Disjunction: Why did the inclusive "OR" become the convention? In How to Prove it by Velleman, for defining disjunctions, he gives the difference between exclusive "OR", and inclusive "OR." 
Given two events $P$ and $Q$, the disjunction is defined for them as:


*

*Inclusive: One of $P$ and $Q$, or both.

*Exclusive: One of $P$ and $Q$, but not both.
Quoting from his book:
"In mathematics, or always means inclusive or, unless specified otherwise, ..." (Velleman, 2006, p.15)
My question is -

Why did the inclusive definition of disjunction become the convention?

Was it coincidental, or is there some aspect to the inclusive definition that makes it more convenient?

The most common case in mathematics is probably when "or both" is obviously impossible, in which case it doesn't matter if you use inclusive or exclusive or.  For example, if we say n = 2 or 3, we know it can't be both.
In cases where it does matter, the inclusive disjunction is radically more likely to be case.  I think it's because any kind of proof by cases will lead to inclusive or.  Suppose you were proving a theorem of the form that "A or B implies C".  If you prove that "A implies C" and "B implies C", then the inclusive or version immediately holds.  If only the exclusive or version holds, then the proof strategy would probably be something like "A and not B implies C", and "B and not A implies C", which is a more complex kind of result.
Off the top of my head, I can't think of that many results where an exclusive or arises.  The only one that comes to mind is the Fredholm alternative.
