# Finding functions such that $F\circ g_k=i_{\mathcal P (\mathbb R)}$

Let $F:(\mathbb R\times \mathcal P (\mathbb R))\to \mathcal P (\mathbb R) \\ F((x,A))=\{y\in \mathbb R| \frac {x+y} 2\in A \}$

Define two different functions $g_k:\mathcal P (\mathbb R)\to (\mathbb R\times \mathcal P (\mathbb R))$ such that $F\circ g_k=i_{\mathcal P (\mathbb R)}, k=1,2$

This is equivalent to showing that $F$ is onto using composition of two different functions.

An easy example would be to take $F(0,\{1,2,3\})=\{2,4,6\}$ so for this example, this function could work $g(A)=\{x\in A| \frac x 2\}$.

But the $x$ in $\frac {x+y} 2$ really complicates things, I don't see a way to recover that $x$ when making the $g$'s so I can't revert the set back to have its original values..

• You have to clarify the MathJax in your question,the domains and ranges of the maps aren't clear. – Mathemagician1234 Feb 4 '15 at 19:30
• What's the rectangle with the gleph symbols mean? – Mathemagician1234 Feb 4 '15 at 19:32
• @Mathemagician1234 rectangle with the gleph symbols?? I think the Latex doesn't show correctly in your browser.. – GinKin Feb 4 '15 at 19:33
• @Mathemagician1234 just to show you I'm not messing with you: i.imgur.com/ejbWEfT.png – GinKin Feb 4 '15 at 19:59