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# MCQs on Types of Proofs

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)))

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Answer: a

Explanation: Definition of direct proof.

Which of the following can only be used in disproving the statements?

a) Direct proof

b) Contrapositive proofs

c) Counter Example

d) Mathematical Induction

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Answer: c

Explanation: Counter examples cannot be used to prove results.

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)))

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Answer: d

Explanation: Definition of proof by contraposition.

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

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Answer: c

Explanation: Definition of vacuous proof.

In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?

a) Direct proof

b) Proof by Contradiction

c) Vacuous proof

d) Mathematical Induction

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Answer: b

Explanation: Definition of proof by contradiction.

A proof covering all the possible cases, such type of proofs are known as ___________

a) Direct proof

b) Proof by Contradiction

c) Vacuous proof

d) Exhaustive proof

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Answer: d

Explanation: Definition of exhaustive proof.

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

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Answer: a

Explanation: Some examples are not valid in proving results.

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

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Answer: c

Explanation: Definition of proof by cases.

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

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Answer: c

Explanation: Definition of trivial proof.

A theorem used to prove other theorems is known as _______________

a) Lemma

b) Corollary

c) Conjecture

d) None of the mentioned

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Answer: a

Explanation: Definition of lemma.