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?
Best Answer
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 k ≥ s 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?