Definition:Extension of Assignment

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Let $\mathcal A$ be a structure for predicate logic.

Let $\sigma$ be an assignment for $\mathcal A$.

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 $\sigma + (y / a)$, is defined by:

$\forall x \in \operatorname{dom} \left({\sigma}\right) \cup \left\{{y}\right\}: \left({\sigma + (y / a)}\right)\left({x}\right) := \begin{cases} a & \text{if $x = y$} \\ \sigma \left({x}\right) & \text{otherwise} \end{cases}$

Note in particular the case where $y \in \operatorname{dom} \left({\sigma}\right)$.

If $\sigma \left({y}\right) = a'$, say, then $\sigma + (y / a)$ overwrites this value to become $a$ instead.