Mathematics

AS: Pure Mathematics

# Proof

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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.