While reading about cardinality, I've seen a few examples of bijections from the open unit interval (0,1) to R, one example being the function defined by f(x)=tanπ(2x−1)/2. Another geometric example is found by bending the unit interval into a semicircle with center P, and mapping a point to its projection from P onto the real line.
My question is, is there a bijection between the open unit interval (0,1) and R such that rationals are mapped to rationals and irrationals are mapped to irrationals?
I played around with mappings similar to x↦1/x, but found that this never really had the right range, and using google didn't yield any examples, at least none which I could find. Any examples would be most appreciated, thanks!
Answer
(1/x)−2 on (0,1/2] and 2−(1/(x−1/2)) on (1/2,1).
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