Supremum And Infimum Of A Set

Supremum and Infimum of a Set: A Comprehensive Guide

Understanding the supremum and infimum of a set is crucial for anyone studying real analysis, calculus, or related mathematical fields. These concepts describe the least upper bound and greatest lower bound of a set, respectively, providing valuable tools for analyzing the behavior of functions and sequences. This article will delve into the definitions, properties, and practical applications of suprema and infima.

What is a Bounded Set?

Before diving into suprema and infima, we need to understand the concept of a bounded set. A set of real numbers is bounded above if there exists a real number M such that x ≤ M for all x in the set. Similarly, a set is bounded below if there exists a real number m such that x ≥ m for all x in the set. A set is bounded if it is bounded both above and below. For example, the interval [0, 1] is bounded, while the set of all natural numbers is bounded below but not above.

Defining Supremum (Least Upper Bound)

The supremum of a set S, denoted as sup(S), is the least upper bound of the set. This means that:

1. sup(S) is an upper bound of S: x ≤ sup(S) for all x ∈ S.
2. sup(S) is the least upper bound: If M is any other upper bound of S, then sup(S) ≤ M.

It's important to note that the supremum may or may not be an element of the set S. If sup(S) ∈ S, then the supremum is also the maximum of the set.

Defining Infimum (Greatest Lower Bound)

The infimum of a set S, denoted as inf(S), is the greatest lower bound of the set. This implies:

1. inf(S) is a lower bound of S: x ≥ inf(S) for all x ∈ S.
2. inf(S) is the greatest lower bound: If m is any other lower bound of S, then inf(S) ≥ m.

Similar to the supremum, the infimum may or may not belong to the set S. If inf(S) ∈ S, then the infimum is also the minimum of the set.

Examples

Let's illustrate these concepts with some examples:

• S = {x ∈ ℝ | 0 < x < 1}: sup(S) = 1 and inf(S) = 0. Note that neither 0 nor 1 are elements of S.
• S = {1/n | n ∈ ℕ}: sup(S) = 1 (and is an element of S) and inf(S) = 0. 0 is not an element of S.
• S = {(-1)^n + 1/n | n ∈ ℕ}: This set is bounded, oscillating between values close to -1 and values close to 1. Finding the precise supremum and infimum requires a deeper analysis of the sequence. However, we can qualitatively understand that there is a supremum and an infimum.

Properties of Supremum and Infimum

Suprema and infima possess several important properties:

• Uniqueness: If a supremum or infimum exists, it is unique.
• Completeness Axiom: The completeness axiom of real numbers guarantees that every non-empty set that is bounded above has a supremum, and every non-empty set that is bounded below has an infimum. This is a fundamental property distinguishing the real numbers from the rational numbers.
• Relationship to Maximum and Minimum: If a set has a maximum, then that maximum is also its supremum. Similarly, if a set has a minimum, that minimum is also its infimum.

Applications

Suprema and infima are fundamental concepts with widespread applications in various areas of mathematics and beyond:

• Calculus: Finding limits of functions and sequences often involves working with suprema and infima.
• Optimization: In optimization problems, suprema and infima represent optimal values.
• Measure Theory: The concept of measure relies heavily on suprema and infima.
• Functional Analysis: Suprema and infima are essential tools in functional analysis.

Conclusion

The concepts of supremum and infimum are cornerstones of real analysis and have far-reaching implications in numerous mathematical fields. Understanding their definitions, properties, and applications is crucial for a strong grasp of advanced mathematical concepts. This guide provides a solid foundation for further exploration of these important topics.