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one of the following statements is true.Identify:-

(A) Every subset of $\Bbb{Q}$ which is bounded above has a unique supremum rational number.
(B)Every subset of $\Bbb{Q}$ which is bounded above has a unique supremum real number.
(C)The set of rational numbers has least least upper bound property.
(D)The set of real numbers does not have the greatest lower bound property.


I am confused because I think both B and D are correct but they say only one option is correct

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2 Answers

They are all wrong.

(A) $\{x\in \mathbb Q\mid x^2<2\}$ is bounded from above (e.g. $42$ is an upper bound), but has no rational number as supremum

(B) $\emptyset$ is bounded from above (e.g. $42$ is an upper bound), but has no real number as supremum (of course this example also works for (A))

(C) Least upper bound property means that every nonempty set that is bounded from above has a least upper bound (in the set). As the exmaple to (A) shows, $\mathbb Q$ does not have this property

(D) $\mathbb R$ is specifically constructed to have the least upper bound property!

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If the exercise indeed says that one of the statements is true, I'd guess it's supposed to be B and they forgot about the "non-empty" part. –  Javier Badia May 26 '13 at 12:41
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Greatest lower bound for reals is $+\infty$, similarly, the supremum of reals is $-\infty$. I assume that whoever gave you that question meant that (B) should be correct(is there a non-empty assumption?), however, as $\emptyset$ is always a subset, and has $-\infty$ as upper bound (which is NOT real by the way), (B) is also false.

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