The method is chosen by the logical shape of the claim and by the easiest definition to expose. A proof method is not a label added after the algebra; it determines the first assumption and the final sentence.
| Situation | Best first choice | Opening move |
|---|---|---|
| The hypothesis already has a useful algebraic definition. | Direct proof | Assume the hypothesis and translate it into its defining form. |
| A universal claim appears too strong. | Counterexample | Search for one admissible value that makes the conclusion false. |
| The negated conclusion gives a useful equation or lowest-terms representation. | Contradiction | Assume the conclusion is false and derive an impossibility. |
| \(\neg Q\) is easier to use than \(P\) in \(P\Rightarrow Q\). | Contraposition | Prove \(\neg Q\Rightarrow\neg P\). |
Direct proof: Assume \(P\). Use definitions and known results to obtain \(Q\). Therefore \(P\Rightarrow Q\).
Counterexample: State an admissible example, verify the hypothesis, show the conclusion fails, then state that the universal claim is false.
Contradiction: Assume the negation of the required conclusion. Derive an impossibility or violate an explicit condition. Reject the temporary assumption and conclude that the original statement is true.
Contraposition: Write the contrapositive accurately, prove it directly, then invoke logical equivalence to conclude the original implication.
| Common failure | Why it fails | Repair |
|---|---|---|
| Starting with the required conclusion | This often assumes what must be proved. | Begin with the hypothesis or the correct indirect assumption. |
| Checking examples | Finite evidence cannot prove a universal statement. | Use definitions to cover every admissible value. |
| Giving a non-admissible counterexample | It does not satisfy the claim’s hypothesis or domain. | Verify every condition before using the witness. |
| Reaching an odd-looking result | A contradiction must be logically impossible, not merely unexpected. | Name the violated condition explicitly. |
Choose a method before solving: (i) if \(n^3\) is even then \(n\) is even; (ii) every quotient of irrational numbers is irrational; (iii) \(\sqrt3\) is irrational.
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