Study Notes
Mathematical proof involves demonstrating the truth of a statement using logical reasoning. Proof by Induction — a method that proves a statement is true for all positive integers by showing it holds for the first case and assuming it holds for an arbitrary case to prove the next. Example: Proving a summation formula or divisibility statement. Proof by Contradiction — a method where you assume the opposite of what you want to prove, and show that this assumption leads to a contradiction. Example: Proving there are infinitely many prime numbers or that is irrational.
Exam Tips
Key Definitions to Remember
- Proof by Induction
- Proof by Contradiction
Common Confusions
- Forgetting to prove the base case in induction
- Misunderstanding the assumption step in induction
Typical Exam Questions
- How do you prove a statement using induction? Show the base case is true, assume true for , then prove for .
- How do you prove a statement using contradiction? Assume the opposite of the statement and show a contradiction arises.
- What is an example of a proof by contradiction? Proving is irrational.
What Examiners Usually Test
- Ability to correctly apply the steps of induction
- Understanding of how to set up a contradiction
- Clarity in logical reasoning and explanation