\begin{definition}[Definition:Assignment for Structure]
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 $A$.
An '''assignment for $\AA$''' is a mapping $\sigma$ such that:
:the codomain of $\sigma$ is $A$
:the domain of $\sigma$ is a subset of $\mathrm{VAR}$
That is, the domain of $\sigma$ contains only variables, and maps them to elements of $A$.
\end{definition}