Question

Describe the main parts of a proof.

Answer 1

A mathematical proof normally has several standard parts. Clear structure helps readers follow the logic and verify correctness. The main parts are:

1. Statement (What you are proving)
- A clear, precise formulation of the theorem, proposition, lemma, corollary or claim, including the domain and any quantifiers.

2. Hypotheses / Given information
- List the assumptions or conditions you are allowed to use. Make explicit which variables are arbitrary and which are fixed.

3. Definitions and notation (if needed)
- Introduce or recall any definitions, notation or conventions used in the proof so there is no ambiguity.

4. Proof strategy / roadmap (optional but helpful)
- A short statement of the approach (e.g., direct proof, proof by contradiction, induction, contrapositive, existence by construction). For longer proofs, an outline of the main steps is useful.

5. Lemmas or claims (if needed)
- State and prove intermediate results (claims or lemmas) that simplify the main argument. Number or label them for clarity.

6. Body of the proof: sequence of logical steps
- A chain of statements each justified by definitions, previously proved results, axioms, or clear reasoning. Each step should follow logically from earlier ones; cite theorems or rules when used.

7. Handling edge or special cases
- If the statement requires separate treatment of cases (e.g., n = 0, n > 0), handle them explicitly.

8. Conclusion
- A clear closing sentence that says the theorem follows from the steps (often written “Therefore…”, “Hence…”, or “This proves the claim.”) and optionally a QED symbol.

9. Verification / remarks (optional)
- Check that all quantifiers and domains were respected, note any limitations, or state corollaries and possible generalizations.

Brief example (direct proof)
- Statement: “The sum of two even integers is even.”
- Given: Let a and b be even integers.
- Definitions/notation: Even means an integer is 2k for some integer k.
- Proof: Since a and b are even, a = 2m and b = 2n for some integers m,n. Then a + b = 2m + 2n = 2(m + n). Because m + n is an integer, a + b is of the form 2k, so it is even.
- Conclusion: Therefore the sum of two even integers is even. QED.

Tips: be explicit about quantifiers and domains, justify each nontrivial step, and choose an appropriate proof method for the statement.