Week 3: continue week 2...

In this week, we continued from the week 2 contents. Followed by implication, converse and counter positive, this week, I learned about equivalence, new symbols conjunction and disjunction, more of negation, exploiting Venn Diagram and Truth Table for proving, logical arithmetic and De morgan's Law.

Since we had a second tutorial, I feel that the material is a lot harder. In particular, when dealing with expressing sentences into mathematical expressions , I struggle. There are some stuff that is not easy to come up with solutions and when it comes to details, I think I still need more practice. I think I was confused about existential quantifier at the beginning. However, after  tutorial, it became organized in my head and now I can comprehend the problems. Although, I still have some problems to work on, I would say I generally understand the material.

When P => Q and Q => P ( P <=> Q), it is called that they are equal. There are few other ways to express equivalence:

• P iff (if and only if) Q
• P is necessary and sufficient for Q
• P => Q, and conversely
• P exactly/precisely when Q

Weirdly, this equivalence also works when implications are vacuously true for both directions as well. For example, 

       

                             
In both directions, the antecedents are false, so implication is true. It is odd and it looks like it is false (to me, when I instantly look at this implication), but this is equivalence.

Conjunction and disjunction is new symbols that came up on this week.   is conjunction and  is disjunction. Conjunction meas 'and' and disjunction means 'or'. Coincidentally, they look similar to intersection and union symbols. Conjunction is used to combine two sentences into a new sentence by claiming that both of the original sentences are true. For conjunction, it is true at least one of them is true.

When I first saw the concept of negation, I was confused with converse. Also, I don't know why but it was hard to find negation of expressions. So when we find the negation, think about the expression that makes original implication false. For example, think about this implication 'all employees are making over 100k.' In order for this implication to be false, I need to find one (or some) employee who are making less than 100k. If the original implication is true, there will be no employee who are making less than 100k, so this would be false. Therefore, the negation is 'there exists some employees who are making less than 100k.'

We can use negation to prove that a statement is true. Because a statement will be true when its negation is false. Not only can we use negation to prove, we can also use Venn Diagram or Truth Table to prove. Venn Diagram might be hard to draw when there's more than 3 of elements to verify. In that case, we can use Truth Table to prove its truth.

Conjunction and disjunction is quite interesting. Logical arithmetic applies to these concepts just like when we calculating numbers. Also, De Morgan's Law, which I learned when I learned about sets, applies to here. This comes very useful when I try to express statements in mathematical symbols or try to interpret them.

My skill to prove or to represent statements to mathematical symbols are not really good yet. When I encounter the problems(such as tutorial problem sets), I still have to think for a good amount of time or sometimes, I can't really figure out how to answer or what kind of answer I have to give. Even though I solve the problems, I'm still not confident if this is right way to do it. I know that I can't be good at for the first attempt but sometimes it little discourages me cause I'm really not sure if I'm doing this right. When I glanced at other students' SLOGs that were posted on the course website, I could find some students who are ahead of the course. I guess the only way to be better myself is keep practicing until I am comfortable with this.

∀x∈R,x2−2x+2=0⇔x>x+5∀x∈R,x2−2x+2=0⇔x>x+5