Prove that an uncountable set X is equivalent to X\Y where Y is a denumerable subset of X How do I prove this?
The problem contains the following "hint": Prove that X/Y contains a denumerable family of pairwise disjoint denumerable subsets. 
I am not sure how that proves cardinal equivalence. My first instinct was to try to show that a bijection exists between X and X\Y, but that may be extremely difficult which would be why the hint was given. 
Any help would be appreciate, and also carefully describe how the solution was found.
Thank you.

Let $g\colon \omega\to Y$ be a bijection.
Find an injection $f\colon \omega\to X\setminus Y$ recursively as follows: Assume we have already found an injective map $f_n\colon n\to X\setminus Y$ (the case $n=0$ being trivial). Then 
$$ x\mapsto\begin{cases}f_n(x)&\text{if }x<n,\\g(x-n)&\text{if }x\ge n\end{cases}$$
defines an injection $\omega\to X$. By assumtion, this is not a bijection, hence there exists some $x_n\notin Y\cup \operatorname{im}f_n$. Define $f_{n+1}\colon n+1\to X\setminus Y$ per $f_{n+1}(n)=x_n$ and $f_{n+1}(x)=f_n(x)$ otherwise.
Ultimately define $f(n)=f_{n+1}(n)$. Now you can define $h\colon X\to X\setminus Y$ per
$$ h(x)=\begin{cases}f(2g^{-1}(x))&\text{if }x\in Y,\\
f(2f^{-1}(x)+1)&\text{if }x\in \operatorname{im}f,\\
x&\text{otherwise}.\end{cases}$$
Why is this a bijection?
