# Definition:Assignment for Structure

## Definition

Let $\mathcal L_1$ be the language of predicate logic.

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

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

An **assignment for $\mathcal A$** is a mapping $\sigma$ such that:

That is, the domain of $\sigma$ contains only variables, and maps them to elements of $A$.

### Assignment for Term

Let $\tau$ be a term of $\mathcal L_1$.

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

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

- $V \left({\tau}\right) \subseteq \operatorname{dom} \left({\sigma}\right) \subseteq \mathrm{VAR}$

That is, the domain of $\sigma$ contains only variables, and at least those which occur in $\tau$.

### Assignment for Formula

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:

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

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