#7 More Proofs

After this week, the learning of proof techniques will almost comes to an end. So I think now is the proper time to sum up main techniques of proofing.

Direct proof : the most common way to carry out a proof,one conclusion followed by another one,just like chains connects together. Straight forward.

Equivalent proof: use contrapositive to prove the original statement. From the truth table, statement and its contrapositive statement hold the same status, i.e. Statement A is true iff contrapositive of A is true.

Proof of existence: useful under disproving the for all... statement or proving the exist statement. Just to find one example.

Proof by contradiction:contains assumption, disprove a statement by proving its negation is true.In personal view, this is the most effective way of proof we've learnt so far. Especially when cannot figuring out whether the statement is true or not.

Proof by cases:divide the antecedent into several parts(all are true) and proof the consequence for each condition. It take times to divide the antecedent.

Most of the times one problem can be solved in multiple ways, if I were writing the proof during exam, I would consider direct proof at first.Then proof by contradiction.