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.