# how to find two right-inverse functions of a function

i am stuck in this problem. i need to find two right-inverse functions of this function:

$h: \Bbb N_0\times \Bbb N \to \Bbb N, (m,n)\mapsto m+n$.

i know that the function h' is a right inverse of a function h if and only if:

$h \circ h' = m+n$.

how can i say this in mathematical way and as for the h function given above? i did this:

$h': \Bbb N->\Bbb N_0\times\Bbb N, m+n\mapsto (m,n)$. $h \circ h' = h(h'(m+n))=m+n$

but now from here, i dont know how to finish the proof and how to show another right-inverse function of h

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For some basic information about writing math at this site see e.g. here, here, here and here. – Julian Kuelshammer Nov 11 '12 at 11:33
Note that just writing $h'(m+n)=(m,n)$ doesn't make it a well-defined function (and that is why you struggle to find another one). Say, what is $h'(5)$? – Martin Argerami Nov 11 '12 at 11:40

As you point out, $h^{-1}$ maps naturals to pairs $(m, n)$. If you think about what $h^{-1}$ is doing, you'll see that you need to find a pair $(m, n)$, $m \in \mathbb{N}_0$, $n \in \mathbb{N}$ such that $m+n$ is the function $h^{-1}$'s input. That is, if $h^{-1}(k) = (m, n)$, then $m + n = k$. An easy approach is to fix some $m$ first, then let $n = k - m$. This will work provided $k > m$. We can then handle the remaning cases separately. This gives $h^{-1}(n) = (0, n)$ as a simple example, since $(0, n) \in \mathbb{N}_0 \times \mathbb{N}$. So that gives one right inverse. Another can be given by, for example,

$$h^{-1}(n) = \begin{cases}(1, n-1), & \mbox{if } n > 1\\(0, 1), & \mbox{if } n = 1\end{cases}$$

Can you devise a suitable function for $m = 2$? For $m = 3$?

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wow, thanks, now i understand. yeah, for $m=2$, $n$ will remain as free choice, so that i can say: $h^-1(2+n)->(2,n)$, right? – doniyor Nov 11 '12 at 11:58
Yes, but you still need to define it for inputs less than or equal to 2. – mike4ty4 Nov 11 '12 at 12:04

The notation for $h\circ h'$ as '$m+n$' is not correct mathematically. Though, I think I understand what you were thinking about.

You need the following, $h':\Bbb N\to \Bbb N_0\times\Bbb N$, this maps any natural number to a pair of natural numbers (if I understand it right, the first element of this pair can also be zero). And what is needed, is,

if for an $s\in\Bbb N$, we have $h'(s)=(u,v)$, then $u+v=s$ holds.

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yes, you are right about what i am thinking about. i think, there are more than 2 right-inverse function, right? – doniyor Nov 11 '12 at 11:48
Well, mike4ty4 already gave you infinitely many right inverses (one for each $m$).. – Berci Nov 11 '12 at 11:52

Just to give a little more clarity. For each $m\in \mathbb{N}$, define $g_{m}:\mathbb{N}\rightarrow \mathbb{N}_{0}\times \mathbb{N}$ as \begin{eqnarray*} g_{m}(n)=\left \{ \begin{array}{cc} (m,n-m)&\text{ if } n\geq m\\ (0,n)&\text{ if } n<m \end{array}\right. \end{eqnarray*}

Then each $g_{m}$ is a right inverse of $h$.

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