Proof
Proof by Deduction
- Understand that proof by deduction involves starting from known truths and using logical reasoning to arrive at a new conclusion.
- Grasp the importance of clearly stating assumptions at the start.
- Appreciate that each step must follow logically from the previous one, with no gaps in reasoning.
Direct and Indirect Proof
- Recognise the difference between direct and indirect proofs.
- In direct proof, the result is proven by a sequence of logical steps.
- With indirect proof or proof by contradiction, an assumption is made that the opposite of what is to be proven is true. If this leads to a contradiction, then the original assertion must be true.
Disproof by Counterexample
- Understand the concept of disproof by counterexample.
- Identify that a single counterexample is enough to disprove a proposition.
- Practice constructing counterexamples to disprove given statements.
Proof by Exhaustion
- Know that proof by exhaustion involves testing all possible cases.
- Be aware this method can be laborious and so is often not the most efficient strategy.
Understanding of Terminology
- Be familiar with the term theorem for a proven mathematical statement of importance.
- Understand axioms, which are statements accepted as true without proof.
- Recognise the term corollary for a result that follows directly from a theorem.
Mathematical Language and Notation in Proofs
- Use precise mathematical language and notation in your proofs.
- Be comfortable with commonly used notation, such as "∀" for 'for all' and "∃" for 'there exists'.
- Use logical symbols like "∧" for 'and', "∨" for 'or', and "¬" for 'not'.