Problem 14 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose \(S_{2}\) is the set of numbers to which \(x\) belongs if and only if \(x\) is positive and \(x^{2}>2\) and \(S_{1}\) consists of all of the other numbers. 1\. Give an example of a number in \(S_{2}\). 2\. Give an example of a number in \(S_{1}\). 3\. Argue that every number in \(S_{1}\) is less than every number in \(S_{2}\). 4\. Which of the following two statements is true? (a) There is a number \(C\) which is the largest number in \(S_{1}\). (b) There is a number \(C\) which is the least number in \(S_{2}\). 5\. Identify the number \(C\) in the correct statement of the previous part.

Step-by-Step Solution

Answer

1 in S1, 2 in S2; every S1 number is less than every S2. Statement (b) is true, with C = \( \sqrt{2} \).

Step 1: Understanding the Set Definitions

Start by understanding the conditions given for each set. The set \( S_2 \) contains numbers that are positive and satisfy the condition \( x^2 > 2 \). The set \( S_1 \) consists of numbers that do not meet the criteria for \( S_2 \). This means \( S_1 \) contains numbers that are either not positive, or their squares are not greater than 2.

Step 2: Determine an Example for S_2

To find a number in \( S_2 \), we need a positive number where \( x^2 > 2 \). For example, 2 is a positive number, and \( 2^2 = 4 \), which is greater than 2. Thus, 2 belongs to \( S_2 \).

Step 3: Determine an Example for S_1

To identify a number in \( S_1 \), find a number that is either non-positive or where \( x^2 \leq 2 \). For example, 1 is a positive number, but \( 1^2 = 1 \), which is not greater than 2. Thus, 1 belongs to \( S_1 \).

Step 4: Comparison between S_1 and S_2

Every number in \( S_1 \) is either non-positive or, if positive, has a square less than or equal to 2. Every number in \( S_2 \) is positive with its square greater than 2. Therefore, every number in \( S_1 \) is less than every number in \( S_2 \).

Step 5: Evaluating the Statements

Consider the statements:(a) A largest number \(C\) in \( S_1 \): Since \( S_1 \) includes all numbers \( x \leq \sqrt{2} \) or non-positive numbers, it does not have a maximum since \( \sqrt{2} \) is not included.(b) A least number \( C \) in \( S_2 \): The smallest number satisfying \( x^2 > 2 \) and \( x > 0 \) is \( x = \sqrt{2} \), thus it is the smallest element in \( S_2 \). Statement (b) is true.

Step 6: Identifying the Number C

From the correct statement in the previous step, identify the number \( C \) as \( \sqrt{2} \), which is the least number in \( S_2 \).

Key Concepts

Set Theory in Mathematics

Set theory is a branch of mathematical logic that deals with collections of objects, known as sets. In this exercise, we are dealing with two sets, denoted as \( S_1 \) and \( S_2 \). Each set is defined based on specific conditions.

• \( S_2 \) includes numbers that are positive and satisfy the condition \( x^2 > 2 \). This means any number in this set must be greater than the square root of 2.
• \( S_1 \), on the other hand, contains all other numbers. These can be negative numbers, zero, or positive numbers where the square of the number is less than or equal to 2.

Set theory helps to simplify problems by categorizing elements based on properties. By understanding how numbers fall into \( S_1 \) or \( S_2 \), it becomes easier to analyze and solve inequalities in calculus.

This kind of arrangement can help determine the relationships between different elements, such as identifying the smallest or largest possible values and comparing elements within and outside these sets.

Positive Numbers

Positive numbers are numbers greater than zero. In our exercise, we need to understand how positivity interacts with other mathematical conditions, like when finding examples in set \( S_2 \).

• A number is considered positive if it lies to the right of zero on the number line.
• Positive numbers in \( S_2 \) must also satisfy that \( x^2 > 2 \), making them greater than \( \sqrt{2} \).
• For \( S_1 \), if a number is positive, it leads to a condition where \( x^2 \leq 2 \).

Negative numbers or zero automatically belong to \( S_1 \), as they do not satisfy the positivity requirement of \( S_2 \).

Understanding the nature of positive numbers is crucial when dealing with inequality problems because it simplifies the comparison of values based on their magnitudes and signs.

Square Root Properties

The square root function is pivotal in this exercise, particularly for understanding the boundary values that dictate membership in \( S_2 \). Let's dive into a few key properties of square roots that are especially relevant:

• The square root of a number \( x \), denoted \( \sqrt{x} \), is a value that, when squared, gives \( x \).
• For non-negative numbers, the square root is always positive. This is vital in understanding constraints like \( x^2 > 2 \).

For instance, because \( x^2 > 2 \) implies \( x > \sqrt{2} \), the smallest positive number whose square is greater than 2 is \( \sqrt{2} \).

This relationship helps in determining boundary points between different sets—\( \sqrt{2} \) is the least element in \( S_2 \), and numbers less than or equal to \( \sqrt{2} \) belong to \( S_1 \).

Understanding these properties allows us to classify numbers accurately and is essential for tackling problems involving inequalities.