Problem 12 - A Primer of Real Analysis

Question

Let \(A=\left\\{a: a \in \mathbb{Q}, a^{3}<2\right\\}\). 1\. Show that if \(a \in A\) and \(ba\), then \(b \notin A\).

Step-by-Step Solution

Answer

1. True, 2. True.

Step 1: Understanding Set A

The set \( A \) consists of all rational numbers \( a \) such that \( a^3 < 2 \). This means \( a \) must be a rational number for which if you cube it, the result is less than 2.

Step 1: Given \( a \in A \), proving \( b \in A \) for \( b < a \)

Assume \( a \in A \), meaning \( a^3 < 2 \). For any rational number \( b < a \), since the function \( f(x) = x^3 \) is strictly increasing over \( \mathbb{Q} \), it follows that \( b^3 < a^3 \). Therefore, \( b^3 < 2 \) and hence \( b \in A \).

Step 2: Given \( a \notin A \), proving \( b \notin A \) for \( b > a \)

Assume \( a otin A \). This implies \( a^3 \geq 2 \). For any rational number \( b > a \), since the cube function is strictly increasing, \( b^3 > a^3 \). Since \( a^3 \geq 2 \), it follows that \( b^3 > 2 \), and hence \( b otin A \).

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers. In simpler terms, they are numbers that can be written in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers, and \( q \) is not zero. This also includes numbers like 7, which can be written as \( \frac{7}{1} \).

Key characteristics of rational numbers:

• They have a finite or repeating decimal representation.
• They are dense on the number line, meaning between any two rational numbers, there exists another rational number.
• Important examples include fractions like \( \frac{1}{2}, \frac{3}{4} \), and integers since any integer \( n \) can be expressed as \( \frac{n}{1} \).

Rational numbers are important in real analysis as they define concepts like limits, continuity, and density which help in exploring the properties of real numbers. When examining functions or sets like our set \( A \), it's crucial to understand how these numbers behave under operations like addition, subtraction, multiplication, and especially functions like exponentiation (cubing in this case).

Inequality

Inequality in mathematics deals with the relative size or order of values. It is used to show that one value is smaller or larger than another, represented by symbols like \(<, >, \leq, \geq\). For example, \( a < b \) means that \( a \) is less than \( b \), while \( a \geq b \) indicates that \( a \) is greater than or equal to \( b \).

Understanding inequalities is pivotal when dealing with expressions like \( a^3 < 2 \) or \( a^3 \geq 2 \), as seen in set \( A \).

• Inequalities help us identify bounds in sets or functions.
• They allow us to make comparisons and decide whether elements belong to certain sets based on conditions.
• Handling inequalities requires understanding how operations like squaring or cubing affect the values; i.e., the cube of a larger rational number is more than that of a smaller one.

When working with sets and functions, manipulating inequalities is often necessary to prove properties like inclusion or exclusion from a set, as seen in this exercise.

Strictly Increasing Functions

A function is said to be strictly increasing when it preserves the order of inputs in its outputs. This means if you have two numbers \( x_1 \) and \( x_2 \) where \( x_1 < x_2 \), then a strictly increasing function \( f \) ensures \( f(x_1) < f(x_2) \).

The function \( f(x) = x^3 \) is strictly increasing for rational numbers because the order of any two numbers, say \( x \) and \( y \), such that \( x < y \), will maintain the order in their outputs: \( x^3 < y^3 \).

• Strictly increasing functions maintain order, crucial in proofs involving inequalities.
• They help in determining property propagation; for example, in descending and ascending regions of graphs.
• This property can be used to show that within a certain range, larger inputs lead to larger outputs, as demonstrated in the solution involving set \( A \).

Understanding strictly increasing functions allows precise reasoning about how changes in input affect output, which plays a vital role in proving characteristics like whether a number is in a set based on its cube.