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

**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 $A$.)