# Discrete Mathematics Questions and Answers – Types of Proofs

This set of Discrete Mathematics Assessment Questions and Answers focuses on “Types of Proofs”.

1. Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove _________
a) ∀nP ((n) → Q(n))
b) ∃ nP ((n) → Q(n))
c) ∀n~(P ((n)) → Q(n))
d) ∀nP ((n) → ~(Q(n)))

Explanation: Definition of direct proof.

2. Which of the following can only be used in disproving the statements?
a) Direct proof
b) Contrapositive proofs
c) Counter Example
d) Mathematical Induction

Explanation: Counter examples cannot be used to prove results.

3. Let the statement be “If n is not an odd integer then sum of n with some not odd number will not be odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “sum of n with some not odd number will not be odd.” A proof by contraposition will be ________
a) ∀nP ((n) → Q(n))
b) ∃ nP ((n) → Q(n))
c) ∀n~(P ((n)) → Q(n))
d) ∀n(~Q ((n)) → ~(P(n)))

Explanation: Definition of proof by contraposition.

4. When to proof P→Q true, we proof P false, that type of proof is known as ___________
a) Direct proof
b) Contrapositive proofs
c) Vacuous proof
d) Mathematical Induction

Explanation: Definition of vacuous proof.

5. In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?
a) Direct proof
c) Vacuous proof
d) Mathematical Induction

Explanation: Definition of proof by contradiction.
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6. A proof covering all the possible cases, such type of proofs are known as ___________
a) Direct proof
c) Vacuous proof
d) Exhaustive proof

Explanation: Definition of exhaustive proof.

7. Which of the arguments is not valid in proving sum of two odd number is not odd.
a) 3 + 3 = 6, hence true for all
b) 2n +1 + 2m +1 = 2(n+m+1) hence true for all
c) All of the mentioned
d) None of the mentioned

Explanation: Some examples are not valid in proving results.

8. A proof broken into distinct cases, where these cases cover all prospects, such proofs are known as ___________
a) Direct proof
b) Contrapositive proofs
c) Vacuous proof
d) Proof by cases

Explanation: Definition of proof by cases.

9. A proof that p → q is true based on the fact that q is true, such proofs are known as ___________
a) Direct proof
b) Contrapositive proofs
c) Trivial proof
d) Proof by cases

Explanation: Definition of trivial proof.

10. A theorem used to prove other theorems is known as _______________
a) Lemma
b) Corollary
c) Conjecture
d) None of the mentioned

Explanation: Definition of lemma.

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