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.