Let $\displaystyle f

0,1) \rightarrow \mathbb{R}$. Since there is a bijective function $\displaystyle f$ from $\displaystyle (0,1)$ to $\displaystyle \mathbb{R}$.

Let us say $\displaystyle f(0.25)=3$ and $\displaystyle f(0.3)=5$. If I remove equal number of elements from both sets, say $\displaystyle (0,1)-\{0.25\}$ and $\displaystyle R-\{5\}$. Now $\displaystyle 0.25 \notin (0,1)$ and $\displaystyle 3 \in \mathbb{R}$ , and $\displaystyle 0.3 \in (0,1)$ and $\displaystyle 5 \notin \mathbb{R}$.

Have we lost the bijective function?

Can we say $\displaystyle

|(0,1)-\{0.25\}|\not=|\mathbb{R}-\{5\}|

$?