Further Mathematics (MEI)

Core Pure

# Proof

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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'.