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