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Determine which of the following sets have the least upper bound property and which have the greatest lower bound property.

(a) $S = (-\infty, 1) \cup [2, 3) \cup (3, 10]$

(b) $S = (-\infty, 1) \cup [2, 3) \cup [3, 10]$

(c) $S = (-\infty, 1) \cup [2, 3) \cup [9, 10]$

I am thinking each of the sets has the least upper bound property, but not the greatest lower bound property because each set is bounded above by 10 and bounded below by -infinity. Am I correct?

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Let's review some definitions:

A set S of real numbers is called bounded from above if there is a real number k such that ks for all s in S. The number k is called an upper bound of S. The terms bounded from below and lower bound are similarly defined.

The supremum of a subset S of a totally or partially ordered set T is the least element of T that is greater than or equal to all elements of S. Consequently, the supremum is also referred to as the least upper bound. The infimum or greatest lower bound is similarly defined.

completeness of the real numbers asserts every nonempty subset of the set of real numbers that is bounded above has a supremum that is also a real number.

In our given problems, the set T in the supremum definition is ℝ. To be bounded above, each set S, do we have a number k ≥ for all s in S? Sure. It's 10. So it is bounded above and by completeness, it has a supremum that is also a real number. So your intuition about having the greatest upper bound property is correct. Now what is the least element of ℝ that is greater than or equal to all elements of S?

Similarly, what is the greatest element of ℝ that is less than or equal to all the elements of S?

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Hint: ;

(a) For the least upper bound consider the following question: What is the supremum of the set $(-\infty,1)$?

(b) For the greatest lower bound consider the following question: What is the infimum of the set $(3,10]$?

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  • $\begingroup$ From what you said, I am thinking that in a, b, & c, there is no least upper bound of the set $(-\infty,1)$, so each S does not have the greatest lower bound property. Also, there is no infimum of the set $(3,10]$, but it exists in the second two Sets, so only the first set S does not have the least upper bound property. Is this correct? $\endgroup$ – MDW Oct 13 '13 at 21:49
  • $\begingroup$ @Michael I guess in your last sentence you mean "only the first set doesn't have the greatest lower bound property". Other than that everything you said is correct. $\endgroup$ – azarel Oct 13 '13 at 21:56
  • $\begingroup$ b is funny as it's just $S=(-\infty,1)\cup[2,10]$. $\endgroup$ – gniourf_gniourf Jul 5 '14 at 14:05

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