Mathematical Proof Techniques for AS

Year 12 Mathematics Direct Proof, Contradiction & Induction Building Rigorous Mathematical Arguments

Starter Question

'The sum of two even numbers is even.' How would you prove this statement is always true? Discuss in pairs for 3 minutes

What is Mathematical Proof?

A logical argument that establishes the truth of a mathematical statement beyond any doubt Uses definitions, previously proven theorems, and logical reasoning Different from examples or intuitive arguments Essential for mathematical rigour and certainty

Direct Proof

Start with known facts and definitions Use logical steps to reach the desired conclusion Example: Prove the sum of two odd integers is even Let first odd integer = 2a + 1, second = 2b + 1 Sum = (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1) Since (a + b + 1) is an integer, the sum is even

Proof by Contradiction

Assume the opposite of what you want to prove Show this assumption leads to a logical contradiction Therefore, the original statement must be true Example: Prove √2 is irrational Assume √2 = p/q where p, q are integers with no common factors Then 2 = p²/q², so 2q² = p² This means p² is even, so p is even Let p = 2k, then 2q² = 4k², so q² = 2k² This means q is also even - contradiction!

Proof by Mathematical Induction

Guided Practice

Work in pairs on these three problems: 1. Direct proof: If a divides b and b divides c, then a divides c 2. Contradiction: Show no integer solution to 2x + 1 = 2 3. Induction: Prove 2ⁿ > n for all n ≥ 1 15 minutes - I'll circulate to help

Reflection & Next Steps

'Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.' - William Paul Thurston Proof techniques are essential tools for mathematical understanding Practice makes perfect - work through examples regularly Next lesson: Advanced proof applications