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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

user9026
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  • If a binary relation is defined as a set of ordered pairs, you could try writing this one as a disjoint union, using set builder notation for each of the sets in the union - assuming the terms of the question allow it. – Calum Gilhooley Sep 13 '20 at 03:29
  • @calum-gilhooley This is not a homework question. I am trying to do some proof and I need this relation there. I have never seen piecewise relation before. I have seen piecewise functions. If we can't use set builder notation, is it ok to just state relation in words like I have done here ? – user9026 Sep 13 '20 at 03:39
  • If it's not a homework question, then you are free to use any established notation. To me, the most natural representation of the set of ordered pairs in the relation is as a disjoint union. That would follow closely the notation you have already used, and it would arguably be clearer than the verbal form, although the latter is also acceptable. – Calum Gilhooley Sep 13 '20 at 03:44
  • @calum-gilhooley Can you just write the relation using disjoint union ? I have never seen that before. – user9026 Sep 13 '20 at 03:49
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    I can't see any reason not to - unless a binary relation is defined as an ordered triple, and even then, the "graph" component of the relation can be written as a disjoint union. – Calum Gilhooley Sep 13 '20 at 03:59
  • Thanks. I will try that – user9026 Sep 13 '20 at 04:01

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