Direct Proof

Abstract

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• A method of proving a mathematical statement by starting with known facts (axioms, definitions, or previously proven theorems) and using logical steps to directly show that the statement in question is true

Applicable only when there is a starting point

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

Example

• Prove that the product of two consecutive odd numbers is always odd

Proof by Construction

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• Also known as Proof by Example
• A form of Direct Proof that proves the existence of a mathematical object by actually providing a concrete example

Important

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

Example

• Prove there is an integer such that it is greater than 2 and (x^2 - 5x + 6 > 0)

Proof by Exhaustion

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• Also known as Proof by Brute-force or Proof by Cases
• List down all the possible cases and check on all cases

Important

Useful there is only a handful of possible cases.

Example

• Prove that 6a is not divisible by 5 for an integer a between 1 and 4 inclusive
• Prove that the difference of two consecutive squares between 30 and 100 is odd
• Prove that there exist irrational numbers p and q such that p^q is rational

Proof by Deduction (演绎推理)

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• Direct Proof that is used when the number of cases is infinite, with the help of Theorem & Axioms to prove something

Important

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 difference of two consecutive squares is always odd
• Prove that the sum of any two even integers is even
• Prove the sum of any two rational numbers is rational