Logical equivalent of not necessary I'm working through "How to think like a mathematician" for uni and there is this part of a question: Rewrite the following as 'if ..., then ...' statements: ... (d) Regular work is not necessary to pass the course. so my first instinct is to say: `"If the course is passed then regular work might have been done"` but that just feels messy and I can't really translate that into symbols i.e. `A implies B` I was thinking that I could exploit the nature of the implies: `"If regular work is done then the course is passed"` but I am not sure if that is entirely equivalent, and just cheesing the question. There aren't any solutions in the book and the online ones don't have this section in them.

I would write it as $¬(A\implies B)$.

Where $A$ means the course was passed and $B$ implies regular work was done.

Notice that $A\implies B$ means that regular work is needed to pass the course.