P2.1 - Proof

A mathematical proof starts from stated assumptions and uses valid logical steps to show that the conclusion must follow for every object in the stated domain.

To prove $2x^2+8\ge8x$ for every real $x$, move everything to one side: $$2x^2-8x+8=2(x-2)^2.$$ Because a real square is non-negative, $2(x-2)^2\ge0$, so the stated inequality follows for every real $x$.

When multiplying, dividing, taking roots or cancelling factors, state the condition that makes the step valid. For example, dividing an inequality by a positive quantity preserves its direction; dividing by a quantity of unknown sign does not.

Proof by exhaustion works when the allowed possibilities can be divided into a finite, complete set of cases. Each case must be checked, and the cases must leave no allowed value uncovered.

To prove $n^2+n$ is even for every integer $n$, exhaust the two parity cases. If $n=2k$, then $n^2+n=2k(2k+1)$, which has a factor 2. If $n=2k+1$, then $n^2+n=(2k+1)(2k+2)=2(2k+1)(k+1)$, also even. Every integer is even or odd, so the proof is complete.

For a literal finite set, the single-digit primes are exactly $2,3,5,7$. The values of $p^2+p$ are respectively $6,12,30,56$, all even. Naming the complete allowed set is what turns these checks into a proof.

A universal claim says a conclusion holds for every allowed input. To disprove it, one counterexample is enough: choose an input that satisfies the claim's conditions but makes its conclusion false.

Consider the claim ‘$n^2-n+1$ is prime for every positive integer $n$’. Take $n=5$, which is in the stated domain. Then $$5^2-5+1=21=3\times7,$$ so the result is composite. Therefore $n=5$ is a counterexample and the universal claim is false.

A useful search targets values likely to expose the weakness, but the final disproof must show the calculation and the failed property. A valid counterexample need not be the smallest one.