Understanding Proof

Proof in mathematics is a method used for verifying that a mathematical statement or theorem is universally true.
It is a logical argument that demonstrates the truth of a statement beyond any doubt, not based on examples or pattern spotting.
Proofs rely on previously proven statements, definitions, axioms, and logical reasoning.
The concept of proof is fundamental to mathematics, used to ensure the validity and integrity of mathematical concepts and relations.
Proofs are inductive, that is, they build upon prior knowledge and understanding.

Types of Proof

Direct proof is the most straightforward type of proof, where statements are proven by a series of logical steps from the initial assumption to the conclusion.
Indirect proof (also known as proof by contradiction) is a method involving initially assuming the opposite of the statement to be proven, and then showing that this leads to a contradiction.
Proof by exhaustion involves proving a statement by testing it in all possible cases. It’s usually used when the number of cases is fairly small.
Proof by induction is a method used to prove a statement or theorum is true for all natural numbers. It involves proving the base case and then assuming the statement is true for the number ‘n’, and then proving it for the number ‘n+1’.

Understand the problem. Make sure you fully comprehend what the statement or theorem is proposing.
Devise a plan. This may involve drawing diagrams, or using known mathematical relationships.
Execute the plan. Perform each step of the plan diligently, ensuring accuracy at each stage.
Self-check. Always review the proof upon completion, checking for any logical errors or missed steps in the argument.

All proofs should be supported by a clear and accurate narrative that explains the logical progression from the initial assumption to the final conclusion.
Precision is vital. Every step must be justifiable with known truths and inferences should follow logically from any assumptions made.
Highlight any points in the proof where an assumption is made or a known truth is used to justify the next step.
The proof should be understandable to someone else reading it, even if they aren’t familiar with the topic - this is a good test to apply to your own work.
Missteps or vague arguments in proofs can lead to false results, so attention to detail when writing a proof is crucial.