\begin{definition}[Definition:Assignment for Structure/Formula]
Let $\LL_1$ be the language of predicate logic.
Let $\mathrm{VAR}$ be the collection of variables of $\LL_1$.
Let $\AA$ be an $\LL_1$-structure on a set $A$.
Let $\mathbf A$ be a well-formed formula of $\LL_1$.
Denote with $\map V {\mathbf A}$ the variables which occur freely in $\mathbf A$.
An '''assignment for $\mathbf A$ in $\AA$''' is a mapping $\sigma$ with codomain $A$, whose domain is subject to the following condition:
:$\map V {\mathbf A} \subseteq \Dom \sigma \subseteq \mathrm{VAR}$
That is, the domain of $\sigma$ contains only variables, and at least those with a free occurrence in $\mathbf A$.
\end{definition}