Week 6 - In depth of proofs

In this week's lecture we learned more examples of proofs. We practiced proofs from for equations to implication to proofs with qualifiers. Due to the Thanks giving holiday at the start of this week, this week's lecture is also very short, but we have learned many quite important types of proofs which made this week's lecture by no means easier than the previous weeks. There are many types of proofs which can mainly categorized into four different proof types, direct proof, contradiction proof, negation proof, case proof.

-Direct proof is the simplest, it is where we proofing by assuming the antecedent is true, and manipulate the antecedent into the form of the consequent, proving the consequent is true. There are also a type of statement called existential, we can prove this statement true simply by proving that there exist one value that satisfies this statement.

-Contradiction proof is another type of direct proof, where we just use the contrapositive of the implication or quantifiers to proof the original statement. Because of the contrapositive is exactly the same as the original statement, we can just simply prove it by following the structure of the direct proof. This types of proof is used when the implication does not give us a clear picture of how to prove it, and by using the contrapositive of the original statement it can show us a much easier way of proving the statement.

-Negation proof is used for proving a false statement. The simplest way of proving that is to negate the statement, and by proving the negated statement true it will show that the original statement is false.

-Case proof is a type of proof where it is best proven using different cases. By using different cases and listing all the possibilities down, it can give us a clear idea of what and how to prove the statement. If the statement has two cases, we prove the statement is true by simply proving both cases are true.