Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2009 / B / 4 / a / ii

The relation $A\subseteq B$ means that the domain of $A$ is a subset of the domain of $B$ and the inclusion map is an $L$-embedding. Thus it contains all named constants, is closed under every named function, and uses the restrictions of the ambient relations.

The stronger relation $A\preccurlyeq B$ means that for every first-order formula $\varphi (\bar{x})$ and tuple $\bar{a}$ from $A$,This includes formulas with quantifiers ranging over the respective domains. Elementary inclusion implies ordinary substructure inclusion, but the latter need only preserve quantifier-free truth. In particular an existential witness available in $B$ need not lie in an arbitrary substructure.