previously proven theorem.
C. A conjecture that hasn't been proven.
D. A proof mistake.

47 Which logical law allows us to simplify ?

A. Simplification
B. Addition
C. Resolution
D. Conjunction

48 To prove the statement "If is an integer and is odd, then is odd" by contraposition, we assume:

A. is odd.
B. is even.
C. is odd.
D. is even.

49 The fallacy of Denying the Antecedent takes the form:

A.
B.
C.
D.

50 Which of the following is an example of a vacuously true statement?

A. If , then the moon is made of cheese.
B. If , then the sky is blue.
C. All prime numbers are odd.
D. There exists a number divisible by zero.
Next MCQ
Unit 2