Dropping empty sets from a collection of sets.

Question

I have a collection of sets $A = \{A_n : n \in \mathbb{N} \}.$ I want to drop all the empty sets, i.e. I want to define the set $B = \{ X \in A : X \neq \emptyset \}.$ Can I define the set $B$ without problem? Somehow this seems problematic to me.

Answer 1

It's not problematic, aside from your overloading of the variable $A$—it's just standard set-builder notation. To avoid re-using the variable $A$, you should instead write something like: $$B = \{ X \in A \mid X \ne \varnothing \}$$ Or you could use some fancy typeface, for instance $$\mathcal{A} = \{ A_n : n \in \mathbb{N} \} \quad \text{and} \quad \mathcal{B} = \{ A \in \mathcal{A} \mid A \ne \varnothing \}$$ To avoid using set-builder notation altogether, you could define $B = A \setminus \{ \varnothing \}$.

Answer 2

Well, the definition you've written is problematic because you're using something that's already been defined (namely, $A$) as something other than what it is; instead, $$\{X\in A: X\not=\emptyset\}$$ would be better. Another option would be $$\{A_n: A_n\not=\emptyset\},$$ but I actually dislike this one a little (I'm being pedantic here) since it doesn't tell you what "$n$" is allowed to be; I would prefer $$\{A_n: n\in\mathbb{N}, A_n\not=\emptyset\}.$$