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


  • 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

Indirect Proof (反证法)

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

Proof by Counterexample (反例法)

Proof by Contradiction (矛盾证明法)


Proof by Contraposition (逆否命证明法)



  • There is no irrelevant details


  • 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