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Unit 1: Logic and Proofs - Practice Quiz

MTH401 — Discrete Mathematics 60 Questions
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1 Which of the following is a proposition?

Propositional logic Easy
A. .
B. is a prime number.
C. Close the window.
D. What time is it?

2 When is the implication false?

Propositional logic Easy
A. When and are false
B. When is false and is true
C. When and are true
D. When is true and is false

3 Which expression is equivalent to by De Morgan's law?

Propositional equivalences Easy
A.
B.
C.
D.

4 Which expression is logically equivalent to ?

Propositional equivalences Easy
A.
B.
C.
D.

5 What does the statement mean?

Quantifiers Easy
A. is true for some
B. is true for every
C. has no truth value
D. is false for every

6 Which expression is the negation of ?

Quantifiers Easy
A.
B.
C.
D.

7 What is a theorem?

Introduction to proof Easy
A. A statement proved to be true
B. An example without justification
C. A question with no answer
D. A statement known to be false

8 To prove directly, what is normally assumed first?

Direct proof Easy
A. Assume that is false
B. Assume that is false
C. Assume that is true
D. Assume that is true

9 In a direct proof that the sum of two even integers is even, how should the integers be written?

Direct proof Easy
A. and
B. and
C. and
D. and

10 What is the contrapositive of ?

Proof by contraposition Easy
A.
B.
C.
D.

11 What is the contrapositive of "If is even, then is even"?

Proof by contraposition Easy
A. If is even, then is even.
B. If is odd, then is odd.
C. If is odd, then is even.
D. If is even, then is odd.

12 How can an implication be proved vacuously?

Vacuous and trivial proof Easy
A. Show that is false
B. Show that is true
C. Show that is false
D. Show that equals

13 How can an implication be proved trivially?

Vacuous and trivial proof Easy
A. Show that is unknown
B. Show that is false
C. Show that is false
D. Show that is true

14 When is proof by cases an appropriate strategy?

Proof strategy Easy
A. When all possible cases can be examined
B. When the conclusion must be assumed
C. When no definitions are available
D. When only one example is known

15 To prove a statement by contradiction, what is assumed initially?

Proof by contradiction Easy
A. Assume that is equivalent
B. Assume that is false
C. Assume that is undefined
D. Assume that is true

16 In a proof that there is no greatest integer, suppose is the greatest integer. Which fact gives a contradiction?

Proof by contradiction Easy
A. is equal to
B. is larger than
C. is equal to
D. is smaller than

17 To prove that statements and are equivalent, which implications must be proved?

Proof of equivalence and counterexamples Easy
A. and
B. and
C. and
D. and

18 Which value is a counterexample to the claim "For every integer , "?

Proof of equivalence and counterexamples Easy
A.
B.
C.
D.

19 From and , a student concludes . What is this invalid argument called?

Mistakes in proof Easy
A. Applying modus ponens
B. Denying the consequent
C. Affirming the consequent
D. Using the contrapositive

20 Why does checking several examples not prove a universal statement?

Mistakes in proof Easy
A. Examples cannot use integers
B. Universal statements lack truth values
C. An unchecked case may be false
D. Every example must be negative

21 Suppose is true, is false, and is true. What are the truth values of and , respectively?

Propositional logic Medium
A. False and false
B. True and true
C. False and true
D. True and false

22 Let mean a student submits the assignment and mean the student receives credit. Which sentence represents ?

Propositional logic Medium
A. Receiving credit is sufficient for submitting the assignment.
B. Submitting the assignment is equivalent to receiving credit.
C. Not submitting the assignment guarantees receiving no credit.
D. Submitting the assignment is sufficient for receiving credit.

23 Which proposition is logically equivalent to ?

Propositional equivalences Medium
A.
B.
C.
D.

24 Simplify the proposition .

Propositional equivalences Medium
A.
B.
C.
D.

25 What is the correct negation of ?

Quantifiers Medium
A.
B.
C.
D.

26 Over the domain of integers, which quantified statement is true?

Quantifiers Medium
A.
B.
C.
D.

27 To begin proving the statement , if is divisible by , then is divisible by , which setup is appropriate?

Introduction to proof Medium
A. Choose an arbitrary integer and assume for some integer .
B. Choose an arbitrary integer and assume for some integer .
C. Choose a particular integer and verify that it is divisible by .
D. Assume every integer is divisible by and derive that it is divisible by .

28 Suppose , , and . Which equation directly establishes that ?

Direct proof Medium
A. for integers
B. for integers
C. for integers
D. for integers

29 In a direct proof that the square of an odd integer is odd, let . Which expression completes the argument?

Direct proof Medium
A.
B.
C.
D.

30 What is the contrapositive of the statement: If is even, then is even?

Proof by contraposition Medium
A. If is not even, then is even.
B. If is odd, then is odd.
C. If is odd, then is odd.
D. If is even, then is even.

31 To prove by contraposition that if is odd, then is odd, which statement should be proved directly?

Proof by contraposition Medium
A. If is even, then is even.
B. If is odd, then is even.
C. If is even, then is even.
D. If is odd, then is odd.

32 Consider the statement: For every integer , if and , then . Why is the implication vacuously true?

Vacuous and trivial proof Medium
A. The conclusion cannot hold for any integer .
B. The hypothesis cannot hold for any integer .
C. The conclusion follows by expanding the expression .
D. The hypothesis and conclusion are both always true.

33 For real , the implication can be proved trivially because which fact holds?

Vacuous and trivial proof Medium
A. The hypothesis holds for every real .
B. The hypothesis is false for every real .
C. The conclusion holds for every real .
D. The conclusion holds only when .

34 To prove for every real number by cases, which cases are most appropriate?

Proof strategy Medium
A. and
B. and
C. integer and noninteger
D. rational and irrational

35 A claim about two integers and is unchanged when and are exchanged. Which assumption is justified by symmetry?

Proof strategy Medium
A. Assume without loss of generality that .
B. Assume without loss of generality that .
C. Assume without loss of generality that .
D. Assume without loss of generality that .

36 In the standard contradiction proof that is irrational, one assumes with . What contradiction is eventually obtained?

Proof by contradiction Medium
A. Both and must be odd.
B. Both and must be even.
C. must be even and odd.
D. must be odd and even.

37 To prove by contradiction that there is no greatest integer, suppose is the greatest integer. Which observation yields the contradiction?

Proof by contradiction Medium
A. is an integer smaller than .
B. is an integer equal to .
C. is an integer greater than .
D. is an integer opposite in sign to .

38 Which proof structure is sufficient to establish ?

Proof of equivalence and counterexamples Medium
A. Prove both and .
B. Prove both and .
C. Prove both and .
D. Prove both and .

39 Which value is a counterexample to the claim: For every integer , if is divisible by , then is divisible by ?

Proof of equivalence and counterexamples Medium
A.
B.
C.
D.

40 A student argues: If an integer is divisible by , then it is even. The integer is even, so is divisible by . What is the error?

Mistakes in proof Medium
A. The student incorrectly negates the hypothesis.
B. The student assumes the conclusion is false.
C. The student incorrectly proves the converse.
D. The student uses an invalid existential witness.

41 For how many truth assignments to , , and is the proposition true?

Propositional logic Hard
A.
B.
C.
D.

42 Which proposition is logically equivalent to

Propositional equivalences Hard
A.
B.
C.
D.

43 What is the correct negation of

Quantifiers Hard
A.
B.
C.
D.

44 Assume the domain is nonempty and does not contain as a free variable. Which equivalence is valid for every choice of predicates?

Quantifiers Hard
A.
B.
C.
D.

45 A theorem claims that holds for every integer . Which result is sufficient by itself to refute the theorem?

Introduction to proof Hard
A. An integer satisfying
B. A proof that is false only
C. A proof that holds conditionally
D. An integer satisfying

46 Which algebraic step is central to a direct proof that for every odd integer ?

Direct proof Hard
A. Write and factor
B. Write and factor
C. Write and expand
D. Write and expand

47 Suppose . Which expression directly establishes that ?

Direct proof Hard
A.
B.
C.
D.

48 Which statement is the contrapositive of: "For integers , if , then "?

Proof by contraposition Hard
A. If , then
B. If , then
C. If , then
D. If , then

49 Which is the contrapositive of the statement "If is irrational, then is irrational or is irrational"?

Proof by contraposition Hard
A. If is rational, then and are rational
B. If or is rational, then is rational
C. If and are irrational, then is irrational
D. If and are rational, then is rational

50 Over the integers, classify the proofs of the two statements: (I) and (II) .

Vacuous and trivial proof Hard
A. Both are vacuous proofs
B. I is trivial; II is vacuous
C. I is vacuous; II is trivial
D. Both are trivial proofs

51 Suppose . Why is the statement true for every predicate ?

Vacuous and trivial proof Hard
A. Its conclusion is true for every
B. Its antecedent is false for every
C. Its conclusion implies its antecedent
D. Its converse is true for every

52 To prove that odd implies both and are odd, which strategy most directly avoids separately analyzing all parity combinations?

Proof strategy Hard
A. Assume the conclusion and use contradiction
B. Prove the contrapositive using evenness
C. Verify several odd numerical examples
D. Prove the converse using divisibility

53 In proving a statement about two real numbers and , when is it valid to begin with "without loss of generality, assume "?

Proof strategy Hard
A. When interchanging and preserves the claim
B. When at least one numerical example has
C. When the desired conclusion contains
D. When the case seems algebraically harder

54 In the standard contradiction proof that is irrational, one writes . Which condition is essential before deducing a contradiction from both and being even?

Proof by contradiction Hard
A. and are both positive
B. and are relatively prime
C. and are consecutive integers
D. and have opposite parity

55 A contradiction proof assumes that there is a greatest rational number . Which construction most directly contradicts the maximality of ?

Proof by contradiction Hard
A. A rational with
B. An integer with
C. An irrational with
D. A rational with

56 Which collection of implications is sufficient to prove that statements , , and are logically equivalent?

Proof of equivalence and counterexamples Hard
A. , , and
B. , , and
C. , , and
D. , , and

57 Which triple is a counterexample to the claim that, for positive integers, if and only if or ?

Proof of equivalence and counterexamples Hard
A. , ,
B. , ,
C. , ,
D. , ,

58 Which formula is logically equivalent to ?

Propositional equivalences Hard
A.
B.
C.
D.

59 Which formula is equivalent to the negation of

Quantifiers Hard
A.
B.
C.
D.

60 A purported proof begins with , obtains , factors to , and concludes after cancellation before deriving . What invalid step enables the false conclusion?

Mistakes in proof Hard
A. Factoring as
B. Multiplying the equality by a nonzero value
C. Replacing an occurrence of by
D. Canceling when
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Unit 2