How to distinguish between exclusive "or" and inclusive "or" when the truth value of p and q cannot be simultaneously true I just started learning Discrete Mathematics using Kenneth H. Rosen's "Discrete Mathematics and Its Application", and I'm having trouble understanding the solution for example 9 in section 1.1.2.
Since the disjunction of p and q (or the inclusive "or") is the proposition that states that either p is true, or q is true, or both p and q are true, if the "or" in the statement is an inclusive "or", then if p and q are both true, the truth value of the statement has to be Truth. Therefore, what the explanation is trying to do (or at least what I think it's trying to do) is to show that when both p and q is true then the truth value for the statement is False which means the "or" must be an exclusive "or" . However, if p and q cannot be simultaneously true, then how can there be a truth value for the statement when p and q are both true? The truth table for the statement, whether with the inclusive "or" or the exclusive "or", will only have 2 cases: p is T and q is F, and p is F and q is T.
I guess what I want to ask is how to distinguish between exclusive "or" and inclusive "or" when the truth value of p and q cannot be simultaneously true?
 A: You have r=I spend all my money on Europe or a car, p=I spend all my money on Europe, and q=I spend all my money on a car. The truth table is
p q r (p or q) (p exclusive or q)
T F T T        T                  
F T T T        T                  
F F F F        F                  

Therefore either of those two translations matches with the values of p, q, and r that we have been given in the problem. So either is valid, because they give the same truthedness.
However say you were given a different problem such as the following (in particular, p and q can both be true, and r has a corresponding truthedness corresponding to it). Coming up with an example in words is another story altogether, so I leave that out of the discussion.
p q r (p or q) (p exclusive or q)
T T F T        F
T F F T        T                  
F T T T        T                  
F F F F        F  

Then neither of these formulae would work because r has been defined in such a way that neither p or q nor p exclusive or q are an accurate translation.
Conclusion
Either the regular or or exclusive or will provide a correct translation in this case because p and q cannot simultaneously happen. Your task is to determine the truthity of r corresponding to each combination of p and q that can happen, and come up with a formula that matches those truth results.
A: Your observation is correct: if we know $P$ and $Q$ can't both be true, then the expressions $P\lor Q$ and $P\oplus Q$ are equivalent (in that they have the same truth value). We could write this as the following theorem:
$$\lnot(P\land Q)\implies(P\lor Q\iff P\oplus Q).$$
Your question "if p and q cannot be simultaneously true, then how can there be a truth value for the statement when p and q are both true?" is a little confusing. The truth functions corresponding to the logical operators are defined for all inputs even if some of the function values are irrelevant in a given situation.
Edit: I looked at the exercise you linked to in the comments, and the answer given in the book is questionable. They ask you to "translate" a sentence of the form "P or Q", and expect you to get $P\oplus Q$ by noticing and incorporating additional information (that P and Q can't both be true). It makes sense that one would want to do this, because saying $P\oplus Q$ is "stronger" or "more informative" than saying $P\lor Q$, but I'd argue that this is more than just a translation from English into logical symbols. If the exercise writer wanted $P\lor Q$ to be an incorrect answer, they should've said "write a logical expression that exactly describes the scenarios that are possible" or something, not just "translate".
