I have to write piecewise relation using set builder notation. $A$ and $B$ are non empty disjoint sets. And there are two bijections $f: A \to I_m $ and $g: B \to I_n$ where $m$ and $n$ are in $\mathbb{N}$. Here, $0 \in \mathbb{N}$. And we have $I_m = \{i \in \mathbb{Z}^+ \, | \, i \leq m \} $ and $I_n = \{i \in \mathbb{Z}^+ \, | \, i \leq n \} $. Now, I have to define the piecewise binary relation $h$ from $A \cup B$ to $I_{m+n}$ as follows. $(x, f(x)) \in h \text{ if } x \in A$ and $(x, m + g(x)) \in h \text{ if } x \in B$. So, how to write this piecewise relation using set builder notation ? Or is there a better way to write it ?
Thanks