Definition:Assignment for Structure/Formula

Definition
Let $\mathcal L_1$ be predicate calculus.

Let $\mathsf{VAR}$ be the collection of variables of $\mathcal L_1$.

Let $\mathcal A$ be an $\mathcal L_1$-structure on a set $A$.

Let $\mathbf A$ be a well-formed formula of $\mathcal L_1$.

Denote with $V \left({\mathbf A}\right)$ the variables which occur freely in $\mathbf A$.

An assignment for $\mathbf A$ in $\mathcal A$ is a mapping $\sigma$ with codomain $A$, whose domain is subject to the following condition:


 * $V \left({\mathbf A}\right) \subseteq \operatorname{dom} \left({\sigma}\right) \subseteq \mathsf{VAR}$

That is, the domain of $\sigma$ contains only variables, and at least those with a free occurrence in $\mathbf A$.

Also see

 * Definition:Assignment for Term