= Solution
The <complete theory> $\operatorname{Th}(A)$ is the <set> of all $L$-sentences true in $A$. The <existential theory of a structure> $\operatorname{Th}_\exists(A)$ consists of those true sentences of the form $\exists\bar x\,\theta(\bar x)$ with quantifier-free $\theta$. The universal <first-order theory>[theory] $\operatorname{Th}_\forall(A)$ uses $\forall\bar x\,\theta(\bar x)$ with quantifier-free matrix. Allow equivalent formulas and empty <quantifier> blocks, or insert dummy variables in nonempty domains. These are sentences in the original language, distinct from diagrams, which add constants naming <Elements> of the structure.
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