Explain what is meant by a substructure of a $\Sigma$-structure $A$, where $\Sigma$ is a first-order signature (possibly including both predicate symbols and function symbols). Show that if $B$ is a substructure of $A$, and $\phi$ is a first-order formula over $\Sigma$ with $n$ free variables, then $[\phi]_{B}=[\phi]_{A} \cap B^{n}$ if $\phi$ is quantifier-free. Show also that $[\phi]_{B} \subseteq[\phi]_{A} \cap B^{n}$ if $\phi$ is an existential formula (that is, one of the form $\left(\exists x_{1}, \ldots, x_{m}\right) \psi$ where $\psi$ is quantifier-free), and $[\phi]_{B} \supseteq[\phi]_{A} \cap B^{n}$ if $\phi$ is a universal formula. Give examples to show that the two latter inclusions can be strict.

Show also that

(a) if $T$ is a first-order theory whose axioms are all universal sentences, then any substructure of a $T$-model is a $T$-model;

(b) if $T$ is a first-order theory such that every first-order formula $\phi$ is $T$-provably equivalent to a universal formula (that is, $T \vdash(\phi \Leftrightarrow \psi)$ for some universal $\psi$ ), and $B$ is a sub-T-model of a $T$-model $A$, then $[\phi]_{B}=[\phi]_{A} \cap B^{n}$ for every first-order formula $\phi$ with $n$ free variables.