Page 54 - HSC Proofs

1. HSC-Proofs - an Introduction

In many other areas of mathematics, the goal is to find $x$. In the HSC Mathematics Extension 2 course, the goal is to prove why a statement about $x$ must always be true. The "Nature of Proof" is arguably the most intimidating topic in the entire syllabus. It historically challenges students because rote memorization is useless here; you must construct logical arguments from scratch. Whether utilizing contradiction, contrapositive, or applying specific inequalities like AM-GM, this booklet trains you to think like a pure mathematician, demanding absolute rigor at every algebraic step.

2. Learning Outcomes & Syllabus Mapping

• The Nature of Proof.
• Understand and construct proofs using direct logic, contrapositive, and contradiction.
• Apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality.
• Apply the Triangle Inequality.
• Construct abstract proofs involving number theory (odd/even, divisibility, primes).

3. Prerequisites

• Flawless algebraic manipulation.
• Mastery of HSC-Induction (Mathematics Extension 1).
• A strong grasp of logic and implication (If $P$, then $Q$).

4. Common HSC Mistakes

The most common mistake is circular reasoning—assuming the result you are trying to prove in line one, manipulating it to something true (like $0 = 0$), and declaring the proof finished. You MUST start from known axioms (e.g., $x^2 \ge 0$) and build towards the required result. Students also frequently confuse the inverse ("If not $P$, then not $Q$") with the contrapositive ("If not $Q$, then not $P$"), applying the wrong logical structure entirely.

5. Sample Worked Problem

6. Exam Strategy & Weighting

Proof constitutes a significant portion of the Mathematics Extension 2 paper. You will likely see a logic question in the multiple-choice section (identifying a contrapositive statement). The true test comes in Question 15 or 16, where you will be asked to prove a complex abstract inequality or a number theory property. The best strategy for inequality proofs is to work on rough paper starting from the result to figure out the path, but then write your final exam answer forwards from a known axiom.

7. Key Definitions / Glossary Summary

• Direct Proof: Starting with known facts and logically deducing the conclusion ($P$ implies $Q$).
• Contrapositive Proof: Proving "If not $Q$, then not $P$", which is logically equivalent to "If $P$, then $Q$".
• Proof by Contradiction: Assuming the statement is false, and showing this leads to a logical impossibility.
• Axiom: A foundational mathematical truth accepted without proof (e.g., the square of a real number is $\ge 0$).

8. Frequently Asked Questions (FAQ)

Q: Can I use mathematical induction for these proofs? A: Yes, if the proposition involves "for all integers $n \ge 1$", induction is almost always the intended method.

Q: Will I be asked to prove things using the Cauchy-Schwarz inequality? A: While Cauchy-Schwarz is not explicitly named in the syllabus, the algebraic techniques used to derive it are highly relevant and often tested in disguised forms.

9. Where to next?

With a solid grounding in pure logic and proof structures, you are ready to apply these exact reasoning skills to the geometric and algebraic challenges found in the HSC-ComplexNumbers and HSC-Math-Extension-2-Book modules.