\begin{definition}[Definition:Extension of Assignment]
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.
\end{definition}