Week 5: Contradiction, Existence, Sequence

This week, I had learn how to prove the statement's existence. During the last week, I worked on how the proof structure outline works. Although it was no easy job memorizing all the step and justification to clarify what happens to each step, it still was interesting how using what I have learned up till now to prove something. Of course, what really motivated me to learn the proof structure even more than any of the week was the term test 1 which took a place on Wednesday. However, I have been kept track of my Slog to review the course materials and come up with some question (mathematical approach) to enhance my knowledge on mathematical reasoning, so somehow I was well prepared for the test.

During week 5, for most of time I have spent my time studying proof, contradiction, and symbolic logic. What I really found difficult during week was the translation of natural language into symbolic logic especially, when I didn't know whether to use a conjunction or an implication. Here is some mathematical approach I came up for this week. 

Let D be the set of all car and people. 
Let P be the set of people.
Let C be the set of Car.
Let B(x,y) interpret as "x bought y".
And let L(x,y) interpret as "x likes y". 

Now using these symbolic logic, I will create a sentence in natural language and translate back into symbolic logic using the condition and definition of each symbol that I have listed above. Now here is the sentence which I have created.

There is a person who didn't  like any of car he/she bought. 

Now to translate the above sentence into the symbolic logic, I need to think about is there needs of implication or conjunction. As I carefully examine the sentence I created, I realized what because of using word "who", that I might need both conjunction and implication to construct a symbolic logic which contains the same meaning. 

                             ∃x∈ D, P(x) ∧ ∀y∈ D, (C(y) ∧ B(x,y)) →  ¬L(x,y) 

The reason why "not like" should go after the implication sign is that the example sentence which I constructed can be interpreted as "All the car a person bought, he didn't like it." Thus, using both concept of conjunction and implication I translated the natural language into symbolic logic.  

Although this week was about how to write a proof, since the class did not learn much information about this due to term test 1, I decide to write something which I had a trouble on during my study for the term test 1. As my reflection on term test 1, I think it went pretty well compare to other questions which I solved for the practice.