\begin{definition}[Definition:Partial Function Equality]
Let $g: \N^k \to \N$ and $h: \N^k \to \N$ be partial functions.
We write:
:$\map g {n_1, n_2, \ldots, n_k} \approx \map h {n_1, n_2, \ldots, n_k}$
{{iff}} either:
:both $\map g {n_1, n_2, \ldots, n_k}$ and $\map h {n_1, n_2, \ldots, n_k}$ are defined and equal
or:
:neither $\map g {n_1, n_2, \ldots, n_k}$ nor $\map h {n_1, n_2, \ldots, n_k}$ are defined.
That is, {{iff}} $\map g {n_1, n_2, \ldots, n_k} = \map h {n_1, n_2, \ldots, n_k}$ wherever either are defined.
Thus, '''$g$ is equal to $h$''', and we can write $g = h$, {{iff}}:
:$\forall x \in \N^k: \map g x \approx \map h x$
Category:Definitions/Mathematical Logic
\end{definition}