# Proof

• ‘Proof’ is a method used in mathematics to establish the certainty of a mathematical statement, formula or an equation.
• Different types of proof techniques you need to understand include direct proof, proof by contradiction (also known as indirect proof), proof by exhaustion, and counter-examples.
• Direct proof entails assuming the truth of the premises, and based on these assumptions, you will show that the conclusion must be true as well.
• Proof by contradiction involves assuming that a statement or theorem is wrong, and then showing that this assumption leads to an absurdity, thus proving the original statement or theorem as true.
• Proof by exhaustion entails checking all possible cases and showing that the conclusion holds true in each one.
• A counter-example is a specific case or example that contradicts a statement or proposition and demonstrates its falseness.
• Another common type of proof is a proof by deduction, which relies on using existing theorems and axioms to demonstrate the truth of a given statement.
• The key step in writing proofs is to determine and note down the key premises or assumptions to start with. This is followed by deducing the necessary subsequent steps until the conclusion is reached.
• When structuring arguments in proving a statement, always be logical, clear, and concise. Ensure each step is connected and follows naturally from the previous one.
• You need to learn to use algebraic methods in proofs. For instance, in proving a formula, you might need to perform mathematical calculations using algebra such as factorising, expanding brackets, and rearranging equations.
• Understand the concept of a mathematical statement and its converse. A converse of a statement changes the direction of the initial statement. For example, the converse of “If it’s raining, then the ground is wet” is “If the ground is wet, then it’s raining”. Not all converses are true, even if their original statements are. This is an essential concept in proof by contradiction.
• Practise creating your own algebraic proofs. This helps strengthen and reinforce your understanding of the algebra concepts and the nature of proof itself. Remember, the more you practise, the better you get at it.