Wednesday, 22 January 2014

functions - Is there a bijection between (0,1) and mathbbR that preserves rationality?



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(2x1)/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 x1/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/(x1/2)) on (1/2,1).


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