### Mathematical Concepts: Supremum and InfimumIn this section, we will discuss the concepts of supremum and infimum within the context of subsets of rational numbers.**Statement:**"A subset of ℚ that contains its supremum but not its infimum."This statement introduces a particular subset of rational numbers (ℚ), which has a few significant properties:1. The subset includes its supremum (the least upper bound).2. The subset does not include its infimum (the greatest lower bound).Let's elaborate on what these terms mean and how they can manifest in a subset.#### Definitions:- **Supremum (sup):** The supremum of a subset $ S $ of ℚ is the smallest rational number that is greater than or equal to every number in $ S $. It is denoted as $ \sup(S) $.- **Infimum (inf):** The infimum of $ S $ is the largest rational number that is less than or equal to every number in $ S $. It is denoted as $ \inf(S) $.### Visual Aid: Understanding Supremum and InfimumAssume we have a subset $ S $ of $ ℚ $ defined as follows:\[ S = \{ x \in ℚ : x < 1 \} \cup \{ 2 \} \]- The **supremum** of $ S $ is 2, and $ S $ contains 2.- The **infimum** of $ S $ would be the least rational number smaller than all elements of $ S $. Since $ S $ extends without bound downwards but never includes its infimum, besides no definite value being the exact least upper bound.#### Graphical Representation:Imagine a number line, where:- The elements of $ S $ are marked.- The number 2 is included ("solid marker") indicating that 2 is in $ S $.- Numbers approaching negative or very small values are members of $ S $, but it never reaches a minimum boundary.This concept is fundamental in advanced mathematics, particularly in the study of real analysis and topology. Understanding how these subsets work within rational numbers helps in exploring more complex mathematical terrains.**Formatting Tools Explored:**The text also includes a screenshot from a Word Processor interface, demonstrating various formatting options:- Text size (12pt)- Paragraph settings- Bold

Supremum of Set :- Let S be set of real number which is bounded above. Then M ∈ R is said to be…

Answered: A subset of Q that contains its…

Math

Advanced Math

A subset of Q that contains its supremum but not its infimum.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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### Mathematical Concepts: Supremum and Infimum

In this section, we will discuss the concepts of supremum and infimum within the context of subsets of rational numbers.

**Statement:**
"A subset of ℚ that contains its supremum but not its infimum."

This statement introduces a particular subset of rational numbers (ℚ), which has a few significant properties:
1. The subset includes its supremum (the least upper bound).
2. The subset does not include its infimum (the greatest lower bound).

Let's elaborate on what these terms mean and how they can manifest in a subset.

#### Definitions:
- **Supremum (sup):** The supremum of a subset $ S $ of ℚ is the smallest rational number that is greater than or equal to every number in $ S $. It is denoted as $ \sup(S) $.
- **Infimum (inf):** The infimum of $ S $ is the largest rational number that is less than or equal to every number in $ S $. It is denoted as $ \inf(S) $.

### Visual Aid: Understanding Supremum and Infimum

Assume we have a subset $ S $ of $ ℚ $ defined as follows:
\[ S = \{ x \in ℚ : x < 1 \} \cup \{ 2 \} \]

- The **supremum** of $ S $ is 2, and $ S $ contains 2.
- The **infimum** of $ S $ would be the least rational number smaller than all elements of $ S $. Since $ S $ extends without bound downwards but never includes its infimum, besides no definite value being the exact least upper bound.

#### Graphical Representation:
Imagine a number line, where:

- The elements of $ S $ are marked.
- The number 2 is included ("solid marker") indicating that 2 is in $ S $.
- Numbers approaching negative or very small values are members of $ S $, but it never reaches a minimum boundary.

This concept is fundamental in advanced mathematics, particularly in the study of real analysis and topology. Understanding how these subsets work within rational numbers helps in exploring more complex mathematical terrains.

**Formatting Tools Explored:**

The text also includes a screenshot from a Word Processor interface, demonstrating various formatting options:
- Text size (12pt)
- Paragraph settings
- Bold

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