Let $U$ be the power set of $\Bbb{Z}$. Let $B= $ {{n}: n $\in \Bbb{Z}$} i.e. the set of all singletons in $U$. $F =$ {$\varepsilon_{\cup}, \varepsilon_{c}$} where $\varepsilon_{\cup}(A,B) = A\cup B$ and $\varepsilon_{c}(A) = A^c =\Bbb{Z}$ \ $ A $
Describe the set $C$ that is generated from $B$ by functions in $F$
I have an idea of what the set $C$ looks like but I'm not for sure. My best guess is $$(\Bbb{Z}-n\Bbb{Z}) \forall n\in \Bbb{Z} $$ But I think $\Bbb{Z}^+$ and $\Bbb{Z}^- $ are not included either. I'm not sure if there are any other subsets that are not included. Please explain. Thank you.
Let $D$ be the set of non-empty finite subsets of $\mathbb Z.$ Let $E=D\cup \{\mathbb Z$ \ $d: d\in D\}.$
We have $C=E.$ You can prove this by showing
(1): $\forall s\;(s\in E\implies s\in C.)$ Therefore $E\subset C.$ (Note:By induction on $m\in \mathbb N,$ if $s\in D$ and $s$ has $m$ members then $s\in C.$)
(2): $\forall s,t\in E\;(s\cup t\in E\land s^c\in E).$ Any subset of $U$ that is closed under unions and complements, and contains $\{n\}$ for all $n\in \mathbb Z,$ must have $C$ as a subset. Therefore $C\subset E.$