Study Notes
Mathematical proofs are essential for verifying the correctness of mathematical statements. They are like solving a jigsaw puzzle where all pieces must fit together logically.
- Deductive Proof — a method where a sequence of logically valid statements leads to a conclusion. Example: Proving by simplifying the left-hand side to match the right-hand side.
- Proof using Contraposition — involves proving that if the conclusion is false, then the premise must also be false. Example: Proving that if then by showing the contrapositive.
- Proof by Induction — a technique to prove statements for all positive integers by proving it for the base case and assuming it for an arbitrary case. Example: Proving using induction.
- Disprove by Counter Example — showing a statement is false by providing a specific example where it does not hold. Example: Disproving for by finding specific values of and that do not satisfy the inequality.
Exam Tips
Key Definitions to Remember
- Deductive Proof
- Proof using Contraposition
- Proof by Induction
- Disprove by Counter Example
Common Confusions
- Confusing the steps of proof by induction
- Misunderstanding the contrapositive in proofs
Typical Exam Questions
- What is a deductive proof? A method where a sequence of logically valid statements leads to a conclusion.
- How do you prove a statement using contraposition? By proving that if the conclusion is false, then the premise must also be false.
- How can a counter example disprove a theorem? By providing a specific example where the theorem does not hold.
What Examiners Usually Test
- Understanding of different proof techniques
- Ability to apply proof methods to solve problems
- Recognizing valid and invalid proofs