Proof
Types of Proof
- Direct Proof: Begin with assumptions then use logical deductions to show a statement is true.
- Proof by Contradiction: Assume the opposite of what you are trying to prove, then demonstrate an absurd or impossible conclusion from this.
- Proof by Exhaustion: Show a statement is true for each member of a finite set.
- Proof by Induction: Suitable for proving statements about natural numbers.
Methods of Proof
- Deductive Reasoning: Each step in the proof is logically deducted from the previous ones.
- Analytic Methods: Rely on analysis and calculation, like algebraic manipulations.
- Geometric Proofs: Use geometric principles to demonstrate truths, often with diagrams.
Logical Deduction
- Understand that a valid argument is one where it is impossible for the premises to be true and the conclusion to be false.
- Recognise the use of implications (if...then... statements) in mathematical proofs.
- Understand contrapositives, where the direction of an implication is reversed and each statement is negated.
Key Concepts
- Understand the concept of a theorem: an important statement that has been proven to be true.
- Recognise the importance of axioms or postulates: self-evident truths that do not need to be proven.
- Identify lemmas: preliminary propositions useful for proving larger theorems.
- Understand that a corollary is a statement that follows with little to no proof required from a previously proven statement.
Proof Writing Guidelines
- Take into account precision and clarity in writing proofs.
- Clear and logical outline: Each step follows from the previous one in an orderly manner.
- Appropriate use of mathematical terminology and notation.
Common Pitfalls
- Beware of circular reasoning, where the thing to be proven is assumed in the proof.
- Avoid undefined terms, ambiguity, or logical fallacies.
- Do not confuse proof by example with a valid proof. This method can only suggest a proof, not replace it.