TMUA Proof Techniques 5

Question 1

Which of the following best describes a 'non-constructive' existence proof?

Accepted Answer

It proves existence without identifying the object explicitly

Answer

A non-constructive proof establishes that something must exist (e.g. via contradiction) without providing an explicit example.

Question 2

Prove or disprove: 'For all integers n, n² − n is even.' The correct conclusion is:

Accepted Answer

True; n² − n = n(n−1) is a product of consecutive integers

Answer

n(n−1) is a product of consecutive integers, so one factor is always even, making the product even.

Question 3

Which rule of inference is: 'P ∨ Q, ¬P ⊢ Q'?

Accepted Answer

Disjunctive syllogism

Answer

Disjunctive syllogism eliminates one option in a disjunction when the other is known to be false.

Question 4

In a proof by cases for an integer n, the usual split is n ≡ 0 (mod 2) and n ≡ 1 (mod 2). What guarantees these two cases are exhaustive?

Accepted Answer

Every integer is either even or odd (the division algorithm)

Answer

The division algorithm guarantees every integer has remainder 0 or 1 when divided by 2, covering all possibilities.

Question 5

Why is 'proof by example' invalid for universal statements?

Accepted Answer

Both A and C

Answer

A single example demonstrates existence but cannot rule out counterexamples; universal proofs require an argument covering every case.

(TMUA) Test of Mathematics for University Admission Practice Test