Take any rational number which is not zero, denote it by r. assume that for any irrational number x, r^(1/x) is rational. then we have a univalent function f:x-> r^(1/x) which maps irrational numbers to rational numbers, which is a contradiction because the set of rationals is countable and the set of irrationals is not - thus for some irrational x, r^(1/x) is not rational, and we have:
(r^(1/x))^x = r
|
|
Posted by ronen
on 2005-01-12 19:10:24 |
Solution by negation
