Mathematical proof

Proof types (Prf1)

• Direct: start from what is given and deduce step by step ('\(n(n + 1)\) is a product of consecutive integers, so it is even').
• Cases: split by parity, by remainder mod \(3\), by sign; the cases must cover everything.
• Contradiction: assume the statement is false (the hypothesis true and the conclusion false), derive an impossibility. Classic: \(\sqrt2\) irrational, infinitely many primes.
• Counterexample: one instance with the hypothesis true and the conclusion false disproves a 'for all' claim; it says nothing about an 'exists' claim.
• Checking small cases (\(n^2 - n + 41\) is prime for \(n = 1, \ldots, 40\) but not \(41\)) is evidence, not proof.

Working with proofs (Prf2-Prf4)

• Ordering steps: the assumption comes first, the contradiction last; each line must use only earlier lines.
• Deducing from given implications: use the contrapositive ('\(f(x) > 0 \Rightarrow g(x) > 0\) and \(g(2) = -1\)' gives \(f(2) \leq 0\)).
• Backwards working (start from the result, reach a truth) is valid only if every step is reversible and the proof is then presented forwards.
• Conjecture from a pattern, then justify algebraically: \(T_n + T_{n+1} = (n + 1)^2\) by expanding.
• A proof for \(\sqrt2\) copied for \(\sqrt4\) breaks where the parity argument no longer applies - identify which step fails.