Mathematical Proof

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 $\sqrt{2}$ 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 $n = k$, then prove for $n = k+1$.
• 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 $\sqrt{2}$ 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