# Union on the empty set and the set containing the empty set

I'm trying to get a clearer sense of some of the consequences the axiom of unions has on the empty set. I understand that $\emptyset = \{\} \not= \{\emptyset\}$.

But assuming the following identities are correct, I don't understand why $\bigcup\emptyset = \bigcup \{\} = \bigcup \{\emptyset\}$.

It's likely that I'm floundering on some minutiae of set theory, but it's making me uncomfortable, and I'd like to know what I'm missing.

• The union of the empty set with any set whatsoever is the other set. That makes sense, right? You have a bucket full of stuff, and you add to it the contents of an empty bucket. The result is the same bucket full of stuff you started with. – user4894 Oct 2 '17 at 22:29

$z \in \bigcup A$ iff there exists $y \in A$ for which $z \in y$. No such $z$ exists for $A = \emptyset$ or $A = \{ \emptyset \}$.

Indeed, for the former, we have no $y$; for the latter, there is a $y$, but it's empty, so there's no $z$.

• Thanks for a very instructive response that helped me get an even better sense of the axiom of union. – innumeratus Oct 2 '17 at 22:42

We can view $\bigcup$ as a (class) function $$\bigcup \colon \mathrm{Set} \to \mathrm{Set}$$

and viewed as such, your question is in instance witnessing that this function is not injective. There are many such examples, e.g. $$\bigcup \{ \{a\}, \{a,b\} \} = \bigcup \{ \{a,b\} \} = \{a, b \}$$ which may, at first, not be very intuitive but readily follows from the definition. With a little practice, this will become second nature in no time.

• This is very interesting. Is this approach to set theory more advanced than the standard undergrad curricula? I'm currently working through Halmos's Naive Set Theory, which I'm supplementing with Hrbacek and Jech's Introduction, and I haven't come across something like a (class) function in the presentation of the axioms. Might you be able to recommend a good source that treats sets theory in this way? – innumeratus Oct 3 '17 at 0:52
• There isn't anything special about using class functions in set theory - they come up every once in a while. I just thought that linking this phenomenon to something you may already be familiar with might ensure you that there isn't anything weird going on here. As far as book recommendations go, I think you already have a good selection there. If you, at a later point, are looking for book recommendation with more specialist topics in mind, you should create a separate question about it - after searching through previous questions here on MSE. – Stefan Mesken Oct 3 '17 at 7:32