Mastering Mathematical Proofs: Year 12
Building Mathematical Rigour Direct Proof, Contradiction & Induction AS Level Mathematics
Starter Question
'The sum of two even numbers is even.' How would you prove this statement is always true? Discuss with your partner for 2 minutes
What is Mathematical Proof?
A logical argument that establishes the truth of a mathematical statement beyond doubt Uses definitions, axioms, and previously proven theorems Different from examples or arguments - must work for ALL cases Essential for mathematical rigour and certainty
Direct Proof Method
Start with given information or definitions Use logical steps to reach the conclusion Each step must follow logically from previous steps Structure: Assumption → Logical Steps → Conclusion
Direct Proof Example
Proof by Contradiction
Assume the opposite of what you want to prove Show this assumption leads to a logical contradiction Since the assumption creates impossibility, original statement must be true Powerful technique for 'impossible' statements
Contradiction Example
Prove: √2 is irrational Assume √2 = p/q (rational) where p, q have no common factors Then 2 = p²/q², so 2q² = p² This means p² is even, so p is even Let p = 2r, then 2q² = 4r², so q² = 2r² This means q is also even - contradiction! Therefore √2 must be irrational
Proof by Mathematical Induction
Prove a statement is true for all natural numbers Base case: Prove true for n = 1 Inductive hypothesis: Assume true for n = k Inductive step: Prove true for n = k + 1 Like dominoes falling - if first falls and each causes next to fall, all fall
Induction Example
Guided Practice
Work in pairs on these three problems: Direct Proof: If a divides b and b divides c, then a divides c Contradiction: Show 2x + 1 = 2 has no integer solutions Induction: Prove 2ⁿ > n for all n ≥ 1 10 minutes - I'll circulate to help
Reflection and Assessment
Which proof technique felt most natural to you? Where might we use proofs in real life? Computer algorithms and cryptography rely on proofs Exit ticket: Write a direct proof that the product of two even numbers is even