Study Notes
Algebraic proof involves demonstrating that a mathematical statement is true in every possible case using logical reasoning and known facts.
- Proof by Deduction — a method of proving a statement by starting from known facts and using logical steps to reach a conclusion. Example: Proving the product of two odd numbers is odd.
- Proof by Exhaustion — a method that involves breaking a statement into smaller cases and proving each case separately. Example: Proving the sum of two consecutive square numbers between 100 and 200 is odd.
- Counter-Example — an example that disproves a statement by showing it does not hold true in all cases. Example: Showing 3n + 3 is not a multiple of 6 for n = 2.
Exam Tips
Key Definitions to Remember
- Proof by Deduction
- Proof by Exhaustion
- Counter-Example
Common Confusions
- Thinking one example is enough to prove a statement true
- Forgetting to cover all possible cases in a proof
Typical Exam Questions
- Prove that the product of two consecutive odd numbers is odd? Use algebraic expressions for odd numbers and show the product is odd.
- Prove that x^2 − 10x + 25 ≥ 0 for all values of x? Complete the square to show the expression is always non-negative.
What Examiners Usually Test
- Ability to construct a logical and structured argument
- Use of algebraic manipulation to prove identities
- Finding counter-examples to disprove statements