Assignment

Last updated Nov 1, 2022

Let $V = {v_{0}, v_{1}, \dots}$ be the Set of Variable Symbols. Let $\mathcal{L}$ be a Language and $\mathcal{M}$ an $\mathcal{L}$-structure. An Assignment is a Function $\sigma : V \to M$ (where $M$ is the Universe of $\mathcal{M}$).

1. Assignment gives us a means to evaluate Terms. Suppose $\sigma$ is an Assignment and $t$ is an $\mathcal{L}$-Term. Then $t^{\mathcal{M}} [\sigma]$ is defined as
   1. $c^\mathcal{M}$ if $t = c \in \mathcal{C}$.
   2. $\sigma(v_{i})$ if $t = v_{i}$, a Variable Symbol.
   3. $f^\mathcal{M}(t_{1}^{\mathcal{M}}[\sigma], \dots t_{n_{f}}^{\mathcal{M}}[\sigma])$ if $t_{1}, \dots, t_{n_{f}}$ are Terms and $t = f(t_{1}, \dots, t_{n_{f}})$.
2. A useful notation is $\sigma[\frac{a}{v}]$ defined as $$\sigma\left \frac{a}{v}\right= \begin{cases} a & \text{if } v_{i} = v \\ v & \text{otherwise } \end{cases}.$$ This will be helpful when we want to override the the assignment of a Variable Symbol because it is a Bound Variable