# Finding a counterexample in model theory

I'm currently reading about models of Set Theory, and I'm working on exercises to better understand the concepts. In Kunen's most recent Set Theory text, he mentions that if we have a transitive model $M$ and if $\cap^M$ is defined, then $\cap$ is absolute. Then, he gives the following exercise:

Describe a two-element non-transitive $M$ that is isomorphic to $\{0,1\}$, such that $\cap^M$ is defined, but $\cap$ is not absolute for $M$, and such that $\subset$ is not absolute for $M$.

I'm having trouble coming up with such a two-element $M$ since I haven't seen many examples of models being used outside the theory. Any help would be greatly appreciated!

Let $M$ consist of two sets $\{A,B\}$, such that $A\not\subset B$, but also $A\in B$ and $B\notin A$. For example, one can take $A=\{\emptyset\}$ and $B=\{A\}$. It follows that $M$ thinks $A$ has no elements (since it has no elements in $M$, and therefore $M\models A\subset B$, even though this isn't true externally to $M$. Similarly, $M$ thinks $A\cap B=A$, since there are no elements in $M$ that are in both $A$ and $B$, and so this intersection has the same elements in $M$ as $A$ has in $M$. But externally, we can see that $A\cap B\neq A$. So $M$ thinks $A$ is empty and $B$ is singleton $A$, so $M$ is isomorphic to $\{0,1\}$.