Mathematical Proof

Abstract

---

• Concise, Polished Mathematical Argument explaining the validity of Mathematical Statement
• There are 2 types of proofs - Direct Proof and Indirect Proof (反证法)

Direct Proof

---

• A method of proving a Mathematical Statement by starting with known facts (Axioms (公理), definitions, or previously proven theorems) and using logical steps to directly show that the statement you want to prove is true

Only Applicable when there is a starting point

Is difficult when the thing we want to proof has an absence of a form like Irrationality of a number, which is number that is hard to be expressed mathematically. In such cases, we can make use of Indirect Proof (反证法) to obtain a starting point

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 Statement, we can proof by providing an example that fulfil the conditions

Indirect Proof (反证法)

---

• 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 Statement

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

---

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