Infimum

Definition

The infimum of a set is the greatest lower bound of that set, meaning it is the largest element that is less than or equal to every element in the set. This concept connects to partial orders because it helps to establish a way to compare elements based on their relationships. Understanding the infimum is essential for analyzing bounded sets and plays a significant role in determining limits and continuity in mathematical contexts.

5 Must Know Facts For Your Next Test

1. The infimum does not need to be an element of the set; it can exist outside the set if there is no greatest lower bound within it.
2. Every non-empty subset of the real numbers that is bounded below has an infimum, as guaranteed by the completeness property of real numbers.
3. In a partially ordered set, the infimum can be found by examining all lower bounds and selecting the largest among them.
4. If a set has a minimum element, then that minimum is also its infimum.
5. The concept of infimum is crucial in calculus, especially when discussing limits and convergence of sequences.

Review Questions

• How does the concept of infimum relate to bounded sets and their properties?
  • The concept of infimum directly relates to bounded sets since it defines the greatest lower bound for those sets. A bounded set has a lower limit, and understanding the infimum allows us to identify this limit even when it might not be an actual member of the set. For instance, if we have a set like {x ∈ ℝ | x > 0}, its infimum would be 0, even though 0 is not part of the set. This shows how the infimum provides insights into the nature of boundedness in sets.
• What role does the infimum play in partially ordered sets and how can it be determined?
  • In partially ordered sets, the infimum serves as a key tool for comparing elements based on their relationships. To determine the infimum, one must identify all elements that act as lower bounds for a specific subset and select the largest among them. This process highlights how elements relate within the order structure and emphasizes the significance of lower bounds in understanding ordering properties.
• Evaluate how understanding the infimum can enhance one's grasp of limits and continuity in mathematical analysis.
  • Understanding the infimum enriches one's comprehension of limits and continuity by providing a framework for evaluating behaviors at boundaries. In calculus, for example, recognizing how sequences converge towards their limits often involves identifying their greatest lower bounds. This connection aids in assessing whether functions approach certain values consistently, which is fundamental for determining continuity and differentiability in analysis. Thus, the concept of infimum not only solidifies foundational principles but also serves as a bridge between discrete sets and continuous functions.

Related terms

Supremum: The supremum of a set is the least upper bound, or the smallest element that is greater than or equal to every element in the set.

Bounded Set: A bounded set is one that has both upper and lower bounds; in other words, there exist elements that serve as limits for all members of the set.

Partially Ordered Set: A partially ordered set is a set equipped with a binary relation that allows some pairs of elements to be compared, but not necessarily all.