Properly Translate Propositional Statement to Discrete Math What is the proper way to write this statement mathematically? "All people are smokers or non-smokers but not both." 
$ H = humans, S = smoker, NS = non-smoker $
My first attempt was:
$$\forall x \in H : x \in S \lor x \in NS$$
But I couldn't convert this into a conditional statement to prove it.  (The original question was "All integers are natural numbers or are negative, but not both.")  So I tried the following: "there is no human who is both a smoker and a non smoker: 
$$ \lnot \exists{x} \in H : x \in S \land x \in NS$$
which is equivalent to "all humans are either a non-smoker or a smoker"
$$ \forall x \in H: x \notin S \lor x\notin NS$$
which is equivalent to "if a person is a smoker then they cannot be a non-smoker" by using the logical equivalence of the implies operator, so now I have a conditional operator to work with for a possible proof.   
$$ \forall x \in H: x \in S \implies x \notin NS $$
Is my second approach correct & what is wrong with the first attempt?  Does the fact that I couldn't manipulate it into a conditional statement indicate an improper translation into discrete math?  
 A: Your first approach was close but was missing something. How do you represent the "not both" part? You don't. In the statement you wrote, $x$ could be both $S$ and $NS$ and it would be true. You need this addition:
$$\forall x \in H : (x \in S \lor x \in NS) \land \lnot(x \in S \land x \in NS) $$
The confusing thing with this problem is that "smoker" and "non-smoker" are mutually exclusive sets in natural language. They are also mutually exclusive in math if you define $NS$ as $\lnot S$, which would be an intuitive and absolutely correct thing to do in a different context. But in the context of this problem, we do not have this definition explicitly. In fact, because the problem explicitly says "but not both" we are guided to choose $S$ and $NS$ as two distinct sets, as you have done.  
Your second definition has a similar shortcoming. You write your definition is 'equivalent to "all humans are either a non-smoker or a smoker"', but it's not. A human can be neither of those things and your second definition would still be true. This is how you need to augment it:
$$\lnot \exists{x} \in H : (x \in S \land x \in NS) \lor (x \notin S \land x \notin NS) )$$
Update
If you wanted to assume that $NS = \lnot S$ (which is absolutely reasonable, but not sure if the problem allows it) then you could simply write your first expression as:
$$\forall x \in H : (x \in S \lor x \in \lnot S) $$
This is basically a tautology, you can write this about any set and any condition.
The problem might have been less confusing if it was expressed like this:

In an island, all islanders are either green-eyed or tall, but not both. Write this statement mathematically.

