Week 7 Review of [Proof]

I am going to review proofs in this week's notes.

1. Proof by cases:

First step is to split argument into different cases then prove the conclusion for each case.

From all the lecture examples and tutorial examples, I think the trick to decide if we need to proof by cases is that see if there is a "\/" (or) show up. 

For example, the example in class, for all nature numbers n , n^2+n is even. We assume n is a nature number, n could be either even or odd,  then there exists a k, n = 2k+1 \/ n = 2k. Then we prove case 1 : n = 2k+1 and case 2: n = 2k. Same as the tutorial question. root(rs) = 0 so either r = 0 \/ s = 0. Then we consider two cases.

Note: See \/ => Proof by cases.

2. Proof <=>:

The trick for this kinda of proof is that when => is true DOES NOT MEAN <= is True. When we want to disprove a poof, we always negate it, and prove the negation is true. The good thing about negate a universal quantifier will become a existential quantifier. To prove a existential quantifier is true, we just need to let xxx= xxx, prove the statement is true.

3. Review proof patterns:

1) introduction rules: negation introduction, conjunction introduction, disjunction introduction, implication introduction, equivalence introduction, universal introduction, existential introduction.

2)elimination rules: negation elimination, conjunction elimination, disjunction elimination, implication elimination, equivalence elimination, universal elimination, existential elimination.