Understanding Supremum and Infimum in Ordered Sets

In the realm of mathematical analysis, particularly in the study of ordered sets such as the real numbers, it is essential to understand the concepts of supremum and infimum. These concepts are fundamental in identifying the bounds of a set and are used extensively in various mathematical proofs and applications.

Supremum and Infimum: Definitions and Examples

Let's delve into the definitions and explore how these bounds work in different scenarios. Consider the sets ( S (-infty, 0) ) and ( T [-infty, 0] ), both subsets of the real line ( mathbb{R} (-infty, infty) ).

S (-∞, 0) does not have a lower bound within ( mathbb{R} ). The set of upper bounds for ( S ) includes all points in ( [0, infty) ). T [-∞, 0] includes the point 0, making it a subset of ( S ) with the added point 0.

For set ( S ), 0 is the least upper bound (supremum) but is not a member of ( S ), indicating that ( S ) has no maximal elements. For set ( T ), however, 0 is both the least upper bound (supremum) and the maximal element, as it belongs to the set itself.

The concept of supremum is about how an ordered set sits within a larger set, whereas the notion of a maximal element is about the set itself.

General Definitions and Procedures

We can generalize the definitions and procedures for finding the supremum and infimum of a set:

Infimum

Pick any number that is less than everything in your set. If no such number exists, the infimum is (-infty). Push this number as far to the right as possible, without crossing any number in your set. If your set is empty, you'll never stop, and the infimum is (infty). The stopping point gives the infimum of the set.

Formally, the infimum (greatest lower bound) of a set is a value that is a lower bound, and no greater value is a lower bound.

Supremum

Pick any number that is greater than everything in your set. If no such number exists, the supremum is (infty). Push this number as far to the left as possible, without crossing any number in your set. If your set is empty, you'll never stop, and the supremum is (-infty). The stopping point gives the supremum of the set.

In the real numbers, every nonempty set that is bounded above has a real-number supremum, and similarly, every nonempty set that is bounded below has a real-number infimum.

Order-Completeness of the Real Numbers

The real numbers are order-complete, meaning that the supremum and infimum of any nonempty set that is bounded above and below, respectively, exist within the real numbers.

In the context of the extended real line, which includes (infty) and (-infty), these values are actual elements of the ordered set, and hence no longer need to be considered as conventions.

Conclusion

Understanding the concepts of supremum and infimum is crucial for advanced mathematical studies and applications. By grasping these definitions and procedures, you can effectively analyze and manipulate the bounds of different sets in your mathematical work.

Keywords: supremum, infimum, bounded sets