= Solution
{c}
The <Tarski-Vaught test> says that $(M,\in)$ is an <elementary substructure> of $(N,\in)$ if and only if every existential statement true in $N$ with parameters from $M$ has a witness in $M$ that satisfies its matrix in $N$. Explicitly, for every <first-order formula> $\psi(x,\bar y)$ and every parameter tuple $\bar a\in M^{|\bar y|}$, require
$$
\boxed{N\models\exists x\,\psi(x,\bar a)\quad\Longrightarrow\quad\exists b\in M\;N\models\psi(b,\bar a).}
$$
The truth of the matrix in this test is evaluated in the larger structure $N$. Under this criterion, $(M,\in)\prec(N,\in)$ means that all <first-order formulas> with parameters from $M$ have identical truth values in the two structures.
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