For All This is False = For Not Some, This is True

What I found rather charming from last week was that there is more than one Mathematical way to say the same thing. While I was commenting on how clean and refined mathematical expressions could be, in such a way that they reduce ambiguity to zero, I found this to be some sort of conundrum (specially when you have an approach to explaining things, and the recipient has another).

Let's focus on one expression that has two different approaches! Let's say for example x and y are courses that belong to C, which is a set that is made by courses taught at UofT. We also have a function P(x, y), where x is a prerequisite of y. The question is, how should we right the following statement?

MAT135 has no prerequisites

We would then have to put it in terms we could understand, correctly using the proper quantifiers. So we could say that:

• For any course x, there is no course that is a prerequisite of MAT135.

∀ x ∈ C, ¬ P(x, MAT135)

•  For not some x, there is a course that is a prerequisite of MAT135. 

¬ ∃ x ∈ C, P(x, MAT135)

This is nothing but a small example of how things could be approached in a different way. While this seems a little obvious since it is a game that could be simplified as Positive - Negative = Negative - Positive, there are some other complex statements that could be described in different ways.

Bottomline, not everything has one single answer. Or perhaps we should say...

∃ x ∈ A, ∃ y ∈ A, ∃ z ∈ Q, S(x, z) ∧ S(y, z) ∧ x ≠ y

Note: Where x and y are answers that belong to A, z is a question that belongs to Q, and S(x, z) is a function such that x is a solution to z.