1. ## Bijection

Hey all,

I know there's a bijection between $\displaystyle \mathbb{R}$ and $\displaystyle \rbrack0,1\lbrack$, but I was wondering if there was one between $\displaystyle \mathbb{R}$\$\displaystyle \mathbb{Q}$ and $\displaystyle \mathbb{R}$\$\displaystyle \mathbb{Q}\bigcap$$\displaystyle \rbrack0,1\lbrack$.

Can anyone help me? Thank you!

2. What do you mean by $\displaystyle \mathbb{R}/\mathbb{Q}$ because that is used to represent a factor ring but the problem is that $\displaystyle \mathbb{Q}$ is not an indeal in $\displaystyle \mathbb{R}$. Thus you mean something else by the symbol $\displaystyle /$ but I do not know what you mean.

3. Sorry for the confusion, the backslash would mean, in this case, $\displaystyle \mathbb{R}$ without $\displaystyle \mathbb{Q}$, or in other words the irrationals. Indeed, I am not trying to represent a factor ring.

4. Originally Posted by juef
Sorry for the confusion, the backslash would mean, in this case, $\displaystyle \mathbb{R}$ without $\displaystyle \mathbb{Q}$, or in other words the irrationals. Indeed, I am not trying to represent a factor ring.
But the cardinality of the irrationals must be the countinuum. Because the cardinality of the rationals is $\displaystyle \aleph_0$.