Representing random variables in a shared and private decomposition

Question

I'm struggling with proving/disproving the following statement.

Let $X$ and $Y$ be two random variables such that $Y \neq g(X)$ and $X \neq g(Y)$ for any function $g$. Then, there are random variables $U$, $C_1$ and $C_2$, and functions $f_1$ and $f_2$ such that:

1. $U$, $C_1$ and $C_2$ are pair-wise independent.
2. $X = f_1(U,C_1)$.
3. $Y = f_2(U,C_2)$.
4. There are no functions $g_1$ and $g_2$ such that $X = g_1(U)$ nd $Y = g_2(U)$.

Answer 1

False. For consider the case $X=Y$.