Proofs

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 $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$ 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 $ac \leq bc$ then $c \leq 0$ 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 $2^0 + 2^1 + ... + 2^n = 2^{n+1} - 1$ using induction.
• Disprove by Counter Example — showing a statement is false by providing a specific example where it does not hold. Example: Disproving $x^2 - 2y > 5$ for $x > 3$ by finding specific values of $x$ and $y$ 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