Structure of Mathematical Proof

Types of Proofs

Direct Proof: Starts with a given proposition and uses logical steps to show that another statement is also true. This type of proof follows the structure ‘if P, then Q’.
Proof by Contrapositive: Shows that ‘if not Q, then not P’, which is the logical equivalent of a direct proof (if P then Q).
Proof by Contradiction (Reductio ad absurdum): Assumes that the statement to be proved is false, and then derives a contradiction, showing that the assumption is incorrect.
Proof by Exhaustion: Involves checking all possible cases. It is usually used when the number of cases is small and manageable.
Proof by Induction: Used to prove statements about natural numbers, where the proposition is shown to be true for a base case (often n=1), and then assuming it is true for n=k, it is shown to be true for n=k+1.

Proposition: The statement that is to be proven. This is often a hypothesis or an assumption.
Axioms: Statements that are assumed to be true without proof. These are the basic principles upon which other logical statements are built.
Theorem: A mathematical statement that has been proven to be true, usually using axioms and previously proven theorems.
Lemma: A smaller, often less important, theorem that is used as a stepping stone to prove a larger theorem.
Corollary: A statement that follows easily from a theorem.

Understand the proposition: Before attempting a proof, understand the statement to be proven completely including its hypotheses and conclusions.
Express in clear language: Aim for clarity, precision, and complete sentences to ensure that every step can be followed easily.
Justify each step: Provide suitable justifications for every step taken in the proof.
Link together the arguments: Every step in the proof should relate logically to the next one.
Check for errors: Review the completed proof for errors before finalising it.