This post does not concern a particular problem or exercise, but instead a peculiarity (for me) in one genre: proofs involving families of sets (that is, sets containing sets as elements). I have noticed that in some statements of theorems which involve families of sets, the hypothesis includes " let F and G be families of sets," whereas in others, the hypothesis is slightly altered to: "let F and G be nonempty families of sets." I have included two theorems (which I have already proven) as examples of this:
1. Suppose F and G are nonempty families of sets. Prove that U(F U G) = (UF) U (UG).
2. Suppose F and G are families of sets. Prove that U(F ∩ G) ⊆ (UF) ∩ (UG).
The difference obviously regards some property about the empty set. My original thought was that being nonempty allowed for the assertion of F or G containing some set- which arises during the course of the proof of 1. That is, x ∈ U(F U G) means there is some A ∈ F U G for which x ∈ A. But then, however, i noticed that the proof of theorem 2. also asserts the existence of some set which is an element of F (and G), without the "nonempty" portion of the hypothesis being present. Could this simply be a mistake on the part of the author, or am i missing a notion here?