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.

We assume it has already been shown that $X$ has a denumerable subset $D$. Then there is a bijection $\beta$ from $\mathbb{N}$ to $D$. (For definiteness, $\mathbb{N}$ starts at $0$.)  
Write $d_i$ instead of $\beta(i)$. Let $Y$ be the set of $d_i$ such that $i$ is even.
We define a mapping  $\varphi$ from $X$ to $X\setminus Y$ as follows:
(1) If $x=d_i$ for some $i$, let $\varphi(x)=d_{2i+1}$;
(ii) If $x\not\in D$. let $\varphi(x)=x$.
It is not difficult to verify that $\varphi$ is a bijection from $X$ to $X\setminus Y$.
Remark: One can modify the argument to show that if $Y$ is a given denumerable subset of $X$, then there is a bijection from $X$ to $X\setminus Y$.
The idea is that we choose a denumerable subset $D$ of $X\setminus Y$. Let $D_0$ be the even index members of $D$, and $D_1$ the odd index members. Send $X\setminus Y$ to $(X\setminus Y)\setminus D_0$ as in the answer above. That makes room to send $Y$ to $D_0$.    
