Intro
Extension 2 proofs are marked on logical structure: state what you assume, what you deduce, and why each step follows. Direct proof, contradiction, and induction each have a standard opening sentence examiners expect. Keywords: formal proof writing, mathematical communication HSC, proving theory. NSW Year 12 Extension 2.
Summary
Direct proofs assume P and deduce Q. Contradiction assumes ¬Q and derives impossibility. Contrapositive proves ¬Q ⇒ ¬P. Use 'Suppose', 'Hence', and 'Therefore' deliberately. Practise rewriting textbook proofs from memory.
Clear proof writing also improves induction and Q16 responses — the same logical connectors apply. Read the NESA glossary for terms like 'hence' and 'show that' which imply specific proof obligations.
Key Points
- Direct: assume hypothesis, chain logical steps to conclusion.
- Contradiction: assume negation of what you want to prove; derive contradiction.
- Contrapositive: prove equivalent ¬Q ⇒ ¬P when direct route is awkward.
- Induction: separate base case, assumption, and step — see induction guide.
- Language markers: 'Suppose that', 'Hence', 'Without loss of generality' (when valid).
- Model answers in the HSC Proofs booklet.
Worked example
Question. Prove that the sum of two odd integers is even.
Solution.
- Let a and b be odd integers. Then a = 2m + 1 and b = 2n + 1 for integers m, n.
- a + b = 2m + 1 + 2n + 1 = 2(m + n + 1).
- Since m + n + 1 is an integer, a + b is divisible by 2 — hence even.
Answer. Sum of two odd integers is even.
Takeaway. Represent odd/even in algebraic form 2k + 1 or 2k at the start — examiners expect explicit structure.
Exam Preparation
Proof marks are communication marks. Each week, write one proof from memory, compare to a model, and highlight missing link words. Pair with induction practice for complete Extension 2 proof coverage.
Swap proofs with a study partner — each marks the other's logical gaps. Peer review trains you to spot missing 'Hence' statements before the HSC. Keep exemplar proofs from the Proofs booklet in a folder of model answers.
- Proof type identification. Label direct vs contradiction before writing.
- Rewrite from memory. One proof per week without notes.
- Peer or teacher review. Focus on 'Hence' gaps where logic jumps.
Proof questions appear across Extension 2 — not only in induction chapters. When asked to prove a statement false, provide a counterexample with verification. Quantifiers matter: 'for all' needs a general argument; 'there exists' needs one example. Read marker feedback from school assessments — 'needs more justification' usually means missing 'Hence' or un stated assumptions.
Mini-FAQ
Can I use bullet points in proofs?
Prefer connected sentences with logical connectors — bullets can obscure flow to markers.
When should I use contradiction?
When direct proof is messy or when proving non-existence (assume exists, derive absurdity).
Is 'WLOG' allowed?
Only when symmetry genuinely lets you simplify a case — misuse costs marks.
Common mistakes to avoid
- Circular reasoning — assuming what you need to prove.
- Missing 'Therefore' links between steps.
- Starting contradiction without stating the assumption clearly.
- Confusing contrapositive with converse (not equivalent).
Proof portfolio: keep five model proofs — direct, contradiction, contrapositive, induction, and counterexample — rewritten in your own words. Before the HSC, recite the opening sentence of each type without notes. Communication marks reward predictable structure more than novel phrasing. Peer-mark each other's proofs weekly to train spotting missing 'Hence' links before the exam.
Induction proofs share structure with general proof writing — study both posts in the same week. Contradiction proofs need a clear opening assumption; without it, markers cannot follow the logic chain.
Rewrite one textbook proof from memory each week and compare line-by-line with the model answer in the Proofs booklet. When markers write 'justify your step', they mean cite a theorem or algebra rule explicitly. Proof writing improves induction and Q16 responses because the same logical connectors — Suppose, Hence, Therefore — apply across Extension 2. Contrapositive proofs are equivalent to direct proofs of the contrapositive statement. Keep a folder of five exemplar proofs you rewrite monthly. Read NESA glossary entries for 'show that' and 'prove' before trial exams.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Proofs booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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All booklets are free for personal and school use under the CC BY 4.0 licence.
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