Proof and Algebraic Methods

Mathematical proof

A proof is a logical argument that shows a statement is always true (or, for disproof, that it is false). At A-Level you must know several methods and use precise reasoning.

Notation and terms

• Integer n; consecutive integers n, n+1; even = 2n, odd = 2n+1.
• ⇒ (implies), ⇔ (if and only if), "for all", "there exists".
• A statement is only proved when shown true in all cases — one example is never a proof.

Proof by deduction

Build a logical chain from known facts to the result, using algebra.

• Example: prove the sum of two consecutive integers is odd. Let them be n and n+1: n + (n+1) = 2n + 1, which is odd (2 × integer + 1). ✓

Proof by exhaustion

Break the statement into a finite number of cases and check each one.

• Example: prove that n² + n is even for n = 1, 2, 3, 4… by testing each relevant case (or by considering n even and n odd separately).

Disproof by counter-example

To disprove a statement, find one example where it fails.

• Example: "all prime numbers are odd" is disproved by 2 (prime and even).

Proof by contradiction

Assume the opposite of what you want to prove, then show this leads to a contradiction — so the assumption must be false and the original statement true.

• Classic examples: proving √2 is irrational, and that there are infinitely many primes.
• Structure: "Assume … Then … which is a contradiction. Therefore …".

Algebraic methods (supporting proofs)

• Completing the square shows an expression is always positive/has a minimum.
• Factor theorem: if (x − a) is a factor of f(x), then f(a) = 0 — used to factorise and prove roots.

Worked example

Prove that the product of two consecutive even numbers is a multiple of 8... (outline): let them be 2n and 2n+2 = 2(n+1). Product = 4n(n+1). Since n(n+1) is the product of consecutive integers, one of them is even, so n(n+1) is even → 4 × even = multiple of 8. ✓

Common mistakes

• Using examples as a proof (they only illustrate, never prove a general statement).
• In contradiction proofs, forgetting to state the assumption clearly.
• Not covering all cases in proof by exhaustion.

Exam tips

• Define your variables generally (e.g. "let n be any integer").
• For "prove/show that", give a complete logical argument, not examples.
• For contradiction, clearly write the assumption and the contradiction reached.

Key facts to remember

• Deduction (algebraic chain), exhaustion (check all cases), counter-example (one case disproves), contradiction (assume the opposite → contradiction).
• One example is not a proof; a single counter-example is a disproof.
• Support proofs with algebra: completing the square, the factor theorem (f(a)=0 ⇔ (x−a) a factor).