TMUA Proof Techniques 4

Question 1

Which of the following is NOT a valid method of mathematical proof?

Accepted Answer

Proof by example

Answer

A single example proves existence but cannot prove a universal statement; only a counterexample can disprove one.

Question 2

To prove ∑_{k=1}^{n} k = n(n+1)/2 by induction, the base case checks:

Accepted Answer

n = 1: 1 = 1(2)/2 ✓

Answer

The standard base case is n = 1: the left side is 1 and the right side is 1(2)/2 = 1, which checks out.

Question 3

If a proof assumes 'let n be an arbitrary even integer' and derives a property, the conclusion holds for:

Accepted Answer

All even integers

Answer

Proving a property for an arbitrary even integer, with no further assumptions, establishes it universally for all even integers.

Question 4

A proof shows: 'Assume for contradiction that there are finitely many primes p₁, p₂, …, pₙ. Consider N = p₁p₂⋯pₙ + 1.' The next key observation is:

Accepted Answer

N has a prime factor not in the list

Answer

N leaves remainder 1 when divided by any pᵢ, so its prime factors are not in the assumed complete list — a contradiction.

Question 5

What is the logical form of modus tollens?

Accepted Answer

¬Q, P ⇒ Q ⊢ ¬P

Answer

Modus tollens: given P ⇒ Q and ¬Q, we conclude ¬P — the basis of many proof by contradiction arguments.

(TMUA) Test of Mathematics for University Admission Practice Test