Logical transforms of theorems I'm trying to transform some theorems to different logical forms but having difficultly for some reason, it should be quite simple but my brain doesn't seem to be functioning lately.
I want to write an implication in terms of various forms to see the different versions of the theorem(s).
In fact I want to transform a definition: We can use the definition of a cts. function at a point as an example:
A function $f:X\rightarrow Y$ is called continuous at $a\in X$ if $\forall N \in N_{f(a)}$ then $f^{-1}(N) \in N_{a}$.
(Here $N_x$ means the set of neighborhoods of x, I can't get mathbb or others to change the font).
The idea is to change implication to the various other forms. The logic is simply $A\rightarrow B$. (So $A = \forall N \in N_{f(a)}$ and $B = f^{-1}(N) \in N_{a}$)
E.g., the contrapositive form $!B\rightarrow !A$ doesn't make any sense because N is defined afterwards:
A function $f:X\rightarrow Y$ is called continuous at $a\in X$ if $f^{-1}(N) \notin N_{a}$ then $\exists N \notin N_{f(a)}$.
of course we can translate/shift this so that we end up with something like(I moved the $\exists$ to try to make the implication work but...)
A function $f:X\rightarrow Y$ is called continuous at $a\in X$ if $\exists N \notin N_{a}$ then $f(N) \notin N_{f(a)}$.
But this too doesn't make sense and doesn't even logically follow from transformations.
I'm simply trying to turn these logical statements in to logical formula to manipulate but something is throwing me off. There are many ways to write an implication and I would like to figure out how to do so and retain the equivalent definition or theorem.
Any ideas?
 A: You haven't actually expresesd continuity correctly: "$\forall N\in N_{f(a)}$" is not a statement, so it can't be the hypothesis of an implication. The right version is $$\color{red}{\forall N(}N\in N_{f(a)}\rightarrow f^{-1}(N)\in N_a\color{red}{)},$$ that is, the entire implication occurs inside the scope of the universal quantifier. The hypothesis and conclusion of that implication are then "$N\in N_{f(a)}$" and "$f^{-1}(N)\in N_a$," respectively.
This is why you're running into trouble with the contrapositive. Applying the contrapositive to the correct version of the definition we get the equivalent property $$\color{red}{\forall N(}f^{-1}(N)\not\in N_a\rightarrow N\not\in N_{f(a)}\color{red}{)},$$ which is perfectly well-formed. Note that we're not touching the outermost "$\forall N$" here; the "contrapositive translation" only applies to the implication itself, not the quantifier structure it sits inside.
(That said, if you want there is a transformation we can apply to that quantifier, namely the duality of $\forall$ and $\exists$: a statement of the form "$\forall xP(x)$" is equivalent to "$\neg\exists x\neg P(x)$." But I don't see how that would be helpful in this context.)
