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How can I show that the set of reals and the set of pairs of reals have the same cardinality?

I know that since reals are uncountable infinite, I can't create a list of reals and talk about the $i^{th}$ real mapping to the $i^{th}$ real pair. So how can I construct a one-to-one and onto mapping $f: \mathbb R \to \mathbb R^2$?

Thank You

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    Try playing around with decimal expansion.2011-10-12
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    http://en.wikipedia.org/wiki/Space_filling_curve2011-10-12
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    It is easier to see with reals in $(0,1)$ and $(0,1) \times (0,1)$. Do you know that all the reals have the same cardinality as the reals in $(0,1)$?2011-10-12
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    Rather than explicitly constructing a bijection between $\mathbb R$ and $\mathbb R^2$, which can get a bit tricky, it may be easier to construct surjections in each direction (one is trivial) and apply the law of trichotomy (which holds for cardinality under AC).2011-10-12
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    This question is closely related: http://math.stackexchange.com/questions/37834/is-mathbbr-equipotent-to-mathbbr22011-10-12
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    @Ilmari: It is as easy to define *injections*, and use the fact that Cantor-Bernstein holds without the axiom of choice as well.2011-10-12
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    @Jeff: The fact that the reals are uncountable does not mean that you cannot create a list of them. It just means that the list will be vastly longer than $\mathbb N$, and you'll need a longer set of indices.2011-10-12

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