Week 4- What is proofs?

This week's lesson reviewed some of the things we  have learned in the previous weeks, like the transitivity of an implication, the relationship between 'for all' and 'there exist', we also learned many new concepts, like limits and proofs.

Limits we learned in this class is extremely similar to the limit we learned in MAT137. Limits are very familiar for me since I have aced this topic in high school math. However what I didn't expect were that what we learned here about limits is completely different from what I learned in high school. In here we are not learning how to solve the limit, but rather proof that the limit exist. For me, who is very confident in solving problems using equations find is extremely hard to understand the concept of proofing and how to proof that the limit exist. The two main symbol we have to know is e(epsilon) and d(delta). These are the variables that indicates the distance between the positions on x-y axis that we want to find the limit is and the distance to the maximum accepted value for the limit.

As you can see on the graph, the area box surrounded by dotted line is where the limit are, and the area could be infinity small or infinity large so that it could prove that a limit does or doesn't exist on that area.

The next thing we learned and the most important for me so far is the concept of proofing. Proof is a very abstract concept for me, unlike solving equations it makes me feel that it lacks an absolute answer to it, which makes me feel don't know where to start on solving the proof. 

In order to prove something is true we must use previously proven statements to support our proof. First we assume the antecedent is true, then if the consequent is true the whole proof is true. But if the  antecedent is false, then this becomes a vacuous truth statement, meaning that the proof will be true. Proofs are 'if... then...' statements means we can also use conjunction and dejunction as well as quantifiers.

Here is an example on how to proof:

      Example: for all x in the set of real numbers, if x>0 then 1/(x+2)<3

      Solution: first we try to make the antecedent looks like the consequent.

                          (x>0)=(x+2>2)=(1/(x+2)<1/2)

               since 1/2<3 we can rewrite the equation as  1/(x+2)<3

               thus, this shows that consequent is true when the antecedent is true, proving this proof is                      correct.

However we have to be careful when we do the converse of this statement, although this statement is true, the converse of this statement is false, because x can be -3 and 1/(-3+2)<3 but x!>0.