Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / B / 6 / i

Let $L_A$ expand $L$ by a constant $c_a$ for each $a\in A$, and let ${\mathcal{A}}_A$ be the canonical expansion interpreting $c_a$ as $a$. Its diagram of a structure is the set of all atomic and negated atomic $L_A$-sentences true in ${\mathcal{A}}_A$. Its elementary diagram is the set of all $L_A$-sentences true there:A substructure inclusion $\mathcal{A}\subseteq \mathcal{B}$ means that $A\subseteq B$, constants have the same interpretations, the operations of $\mathcal{B}$ restrict to those of $\mathcal{A}$, and each relation restricts to the corresponding relation on $A$.

An elementary substructure additionally preserves the truth of every logical formula with parameters from $A$:Here the right-hand condition includes the substructure requirements by applying it to atomic logical formulas and their negations.