elementary set theory - Proof about cardinality of sets $|R-Z|=|R|$

Prove that $|R-Z|=|R|$ where $R$ is reals and $Z$ is integers.

New approach
Maybe the problem is reduced to finding a bijection between (0,1) and [0,1] if we find a bijection between these intervals, we can do the same thing for the other ones

What about this possible bijective function f(x)=1/(x-1)x

Thank you so much for your help, I have more problems like this to work on, so need more help.

Answer

Assume it is known that any open, nonempty subset of $\mathbb{R}$ is uncountable - this step is where your work may be. The rest is easy:

Let $S = \mathbb{R} \setminus \mathbb{Z}$.

First remark that $S \subset \mathbb{R}$, so we must have that $|S| \leq |\mathbb{R}|$.

The open interval $(0, 1)$ is a subset of $S \subset \mathbb{R}$, so $|(0, 1)| \leq |S| \leq |\mathbb{R}|$.

But $|\mathbb{R}| = |(0, 1)|$, and we're done.

NOTE: to see that $|\mathbb{R}| = |(0, 1)|$, we might construct a bijection $f:(0, 1) \longrightarrow (0, \infty)$ such that $f(x) = \frac{1}{1 - x} - 1$, and reason that this bijection can be easily extended to a bijection between $(0, 1)$ and $\mathbb{R}$.