Family of subsets which is closed under finite disjoint unions & complementation, but not a field

Question

I want to find an example of a family of subsets $\mathcal{F}$ of a set $X$ such that $X$ belongs to the family, it is closed under complementation and finite disjoint unions, but $\mathcal{F}$ is not a field on $X$. However, I'm not sure on how to proceed.

I was thinking of considering a finite set $X$, however I think that it should be infinite for the counterexample to exist. I can find examples of families that are not sets, but they will not be closed under complementation. What would be a good example of this?

Thanks for the help.

Answer 1

Hint: Consider $X=\{1,2,\ldots,2n\}$ for some $n \in \mathbb{N}$ and $$\mathcal{F} := \{A \subseteq X; \sharp A \, \text{is even}\}.$$

($\sharp A$ denotes the cardinality of the set $A$.)

Answer 2

Consider $\mathcal(S) = \{\emptyset, \Bbb N\} \cup \{X\subseteq \Bbb N\mid \lvert X\rvert\ge 2\}$.