TMUA Proof Techniques 3

Question 1

Prove: if n² is even then n is even. The most efficient technique is:

Accepted Answer

Proof by contrapositive

Answer

Contrapositive: if n is odd then n² is odd, which follows immediately from (2k+1)² = 4k²+4k+1.

Question 2

Which of the following correctly states the well-ordering principle used in many proofs?

Accepted Answer

Every non-empty set of positive integers has a smallest element

Answer

The well-ordering principle states every non-empty subset of the positive integers contains a least element.

Question 3

To prove ∀n ∈ ℕ, n(n+1) is even, a direct proof uses:

Accepted Answer

n and n+1 are consecutive so one must be even

Answer

Among any two consecutive integers n and n+1, exactly one is even, so their product is divisible by 2.

Question 4

Which logical equivalence justifies replacing a proof of P ⇒ Q with a proof of ¬Q ⇒ ¬P?

Accepted Answer

Contrapositive equivalence

Answer

The contrapositive P ⇒ Q ≡ ¬Q ⇒ ¬P is a tautology, so both statements have identical truth values.

Question 5

A proof by exhaustion is appropriate when:

Accepted Answer

The domain is finite and all cases can be checked

Answer

Proof by exhaustion works by verifying the statement for every element in a finite domain.

(TMUA) Test of Mathematics for University Admission Practice Test