Problem 12

Question

Argue that if $S_{1}$ and $S_{2}$ are two sets of numbers and every number is in either $S_{1}$ or $S_{2}$ and every number in $S_{1}$ is less than every number in $S_{2}$ then it is not true that there are numbers $L_{1}$ and $L_{2}$ such that $L_{1}$ is the greatest number in $S_{1}$ and $L_{2}$ is the least number in $S_{2}$. Is this a contradiction to the Completeness Axiom?

Step-by-Step Solution

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Answer

No, it is not a contradiction to the Completeness Axiom since it is about the impossibility of distinct maximum and minimum given the set conditions.

1Step 1: Understand the Completeness Axiom

The Completeness Axiom states that every non-empty set of real numbers that is bounded above has a least upper bound (supremum), and every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum). This applies to sets of real numbers.

2Step 2: Analyze the Problem Statement

Given sets $S_{1}$ and $S_{2}$ such that every number is in either $S_{1}$ or $S_{2}$, and every number in $S_{1}$ is less than every number in $S_{2}$, we must argue that it is not possible for $L_1$ to be the greatest number of $S_1$ and $L_2$ to be the least number of $S_2$.

3Step 3: Establish That $S_1$ and $S_2$ Are Non-Empty Sets

If $S_1$ and $S_2$ cover all numbers, then each set must be non-empty. $S_1$ has at least one number which is an upper bound, and $S_2$ has at least one number which is a lower bound.

4Step 4: Consider the Absence of Maximum and Minimum

If $L_1$ were the greatest number in $S_1$ and $L_2$ the least in $S_2$, then $L_1 < L_2$. Therefore, there could be numbers between $L_1$ and $L_2$ that don't belong to $S_1$ or $S_2$, which contradicts the problem statement that every number belongs to one set.

5Step 5: Connect to the Completeness Axiom

The issue is not with the Completeness Axiom, but rather with an incorrect assumption—having discrete maximum and minimum $L_1$ and $L_2$ creates a gap not filled by $S_1$ or $S_2$. Completeness isn't violated, but the existence of distinct maximum and minimum is logically impossible under these conditions.

Key Concepts

SetsReal NumbersSupremum and InfimumGreatest Lower BoundLeast Upper Bound

Sets

In mathematics, a set is simply a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics, and they are used everywhere from basic algebra to advanced theoretical physics. A set can contain any type of object, including numbers, letters, symbols, and even other sets. The concept of a set allows us to group objects together based on specific criteria, making it easier to study their properties.
When considering sets of numbers like $S_1$ and $S_2$, they can represent collections of numbers that satisfy certain conditions. In this particular problem, sets $S_1$ and $S_2$ contain all numbers, where each number in $S_1$ is less than every number in $S_2$. This is an illustration of how sets can be used to represent intervals or divisions of the real number line.

Real Numbers

Real numbers include all the numbers on the number line, encompassing rational numbers (like fractions and whole numbers) and irrational numbers (like $\pi$ and $\sqrt{2}$). These numbers fill the entire continuum of the number line without any gaps.
Real numbers can be categorized into several types, such as integers, fractions, and more. They are useful in various areas of mathematics because they allow precise measurements and calculations.

Rational numbers: Can be expressed as a ratio of two integers.
Irrational numbers: Cannot be expressed as a simple fraction and have non-repeating decimals.

Understanding the properties of real numbers is essential, especially when discussing concepts like supremum and infimum, which pertain to the bounds within the set of real numbers.

Supremum and Infimum

The terms 'supremum' and 'infimum' are important concepts when discussing the bounds of a set. The supremum (or least upper bound) of a set is the smallest real number that is greater than or equal to every number in the set. Conversely, the infimum (or greatest lower bound) is the largest real number that is less than or equal to every number in the set.
These concepts are crucial when examining the properties of sets, as they allow us to determine the range within which all numbers in a set fall.

Supremum: $\sup(S)$, requires the set to be bounded above.
Infimum: $\inf(S)$, requires the set to be bounded below.

In the context of sets $S_1$ and $S_2$, determining the supremum and infimum helps in understanding why there can be no distinct greatest or least numbers under given conditions.

Greatest Lower Bound

The greatest lower bound, or infimum, is the highest number that is still less than or equal to all numbers in a set. It acts as a floor for the values within the set, ensuring that no number in the set is below it.
Finding the infimum of a set is particularly useful when dealing with scenarios where a direct minimum does not exist but we can approximate it from below. In mathematical notation, if $L$ is the infimum of set $S$, we write: $L = \inf(S)$.
The concept of the greatest lower bound ensures we have a very comprehensive method for defining the boundary of a set from below, especially in sets where a minimum might not be readily identifiable. This principle applies directly to $S_1$ and $S_2$ in determining their limits without expressly identifying greatest or least numbers.

Least Upper Bound

The least upper bound, or supremum, performs a similar role to the infimum but in reverse; it is the smallest number greater than or equal to all numbers in the set. It provides a ceiling for all the values within the set, ensuring no number in the set exceeds it.
In mathematical terms, if $U$ is the supremum of set $S$, it is denoted as $U = \sup(S)$. This concept is vital when a maximum might not be directly found, but an upper approximation can be derived from the above.
This concept becomes crucial when examining sets $S_1$ and $S_2$. The absence of specific maximum or minimum numbers within $S_1$ or $S_2$ leads us to rely on these abstract boundaries to define the extent of the sets.

Problem 11

Problem 13

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