# Definition:Extension of Assignment

## Definition

Let $\AA$ be a structure for predicate logic.

Let $\sigma$ be an assignment for $\AA$.

Let $y \in \mathrm{VAR}$ be a variable.

Let $a \in A$ be arbitrary.

Then the extension of $\sigma$ by mapping $y$ to $a$, denoted $\sigma + \paren {y / a}$, is defined by:

$\forall x \in \Dom \sigma \cup \set y: \map {\paren {\sigma + \paren {y / a} } } x := \begin{cases} a & \text{if } x = y \\ \map \sigma x & \text{otherwise} \end{cases}$

Note in particular the case where $y \in \Dom \sigma$.

If $\map \sigma y = a'$, say, then $\sigma + \paren {y / a}$ overwrites this value to become $a$ instead.