Well-Ordered
A non-empty set of real numbers is called if every non-empty subset of has a least element.
The set of positive integers is well-ordered.
Is the set of positive rational numbers that can be written in the form , where is a positive integer, ?
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1 solution
Every subset of of a set of rational numbers which can be written in the form 2 a , where a is a positive integer, can be related to a set containing the same number of elements, each doubled. This set will be a subset of the set of positive integers and will have a least element. Since ordering of the numbers does not change by doubling all of them, the original set too will have a least element.