Week 4: Implication/Disjunction, Mixed Quantifiers, Proof

This week, I learnt further information regarding implication and disjunction. In addition to that, I got to realize how the sentence structure can affect the meaning which the text delivers to the readers. For instance, in Danny's PowerPoint slide, "for all, one...one for all", Danny told the class that the def for all_exist (X, Y) can only be true and not the other way around in some case, meaning the converse of the implication can be false depending on how the claim is arranged.

Let's use these two claims which will be different from Danny's example to improve my understanding on the difference in the arrangement of "for all" and "exists". This will be my mathematical approach to review my material for this week. 

∀x∈ X, ∃y∈ Y, x/y = 2

∃y∈ Y, ∀x∈ X , x/y = 2

X = Y = {2, 4, 8, 15}

The first claim can be interpret as "For all x, there exists y which can be used to make x divided by y to obtain 2. 

Now the when we look at the second claim, this can be interpreted as "There is y, which for all x can be used to obtain x divided by y equals to 2."

The first claim is the only right case, since you can choose one element of x and one element of y's result can lead to x/y = 2. Whereas  the second claim gives the condition which when you choose one element in set X, all element in set Y should bring the result of x divided by y equals to 2. Thus, depending on how you arrange universal and existential claim, the sentence or the equation may or may not make a sense. 

One thing I found difficult during this week was double quantifiers. Danny told us that there are three ways in which you can show that subset of Cartesian product N x N (Here, the letter N represents Natural number and this is one of example Danny went through) is non-empty. At first, I had no idea what Danny was talking about in his slide "double Quantifier". However, soon I realized that when there are two same types of claim such as two existential claims, the order of the claim does not affect the information deliver to the audience. Then later, Danny answered that unlike the "for all" and "exists" problem which I wrote out above, within the double quantifiers, the arrangement does not affect the logic. 

As my reflection, I feel like the course material is getting complicated as the time passes by. That is why I think keep reviewing the materials that I learnt in the class every day to prepare myself to face even more complicated logic problem. This week, I have been really busy with my assignment 1 which had become a good exercise for me to review the course materials from week 1 to week 3.