Mathematical Proof

Abstract

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• Concise, Polished Mathematical Argument explaining the validity of Mathematical Statement to skeptic
• There are many Proof Methods

Direct Proof

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• Is difficult when the thing we want to proof has an absence of a form like Irrationality of a number, which is number that does

Proof by Deduction (演绎推理)

• Direct Proof
• Used when the number of cases is infinite
• Use Theorem & Axioms to proof something
• Usually takes the form of - To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property

Example

• Prove that the sum of any two even integers is even
• Prove the sum of any two rational numbers is rational

Proof by Exhaustion/Brute-force/Cases

• List down all the possible cases and check on all cases
• Useful there is a handful of possible cases

Proof by Construction/Example

• A form of Direct Proof
• For Existential, we can proof by providing an example that fulfil the conditions

Indirect Proof (反证法)

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• When Direct Proof is hard to derive, we can try indirect proof

Proof by Counterexample (反例法)

• An example that shows that a Mathematical Statement isn’t always true. Useful for Universal

Proof by Contradiction (矛盾证明法)

• Indirect Proof
• Proof the negation is true to proof the given Mathematical Proof false, vice versa
• Useful when it is hard to use Direct Proof, where the negated Mathematical Statement has form to proof

Example

• Theorem 4.6.1
• Proof square root of 2 is irrational

Proof by Contraposition (逆否命证明法)

• Use when the Contrapositive (逆否命题) is easier to proof

Terminologies

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Concise

• There is no irrelevant details

Polished

• Should be the final drift

Without Loss Of Generality (WLOG)

• Used before an assumption in a proof which narrows the premise to some special case
• And implies that proof for that case can be easily applied to all other cases
• To remove very similar proof, for example, a & b are two consecutive odd number. We need to proof the product of the 2 consecutive odd numbers is always odd
• we need to proof it correct for both a<b & b<a cases, we can remove proof for one of the cases using WLOG