Expressing generality — the language of proof

Why testing examples proves nothing, how 2n, 2n + 1 and n, n + 1, n + 2 encode "any even", "any odd" and "consecutive", and when one counterexample settles everything.

Algebra · Algebraic proof

Expressing generality — the language of proof

Why testing examples proves nothing, how 2n, 2n + 1 and n, n + 1, n + 2 encode "any even", "any odd" and "consecutive", and when one counterexample settles everything.

Why it works

"Show this works for 3, for 7, for 100" is evidence, not proof — the claim is about EVERY number, and no pile of examples covers them all. Algebra's letters do what examples can't:

n

stands for any integer at once, so one calculation with

n

is every calculation simultaneously.

The vocabulary — the standard encodings, with

n

any integer:

any even number: $2n$ (a multiple of 2, by construction);
any odd number: $2n + 1$ (one more than an even);
consecutive integers: $n, \; n + 1, \; n + 2, \ldots$ ;
consecutive EVEN numbers: $2n, \; 2n + 2, \; 2n + 4$ (they step

by 2);

consecutive ODD numbers: $2n + 1, \; 2n + 3$ ;
any multiple of $k$ : $kn$ .

Two INDEPENDENT numbers need two letters: "any two odd numbers" is

2n + 1

and

2m + 1

— writing both as

2n + 1

silently claims they're EQUAL, which shrinks the theorem. (Consecutive odds genuinely share the letter:

2n+1, 2n+3

.)

A tiny demonstration. The sum of two consecutive integers:

n + (n + 1) = 2n + 1

— one more than a multiple of 2 — odd, for every

n

at once. That's the whole game: encode, expand, read off the structure.

Disproving needs only ONE counterexample. "The sum of two primes is always even" dies on a single case:

2 + 3 = 5

, odd. To wreck a "for all" claim, one honest failure is a complete proof — hunt near the special cases (the prime 2, zero, negatives,

n = 1

).

Why the exam cares. Proof questions award marks for the ENCODING (right general forms), the ALGEBRA (expansion/factorising), and the CONCLUSION (a sentence connecting the result to the claim). Miss the encoding and nothing after it can score.