None of the sets , and is bounded above. The is bounded below; is a lower bound for and so is any number less than . In fact, is the larges or greatest lower bound.

Question: How could there be a number less than when is the smallest number in ?

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- Sep 5th 2010, 02:15 PMnoviceUpper Bound and Lower Bounds
None of the sets , and is bounded above. The is bounded below; is a lower bound for and so is any number less than . In fact, is the larges or greatest lower bound.

Question: How could there be a number less than when is the smallest number in ? - Sep 5th 2010, 02:34 PMPlato
First, be careful. Your text seems to imply that . But in some texts .

Using the convention that if then .

In other words 1 is the first element in .

It depends on the superset as to how we answer your question.

In the real numbers, any number less than 1 is a lower bound for .

But in the set there is only one lower bound, 1.

Does that help? - Sep 5th 2010, 02:37 PMemakarov
When one talks about a lower bound, it has to be in the context of some set P and its subset S. An x in P is a lower bound of S if x <= y for all y in S. To talk about a lower bound of one has to designate a superset of .

- Sep 5th 2010, 03:05 PMnovice
- Sep 5th 2010, 06:48 PMnovice
Here is another example:

Quote:

The set is not bounded above. Among its many lower bounds, 0 is the greatest lower bound.

Well, I think I figured it out on my own soon after I typed in the question.

At any rate, thank you for your time.