Proper use of "without loss of generality" I'm trying to understand exactly when I can assume something "without loss of generality in a proof." The technical explanation, I believe, is that it's allowed provided that the case that I'm omitted "reduces" to the one I prove, either through interchanging of labels or through a rather trivial extension. I can really only understand this through examples, though.
The example I have in mind, where I'm not totally sure I can use it, is as follows. Suppose I have countable (as in, finite or countably infinite) sets $X_1, \ldots, $ and want to show that $\bigcup\limits_{i \in \mathbb{N}} X_i$ is countable. I want to say "without loss of generality, suppose $X_i \neq \emptyset$ for all $i$," the justification being that if I take $I = \{j \in \mathbb{N} \mid X_j = \emptyset\}$, then I have
$$ 
\bigcup\limits_{i \in \mathbb{N}} X_i = \bigcup\limits_{i \in \mathbb{N}} X_i \setminus \bigcup\limits_{i \in I} X_i,
$$
i.e., they contribute nothing at all to the union, so having in the union is somewhat "harmless." I don't know if I'm sacrificing generality by making this assumption, though.
I would appreciate any help on understanding this.
 A: 
I can really only understand this through examples, though.

There's nothing wrong with this!  "Without loss of generality" is an imprecise term and can be used in a wide variety of contexts, so there's no fixed rules on exactly how it can be used or what exactly it means.  Ultimately, it is intentionally introducing an easy-to-fill gap in a proof and relying on the reader to fill the gap, giving them a hint that it can be done by some sort of symmetry argument or easy reduction to a special case.  Your understanding of the meaning seems pretty much correct.
As for the specific example you're asking about, to justify the "without loss of generality" you have to actually fill in the gap in the proof that it introduces.  That is, you have to prove the statement in the general case (where some $X_i$ may be empty) using the specific case (where you assume they are all nonempty).  You should never use "without loss of generality" unless you yourself know how to fill in the gap in the proof and trust that your reader can do so as well.  There are various ways to do that in this case; here is one.  Let $Y_i=X_i\cup\{0\}$ for each $i$.  Then each $Y_i$ is nonempty, and is still countable.  So now we are in the specific case where none of our sets are empty and we can deduce that $\bigcup_{i\in\mathbb{N}}Y_i$ is countable.  But $\bigcup_{i\in\mathbb{N}}Y_i=\bigcup_{i\in\mathbb{N}}X_i\cup\{0\}$, so $\bigcup_{i\in\mathbb{N}}X_i$ is a subset of a countable set and thus countable.
Here, the essence of the "without loss of generality" is that if you changed the setup in some simple way to make the assumption true (e.g. added an element to the $X_i$'s so they can't be empty), then this would still give a conclusion that would be strong enough to deduce what you originally wanted (since you would only enlarge the union, and a subset of a countable set is still countable).
A: I think there is a subtle distinction between your case and one where I would use WLOG in my writings.
For me, WLOG is for when you have a number of cases that are dealt with in a nearly identical manner, so that you'd really like to treat them as a single case.  For instance, if I am dealing with a rectangular prism for some reason, I can assume WLOG that no side is longer than the length and no side is shorter than the height.  I can do that because I can reorient the box to turn six potential cases into one.
Your example is different, because we deal with empty sets in a different way than we deal with non-empty sets.  I agree with your instinct to wave them away with a single phrase as it is a trivial case.  But my personal writing style would be to say something like this:

If any of the terms of the family $X$ were empty, they would obviously contribute nothing to $\bigcup_{i\in I}X_i$.  Therefore, for the remainder of this paragraph, let us assume that every term $X_i$ is non-empty....

That being said, I doubt many people would raise their eyebrows if you used WLOG to describe this circumstance, and none would be so troubled as to let it harm their understanding of your larger argument.  But I learned that my two favorite undergraduate professors have passed on since the last time I looked, and this seems like the sort of rhetorical pedantry that they would love to point out.  ^_^
