Definition:Union Mapping

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$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$
$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$

Let $f_1$ and $f_2$ be combinable, that is, that they agree on $S_1 \cap S_2$.

Then the union mapping (or combined mapping) $f$ of $f_1$ and $f_2$ is:

$f: S_1 \cup S_2 \to T_1 \cup T_2: f \left({s}\right) = \begin{cases} f_1 \left({s}\right) : & s \in S_1 \\ f_2 \left({s}\right) : & s \in S_2 \end{cases}$

Also see