Proof

Section 1

1 spec leaf

Notes and three levels of exam-style practice for each registered specification leaf in this section.

Open the printable pack

1.1

Understand and use the structure of mathematical proof, from assumptions through logical steps to a conclusion; use proof by deduction, exhaustion, disproof by counter example, and contradiction (irrationality of √2, infinity of primes).

A proof begins with stated assumptions, uses valid implications at every step and ends with a conclusion that matches the claim.
Choose the proof form deliberately: deduction for a general algebraic chain, exhaustion for finitely many cases, a counterexample to disprove a universal claim, or contradiction by assuming the opposite.
In the classic contradiction for $\sqrt2$ , writing $\sqrt2=a/b$ in lowest terms leads to both $a$ and $b$ being even, contradicting the lowest-terms assumption.
Checking several examples is not a proof of a universal statement; in contradiction proofs, also make clear exactly why the derived result conflicts with the assumption.

Tier 1 · Easy

2 marks

ORIGINAL

Disprove the claim that $n^2+n+41$ is prime for every non-negative integer $n$ .

Tier 2 · Standard

4 marks

ORIGINAL

Prove by exhaustion that $n^2+n+2$ is even whenever $n$ is an integer.

Tier 3 · Hard

5 marks

ORIGINAL

Suppose someone lists all primes as $p_1,p_2,\ldots,p_k$ . Construct an integer from this list and use contradiction to prove that the list cannot be complete.