Question

Let \(A\) and \(B\) be subsets of an ordered field \(F\). Assuming that the required lub and glb exist in \(F\), prove that (i) if \((\forall x \in A)(\forall y \in B) x \leq y,\) then \(\sup A \leq \inf B ;\) (ii) if \((\forall x \in A)(\exists y \in B) x \leq y,\) then sup \(A \leq \sup B\) (iii) if \((\forall y \in B)(\exists x \in A) x \leq y,\) then \(\inf A \leq \inf B\).

Short answer

(i) \( \sup A \leq \inf B \); (ii) \( \sup A \leq \sup B \); (iii) \( \inf A \leq \inf B \).

Step-by-step solution

Understanding Supremum and Infimum

For any subset \( S \) of a field \( F \), the supremum (or \( \sup \)) of \( S \) is the smallest element in \( F \) that is greater than or equal to every element of \( S \). Similarly, the infimum (or \( \inf \)) of \( S \) is the largest element in \( F \) that is less than or equal to every element of \( S \). These concepts will guide the proofs below.

Prove (i): Supremum of A is Less Than or Equal to Infimum of B

Given \((\forall x \in A)(\forall y \in B) x \leq y,\), by definition, every element in \( A \) is less than or equal to every element in \( B \). Therefore, the largest element \( \sup A \) must also be less than or equal to any element in \( B \), including the smallest of them, which is \( \inf B \). Thus, \( \sup A \leq \inf B.\)

Prove (ii): Supremum of A is Less Than or Equal to Supremum of B

Given \((\forall x \in A)(\exists y \in B) x \leq y,\) for each \( x \in A \), there exists at least one \( y \in B \) such that \( x \leq y \). Consequently, the largest element \( \sup A \) must be less than or equal to the largest element in \( B \), which is \( \sup B \). Therefore, \( \sup A \leq \sup B.\)

Prove (iii): Infimum of A is Less Than or Equal to Infimum of B

Given \((\forall y \in B)(\exists x \in A) x \leq y,\), for each \( y \in B \), there exists at least one \( x \in A \) such that \( x \leq y \). Hence, the smallest element \( \inf A \) must be less than or equal to the smallest element in \( B \), which is \( \inf B \). Therefore, \( \inf A \leq \inf B.\)

Key concepts

Supremum

The concept of a supremum, often referred to as the least upper bound (lub), is a crucial element in the study of ordered fields. An ordered field is a mathematical setting where the concepts of order, such as less than or greater than, can be applied to its elements. When we talk about subsets within this ordered field, the supremum relates to the idea of having an "upper limit" for these subsets.

For a subset \( S \) of an ordered field \( F \), the supremum is defined as the smallest element in \( F \) that is greater than or equal to every element of \( S \). This means, even though there can be many upper bounds for \( S \), the supremum is the smallest among them. It's the closest number that caps the set from above without being smaller than any element in \( S \).

• The supremum may or may not include elements from \( S \). It simply needs to be greater than or equal to them.
• If \( S \) has a maximum, then the maximum and the supremum are the same.
• If \( S \) is unbounded above, \( \sup S \) doesn't exist in \( F \).

Understanding the supremum is essential, especially when comparing different subsets within an ordered field using relations like those given in the original problem. By knowing that a supremum exists for a subset, you can make meaningful comparisons between different sets' boundaries.

Infimum

The infimum, also known as the greatest lower bound (glb), is another key concept when working with ordered fields and subsets. Just as the supremum gives the least boundary above, the infimum provides the greatest boundary below.

For any subset \( S \) within an ordered field \( F \), the infimum is defined as the largest element in \( F \) that is less than or equal to every element of \( S \). In simple terms, it serves as the most stringent lower cap for the elements of \( S \).

• The infimum, like the supremum, may not be a member of \( S \), but it must be less than or equal to each element in \( S \).
• If \( S \) has a minimum, it coincides with the infimum.
• When \( S \) is unbounded below, an infimum does not exist for \( S \) in \( F \).

In the context of proving inequalities between bounds of different subsets, the infimum is especially useful when examining how low one set can be relative to another. It helps in logically concluding whether one subset can be bounded by another subset's infimum.

Subsets

In the realm of mathematical fields, subsets play a pivotal role in defining structures and making comparisons. A subset is essentially a collection of elements all of which must belong to a larger set—in this case, an ordered field.

When we examine subsets within an ordered field, each subset is characterized by its elements' properties, such as having certain bounds, either upper or lower.

• Subsets can have relationships established between them based on the elements' order.
• An important aspect of subsets in an ordered field is whether they have bounds, like supremum and infimum, which help in comparative studies.
• Understanding the nature of subsets is crucial for proofs, particularly those involving relations or inequalities, as they often compare the bounds of different subsets to demonstrate certain properties.

Consider the original exercise: comparisons across subsets are made using universal and existential quantifiers ("for all" \( \forall \) and "there exists" \( \exists \)). This captures the essence of why subsets are indispensable, highlighting their use in structured logical arguments to reveal deeper relationships within the ordered field.