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

Week 2: Quantifiers, Implication

This week, I learned about quantifiers, implication and verifying implication. This part wasn't completely surprising since it is the same contents as current Calculus class. It was helpful to have same part in two different classes at the same time. It gave me a better understanding of material.

When verify quantified claims, if it is an universal claim, show there is no 'counter-example'. In other words, if there is at least one counter-example, an universal claim becomes false. But if it is an existential claim, show there is at least one example then it is true. When there is no example, it is a false claim.

Implication is 'If P, then Q.' P is called the antecedent and Q is called the consequent. The converse of implication is 'If Q, then P.' and there is no connection between the implication and the converse. Its negation is 'If P, then not Q.' and contrapositive is 'If not Q, then not P.'

Implication can be verified in a similar manner with verifying quantified claims. Following us 'Truth Table'.

P	Q	P -> Q
T	T	T
T	F	F
F	T	T
F	F	T

When the antecedent is false, the whole implication is true. This is because 'P' is false so it is an empty set. If it is an empty set there is no counter-example can be found. Therefore, the implication is true. When I first read this part, it was hard to accept this. What I thought is that implication should be false because 'P' is false so whole statement cannot be established. It is quite still uncomfortable but at least I understood why the implication is true when 'P' is false.

Week 1: Sets

In the first week of lecture, I had a difficulty to accept the concept of 'not all' and 'not any' and I couldn't connect the concept of building set and using boolean even though I read the material before the class. Honestly, I didn't understand the lecture right away so I could not write anything on the sheet Professor handed out.

After the lecture, I reviewed what I learned today and try to understand the concept step by step. When I just read the note, it wasn't clear to me what these all 'def's mean. So I decided to use examples on python. I wrote the random sets of S1, S2 in python. I tried given 'def's using True and False as elements in a set. After I tried couple of examples, I could see the big picture and could figure out what the comments could have been in the lecture. Following is the one example:

 def q0(S1, S2):

     ''' (set, set) -> bool

     Return whether ...

     '''

     return not all({x in S2 for x in S1})

It would return 'True' when  all elements in S1 are not in S2.

In other words, def q0 means:

Some elements of S1 is not an elements of S2.

At first, I was not familiar with expressions, however, after reviewing the material thoroughly it did not feel too complicated. As a result, I could solve the first week tutorial problems without having difficulties.