To Conjugate or to Negate. That is the Statement - Week 3

Translating English into logical symbols is not a simple feat and once having translated a simple sentence, I am starting to feel pretty good about myself. However it is one thing to memorize a translation, i.e, to know that if someone says if this, then this, I can use an implication, but it is another to comprehend the meaning behind the sentence and then translate it onto paper. My TA had a fantastic suggestion to conjugate or negate any sentence that I am trying to translate (and in this case I am using the inclusive or). Here is one of the examples from Exercise #2 that was particularly challenging at first. However, I tried to come up with a sentence that made the most sense to me (which was the opposite meaning of the statement given) and then apply a negation afterwards.

This is the statement I decided to take on as a challenge (and challenge me it did):

“No course has more than two prerequisites”

Here is how I decided to translate it.

Ø ( $x e C, $y e C, $z e C, $w e C, (P(x,y) Ù P(w,y) Ù P(z,y)) Ù ( x≠w Ù x≠ z Ù w≠z)

Now I tried to push the negation partway through:

"x e C, "y e C, "z e C,  "z e C, Ø((P(x,y) Ù P(w,y) Ù P(z,y)) Ù (x≠w Ù x≠z Ù w≠z))

The entire statement written out looks a bit overwhelming, so I decided to break it down into smaller parts. Here is the latter half of the statement that I now need to negate:

Ø((P(x,y) Ù P(w,y) Ù P(z,y)) Ù (x≠w Ù x≠z Ù w≠z))

This follows the negation pattern that we saw in class this week:

(P(x) Ù Q(x)) Û (P(x) Þ ØQ(x))

For my example, using this type of relationship, it would translate to the following:

Ø((P(x,y) Ù P(w,y) Ù P(z,y)) Ù (x≠w Ù x≠z Ù w≠z)) Û ((P(x,y) Ù P(w,y) Ù P(z,y)) Þ Ø(x≠w Ù x≠z Ù w≠z))

I am almost done. I just want to negate the final component

Ø((x≠w Ù x≠z Ù w≠z)) Û (x=y Ú x=z Ú w=z)

Whew, that was exhausting! Finally, the finished product:

"x e C, "y e C, "z e C,  "z e C, ((P(x,y) Ù P(w,y) Ù P(z,y)) Þ (x=y Ú x=z Ú w=z)

While it took several steps to complete, performing this action actually made the symbols and their relationships with each other more intuitive. So if I want to do well in this class, the question of whether or not to conjugate or to negate is easily answered…the answer is yes. And that is a statement

                                                                        

The Venn of my Existence: Week 1 & Week 2

Understanding the complexities of discrete math and mastering the “new language” of logic and mathematical notation is not for the faint of heart. However, the approach going in must be that of learning any new language. At first, the flow is unnatural, the construction of sentences are slow, and the statement which you thought was formulated perfectly is clearly not conveying anything significant (the confused expression of your peers signifies this is the case).  With perseverance and a lot of hard work, perhaps one day I, along with my fellow classmates, can be fluent in logic. Until that point in time, I request that Rosetta Stone get to work on a Logic Language Learning program. I think we all may need it.

Now on to the real part of the blog. Before I am even attempting sentences, I have gone through the interesting experience of trying to verify and falsify statements using Venn Diagrams. Although at first it seems simple, the complexity comes when trying to isolate the minimum values needed to verify or falsity a particular statement. To make the statement “every x in A is also in B” be true, all elements that are in A must also be contained in B, and to falsify it just takes one counter-example. However, the minimal requirement to demonstrate this in a Venn Diagram is to show that the part of A that exists outside the intersection of A & B is empty. I find it confusing that is not a necessity to have the intersection also be occupied in this case. Furthermore, I also realize this changes once real observations are given. Once a list has been given, there is an obligation to show that the intersection is occupied with those observations.  Overall, understanding the minimal requirements for proving a statement true or false statements has been a trying exercise.

For me, the challenge this week has been learning how to eliminate superfluous information and focus solely on the relevant information needed for a particular problem. While this may in theory seem simple, for me, it is definitely proving to be a challenge.