= Solution
A <first-order structure> consists of a nonempty domain $A$, an interpretation $c^A\in A$ of each <constant symbol>, a total function $f^A:A^n\to A$ for each $n$-ary <function symbol>, and a relation $R^A\subseteq A^n$ for each $n$-ary <predicate symbol>. <Logical equality> is interpreted as actual identity.
An <assignment in a first-order structure> is a map $s$ from the variables into $A$. It extends recursively to terms: $x^A[s]=s(x)$, $c^A[s]=c^A$, and $f(t_1,\ldots,t_n)^A[s]=f^A(t_1^A[s],\ldots,t_n^A[s])$. Atomic truth is determined by membership of the resulting tuple in $R^A$, or equality of the two term values. <Negation> reverses the <truth value>, and an implication is false exactly when its antecedent is true and its consequent false.
For a <universal quantifier>,
$$
A,s\models\forall x\theta\quad\Longleftrightarrow\quad A,s[x\mapsto a]\models\theta\text{ for every }a\in A.
$$
These clauses define a valuation on every formula by <structural recursion>. Only free-variable values affect a formula's truth; consequently a sentence has the same <truth value> under every assignment. This is a semantic assignment of domain elements, not a programming assignment operation.
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