Bijection

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February 21st 2006, 09:10 AM
juef

Bijection

Hey all,

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

Can anyone help me? Thank you!
February 21st 2006, 02:51 PM
ThePerfectHacker

What do you mean by $\mathbb{R}/\mathbb{Q}$ because that is used to represent a factor ring but the problem is that $\mathbb{Q}$ is not an indeal in $\mathbb{R}$ . Thus you mean something else by the symbol $/$ but I do not know what you mean.
February 21st 2006, 03:29 PM
juef

Sorry for the confusion, the backslash would mean, in this case, $\mathbb{R}$ without $\mathbb{Q}$ , or in other words the irrationals. Indeed, I am not trying to represent a factor ring. :)
February 22nd 2006, 02:42 PM
ThePerfectHacker

Quote:

Originally Posted by juef

Sorry for the confusion, the backslash would mean, in this case, $\mathbb{R}$ without $\mathbb{Q}$ , 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 $\aleph_0$ .

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