Find two sets of rational numbers L and R (both does not equal the empty set) such that L union R =Q for each x belong L and y belong R, x<y and set L does not contain a largest number and R does not contain a smallest number.
Find two sets of rational numbers L and R (both does not equal the empty set) such that L union R =Q for each x belong L and y belong R, x<y and set L does not contain a largest number and R does not contain a smallest number.
Hello tigergirlAny irrational number will separate the rationals into two such subsets; for example, . So you could define the sets as:
and
Then clearly
And does not contain a largest number, since for every , you can always find which is greater than , simply by making the mean of and ; i.e. .
Similarly, of course, does not have a smallest number.
Finally, , since every rational number is in one or other set.
Grandad