Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 144 / 1 / a

The Tarski-Vaught test states that a substructure $M\subseteq N$ is an elementary substructure if and only if every formula $\phi (x,\bar{y})$ and tuple $\bar{a}\in M$ satisfyNecessity follows immediately from elementarity. Conversely, assume the witness condition. Induct on formulas to prove that $M\models \psi (\bar{a})$ exactly when $N\models \psi (\bar{a})$ for every $\bar{a}\in M$. Atomic formulas agree because $M$ is a substructure, and Boolean connectives follow by induction. A witness in $M$ is also one in $N$; a witness in $N$ can be replaced by one in $M$ by hypothesis, after which induction applies to the matrix. Universal formulas follow by negation. Thus the witness condition is equivalent to $M\preccurlyeq N$.