Supremum

Definition

The supremum of a set is the least upper bound of that set, which means it is the smallest number that is greater than or equal to every element in the set. This concept plays a crucial role in understanding limits and bounds, especially when dealing with ordered sets. The supremum can be an element of the set or it can be a limit point that is not actually included in the set itself.

5 Must Know Facts For Your Next Test

1. The supremum may not be an element of the set, particularly if the set is unbounded or open-ended, such as the interval (0, 1) which has a supremum of 1.
2. In a finite set, the supremum is simply the maximum element of that set.
3. Suprema are crucial for defining concepts like convergence and continuity in mathematical analysis.
4. The existence of a supremum is guaranteed in complete lattices, which means every subset within them will have a least upper bound.
5. The properties of suprema are essential for proving key results in order theory, including properties related to Zorn's lemma.

Review Questions

• How does the concept of supremum relate to upper bounds and why is it important in understanding ordered sets?
  • The supremum is the least upper bound, meaning it’s the smallest number that meets or exceeds all values in a given set. While an upper bound can be any number greater than or equal to all elements, the supremum specifically identifies the smallest such bound. This distinction is important because it helps define limits and helps establish continuity and convergence within ordered sets.
• In what ways does Zorn's lemma utilize the concept of supremum to establish results about partially ordered sets?
  • Zorn's lemma states that if every chain (a totally ordered subset) in a partially ordered set has an upper bound, then the entire set contains at least one maximal element. The idea of supremum comes into play as it ensures that each chain can indeed have a least upper bound. This connection reinforces how suprema help maintain structure within partially ordered sets and are critical for deriving conclusions about their maximal elements.
• Evaluate how the properties of supremum contribute to understanding completeness in mathematical analysis and its implications on other areas of mathematics.
  • Completeness in mathematical analysis implies that every bounded set has a supremum. This property is foundational because it ensures limits exist for sequences and functions within real numbers. The implications extend into various fields such as calculus and topology, where knowing whether functions converge or whether sets have least upper bounds can influence proofs and theorem development. Understanding supremum helps clarify continuity and differentiability, serving as a bridge between abstract theory and practical application.

Related terms

Infimum: The infimum of a set is the greatest lower bound of that set, meaning it is the largest number that is less than or equal to every element in the set.

Upper Bound: An upper bound of a set is any number that is greater than or equal to every element in that set.

Complete Lattice: A complete lattice is a partially ordered set in which every subset has both a supremum and an infimum.