= Solution
A <first-order formula> $\varphi(x_1,\ldots,x_n)$ is an <absolute formula> for membership structures $A$ and $B$ if, for every tuple $\vec a\in(A\cap B)^n$,
$$
\boxed{(A,\in)\models\varphi(\vec a)
\quad\Longleftrightarrow\quad(B,\in)\models\varphi(\vec a).}
$$
The same elements interpret the free variables in both structures; bound variables range over their respective domains. Thus absoluteness requires both directions, rather than only preservation of truth from the smaller structure to the larger one.
Back to article page
