TMUA Proof Techniques 2

Question 1

Which method proves a statement by assuming the negation of the conclusion and deriving a contradiction?

Accepted Answer

Proof by contradiction

Answer

Proof by contradiction assumes the negation is true and shows this leads to a logical impossibility.

Question 2

To prove P ⇒ Q by contrapositive, what do you actually prove?

Accepted Answer

¬Q ⇒ ¬P

Answer

The contrapositive of P ⇒ Q is ¬Q ⇒ ¬P, which is logically equivalent to the original statement.

Question 3

In a proof by induction, the inductive step proves which of the following?

Accepted Answer

P(k) ⇒ P(k+1)

Answer

The inductive step assumes P(k) (the inductive hypothesis) and deduces P(k+1).

Question 4

Which of the following is a valid counterexample to the claim 'All prime numbers are odd'?

Accepted Answer

2

Answer

2 is prime and even, disproving the claim that all primes are odd.

Question 5

A proof that every integer n satisfies n² ≥ 0 is best classified as:

Accepted Answer

Direct proof using algebraic identity

Answer

Writing n² = (n)(n) and noting a product of two numbers with the same sign is non-negative is a direct algebraic proof.

(TMUA) Test of Mathematics for University Admission Practice Test