Further Mathematics

Proof

# Proof

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Proof

Basics of Proofs

**Proofs**are mathematical statements that are shown to be true using logical reasoning and previously established theorems.- A proof can be as simple as demonstrating that for a given equation, the left-hand side equals the right-hand side.
- More complex proofs can involve a series of calculations or logical deductions.
- Remember that a proof must show that something is true for
*all*relevant cases - just giving example cases isn't sufficient.

Direct Proof

- A
**direct proof**is made by a logical sequence of statements that lead to the conclusion being proved. - It often involves the application of specific rules or properties, such as the properties of integers or relational operators.
- Often, direct proofs start with a premise, and then apply a series of logical steps to reach a conclusion.
- Always write your steps clearly for direct proof and justify each step.

Proof by Induction

- An
**inductive proof**is a method used mainly to establish propositions about all natural numbers, or about all members of an infinite set. - Two steps:
**Base Step**- showing it is true for the first case (usually n=1 or n=0);**Inductive Step**- assuming it is true for some arbitrary case, then showing it is also true for next case. - It's a two-stage process - if both stages are correct, then the proposition is considered to be proved.
- Use the rules of algebra to manipulate your equations in the inductive step.

Contrapositive Proof

- A
**contrapositive proof**uses the law of contrapositive: if P implies Q, then NOT Q implies NOT P. - Usually used when the original statement is difficult to prove directly.
- Essentially proving the opposite can lead to the same result.
- Useful to make the difficult aspects of the proof accessible.

Proof by Contradiction

- Also known as
**reductio ad absurdum**, this is used when a proposition is shown to be true because its negation leads to a contradiction. - Begin by assuming that the statement you want to prove is false, then work from there until you get a contradiction.
- Strong method to prove statements in mathematics.
- Be clear and concise when writing out a proof by contradiction.

Proof Using Counter Examples

- In a proof by
**counterexample**, you disprove a statement by showing an example where it isn't true. - Only applicable when disproving universal statements - not for proving them.
- Always verify that your counterexample is both valid and relevant to the given supposition.