Logic Negation Symbols $\def\nn{\mathord{\sim}}$ 
This is a rather simple question but I can't find an exact answer on it.
In examples, I've seen $\nn$ and $\lnot$. These fall under ‘negation’. If they both fall under negation, does that mean that I can substitute them for each other?


*

*$p$: Sam goes to work.

*$q$: Max goes to work.


So, if I were to write: ’Neither Sam goes to work nor Max goes to work,’ could I write it as $\lnot(p \lor q)$ and also $\mathord{\sim}(p \lor q)$?  Or, if I had ’It is not the case that if Sam goes to work then Max goes to work’, could I write $\nn(\nn p \lor q)$ or $\lnot(\lnot p \lor q)$?
 A: First, your translations are correct.
Second, there is no difference between the meaning of "$\sim$" vs. "$\lnot$":   
$\qquad\qquad\sim(\sim p \lor q)\;$ is precisely the same statement as is $\,\lnot(\lnot p \lor q)$.
Both symbols are used, depending on the preference of the user or depending on context, to denote negation: that is, to assert "$\mathrm{not}\,\left[(\mathrm{not} \;p) \lor q\right]$". 
It's really no different than the fact that we can say, for example, $\;9 \div 2$, or $\;\dfrac{9}{2}$ to denote "$9$ divided by $2$." 
You can use one or use the other (and there are a couple other ways of denoting negation of a proposition), but don't "intermix" their use: that can lead to confusion on the part of the reader. Just choose one (you'll probably want to use the symbol used in your text, or by an instructor), and be consistent in that context. 

See also Wikipedia's list of logical symbols to see some alternatives for other connectives.
A: Yes, these symbols both represent the same thing, so all of your examples are valid.
However, I suggest using only one at a time for clarity's sake.  That is, don't write: ~(¬p ∨ q).  Mixing the symbols may cause people to think that you're using one to represent something other than negation.
A: Wrong!! The correct way to write negation is obviously $\overline{p}$.
Heretics...
