Does the Supremum Have to Be in the Set?

Key Takeaways

• The supremum of a set does not necessarily have to be an element of the set itself. It is the smallest upper bound.
• The supremum is the least upper bound – the smallest number greater than or equal to all elements in the set.
• Examples illustrate cases where the supremum is within the set and cases where it is not.
• Understanding key concepts like upper bound and least upper bound clarifies the supremum's relation to the set.
• While the supremum may or may not be in the set, the maximum must be an element of the set.

Introduction

When working with sets of real numbers, we often want to find the greatest value or upper limit of the set. This concept is formally captured in the idea of the supremum. But does this supremum or least upper bound necessarily have to be contained within the set itself?

This article will provide a comprehensive analysis of the properties and definitions relating to the supremum in order to evaluate whether or not it must be an element of the given set. We will examine key examples that illustrate cases where the supremum is part of the set and cases where it is not. Core concepts such as upper bounds and least upper bounds will also be explored to fully understand the relationship between a set and its supremum.

Gaining clarity on whether or not a set must contain its supremum is valuable for properly using and interpreting this concept across algebra, calculus, and other areas of mathematics. The detailed exploration below will build intuition and demystify when and why the supremum may or may not be in the set it bounds. Let's dive in!

What Is the Formal Definition of the Supremum??

To determine if the supremum must be in the set, we first need to understand what the supremum actually is. Formally, the supremum of a set S is defined as the least upper bound of S.

Breaking this definition down:

• An upper bound of a set S is a number b such that b ≥ s for all s in S. In other words, b is greater than or equal to every element in the set.
• The least upper bound is the smallest number that qualifies as an upper bound. There may be multiple upper bounds, but the least upper bound is the smallest one.
• The supremum, denoted sup(S), is simply the least upper bound of the set S.

So in summary, the supremum is the smallest upper bound of a set – the smallest number that is greater than or equal to all elements in that set.

Does the Least Upper Bound Have to Be in the Set?

Knowing the formal supremum definition, we can now evaluate whether the least upper bound or supremum must be contained within the set it bounds.

The key realization is that the least upper bound of a set does not necessarily have to be an element of that set. While it serves as an upper limit or greatest value of the set, the supremum may or may not be a member of the set itself.

Let's look at some examples to build intuition:

Example 1: Supremum Within the Set

Consider the set S = {1, 5, 7, 9, 10}.

The number 10 is an upper bound of S because it is greater than or equal to all elements (1, 5, 7, 9, 10).

Also, 10 is the least upper bound because there is no smaller number that bounds all the elements.

Therefore, the supremum of S is 10. In this case, the supremum 10 is contained in the set S.

Example 2: Supremum Not in the Set

Now consider the set T = (0, 1) which contains all real numbers between 0 and 1.

The number 1 serves as an upper bound of T because 1 ≥ x for all x in T.

Also, 1 is the least upper bound of T since there is no smaller upper bound.

Therefore, the supremum of T is 1. However, 1 is not contained within the set T which only includes numbers between 0 and 1. The supremum is not in the set in this case.

These examples demonstrate that the supremum may or may not be an element of the set itself – it just needs to serve as the least upper bound.

Why Doesn't the Supremum Have to Be in the Set?

We have seen through examples that the supremum does not necessarily have to be part of the set. But why is this the case logically?

The key realization stems from the definitions of upper bound and least upper bound. An upper bound only needs to be greater than or equal to the set elements, not necessarily a member of the set. And the least upper bound is simply the smallest upper bound, regardless of set membership.

So being in the set is not a requirement to be an upper bound or the least upper bound. All that matters is comparison to the set elements. This allows the supremum to potentially be any real number that satisfies the least upper bound property, regardless of set membership.

How Is the Supremum Different from the Maximum?

An important distinction is that while the supremum does not necessarily have to be in the set, the maximum does.

The maximum of a set is the greatest element in the set. So by definition, the maximum must be a member of the set.

In contrast, the supremum only needs to bound the set, not be inside it. The supremum and maximum are equal when the greatest set element also happens to serve as the least upper bound. But the concepts are distinct in the case where the upper bound lies outside the set.

So in summary:

• Supremum: Least upper bound of a set – may or may not be in the set
• Maximum: Greatest element in a set – must be in the set

How Can I Find the Supremum of a Set?

Now that we understand the supremum definition and its relationship to the set, here are some guidelines for finding the supremum of a given set:

• Identify all upper bounds of the set
• Determine the smallest of those upper bounds
• This least upper bound or minimum upper bound is the supremum
• Check if the supremum is inside or outside the set

You can also work backwards from set elements:

• Sort set elements in increasing order
• The supremum is the smallest number greater than or equal to the maximum set element
• Verify this least upper bound bounds all set elements
• Check if it is inside or outside the set

Practice with examples will build skills in calculating and interpreting suprema.

When Is the Supremum in the Set vs Outside It?

Through many examples, we can make some general observations about when the supremum tends to lie inside or outside the set:

• If the set has a maximum element, then the supremum is likely that maximum and is in the set.
• If the set has no maximum, the supremum is likely outside the set. For example, open intervals like (0, 5).
• The supremum of a closed interval [a, b] is b, which is in the set.
• For bounded sets with no maximum, like (0, 1), the supremum is the least upper bound outside the set.
• For unbounded sets like all reals or (1, ∞), the supremum does not exist.

So in closed intervals and sets with maximum elements, the supremum tends to be in the set. But for open intervals, open sets, and sets with no maximum element, the least upper bound sits outside the set.

Examples Illustrating Supremum Within and Outside a Set

To further develop intuition for when the supremum is in the set vs outside it, let's work through some detailed examples:

Supremum Within the Set

Consider the closed interval S = [2, 5].

The upper bounds of S are numbers ≥ 5, since 5 is the maximum of S. The least upper bound is 5 itself.

Since 5 is also the maximum element of S, the supremum 5 lies within set S.

As another example, take the set T = {1, 3, 7, 8, 9, 10}.

The upper bounds of T are numbers ≥ 10, with 10 being the least upper bound. 10 is also the maximum of set T.

So again in this case, the supremum 10 is an element of the set T.

Supremum Outside the Set

Now consider the open interval U = (2, 5)

The upper bounds of U are numbers ≥ 5. The least upper bound is 5.

However, 5 is not within the interval U which only contains values between 2 and 5.

So here, the supremum 5 lies outside set U.

As another example, take the set V = {1/n | n is a natural number}.

The upper bounds of V are numbers ≥ 1, with 1 being the least upper bound.

But V contains only fractional values less than 1. So the supremum 1 is not in set V.

These examples demonstrate cases where the least upper bound sits outside the given set as its supremum.

Why Is Understanding Supremum Important?

Now that we have thoroughly analyzed the relationship between a set and its supremum, it is worth highlighting why this concept is so important across mathematics:

• Supremum gives the least upper bound or upper limit of a set, providing important information about the scope of values.
• It provides insights on whether a set is open or closed based on whether the supremum is inside or outside the set.
• Understanding supremum aids in problems involving maximum/minimum values, optimization, and real analysis.
• It is a prerequisite for learning concepts like least upper bound property of real numbers and completeness of the reals.
• Supremum has applications in probability, measure theory, topology, numerical analysis, and more advanced areas.

In short, grasping supremum theory develops mathematical maturity and paves the way for higher-level quantitative reasoning and analysis. The foundational skill of finding suprema is valuable across many mathematical domains.

Conclusion: Key Takeaways on Supremum and Set Relationship

In summary, we have demonstrated that:

• The supremum (least upper bound) of a set does not necessarily have to be an element of that set.
• The supremum only needs to be greater than or equal to the set elements to qualify as their least upper bound. Set membership is not required.
• Whether the supremum is inside or outside the set depends on properties of the specific set.
• Closed sets and sets with maximums typically have internally contained suprema. Open sets and sets without maximums have external suprema.
• Understanding the nuances around if and when the supremum is in the set provides valuable insight on bounding and limits of sets.

Grasping supremum concepts deepens mathematical maturity and paves the way for advanced analysis and applications. I hope this detailed walkthrough has demystified when and why the supremum may or may not live within its associated set!

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