According to the definitions in the appendix A.1 of Principles of Program Analysis,
$L = (\mathcal{P}(S), \subseteq)$ for $S = \{1, 2, 3\}$ is a complete lattice.
Furthermore, neither $X = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} \subseteq \mathcal{P(S)}$ or $Y = \{\{1\}, \{2\}\} \subseteq \mathcal{P(S)}$ are considered Moore families of $L$.
I can understand why the set $Y = \{\{1\}, \{2\}\}$ is not a Moore family.
For example, given that $Y' = \emptyset \subseteq Y$ holds (since this true for every set), then my understanding is the following must also hold:
$$\bigcap{Y'} \in Y$$
And thus the greatest lower bound for $Y'$ would be in $Y$ (as must be the case for all possible $Y' \subseteq Y$).
However, $\bigcap{Y'} = \bigcap{\emptyset} = \emptyset \notin Y$ and so we have our counter example for $Y$, and hence $Y$ is not closed under greatest lower bounds.
For, $X$, though, I'm having trouble finding a counter example. Since $\emptyset \in X$ is true, I can't see how any possible intersection of the sets in $X$ would yield a result that is in turn not in $X$.
What am I missing here?