Problem 13 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Let \(S_{2}\) denote the points of the \(X\) -axis that have positive \(x\) -coordinate and \(S_{1}\) denote the points of the \(X\) -axis that do not belong to \(S_{2} .\) Does \(S_{2}\) have a left most point?

Step-by-Step Solution

Answer

No, S2 does not have a leftmost point because it includes all positive real numbers without a minimum.

Step 1: Define Sets S1 and S2

The set \( S_2 \) is defined as points on the X-axis with positive x-coordinate. This means \( S_2 = \{ x \in \mathbb{R}^+ | x > 0 \} \). The set \( S_1 \) is defined as points on the X-axis that do not belong to \( S_2 \), which means \( S_1 = \{ x \in \mathbb{R} | x \leq 0 \} \).

Step 2: Analyze S2 for a Left Most Point

To have a left most point, there must exist a specific smallest element in the set \( S_2 \). However, the set \( S_2 \) includes all positive real numbers, and there is no smallest positive real number; given any small positive number, you can always find a smaller positive number closer to zero.

Step 3: Conclusion Understanding the Property of S2

Since there is no smallest element for the set \( S_2 \), \( S_2 \) does not have a leftmost point. The absence of a smallest positive value among real numbers defines that the set extends continuously towards zero without reaching it.

Key Concepts

Real Numbers

In mathematics, real numbers are incredibly significant and form the cornerstone of many mathematical concepts. Essentially, real numbers include all the numbers that can be found on the number line. This encompasses both rational numbers (like 3 or 0.75) which can be expressed as fractions, and irrational numbers (like \( \sqrt{2} \) or \( \pi \)) that cannot be expressed as a simple fraction.

To illustrate, consider the number line. It stretches infinitely in both directions and each point on this line corresponds to a real number. This includes negative numbers, zero, and positive numbers.

Real numbers possess certain properties:

• Density: Between any two real numbers, there exists another real number.
• Completeness: No holes exist on the real number line. Every point corresponds to a real number.
• Limitless: The set of real numbers is infinite; not countable like integers or natural numbers.

In the context of set \( S_2 \) from the exercise, we are specifically looking at positive real numbers. This set is infinite and continuous, meaning it has no smallest element as closer inspection always reveals a smaller number than the given one.

Sets in Mathematics

Sets are a basic concept in mathematics which refers to a collection of distinct objects, considered as an object in their own right. The objects in a set can be anything: numbers, characters, or even other sets. Sets are usually defined by listing the elements or by stating a property that the elements must satisfy.

Let's look at the example from the exercise:

• Set \( S_2 \): Contains all points on the X-axis with a positive \( x \)-coordinate. Mathematically, this is \( S_2 = \{ x \in \mathbb{R}^+ | x > 0 \} \).
• Set \( S_1 \): Consists of all points on the X-axis not in \( S_2 \), meaning \( S_1 = \{ x \in \mathbb{R} | x \leq 0 \} \).

Sets have noteworthy properties like unions, intersections, and complements.

• Unions: Combining distinct sets to form a new set.
• Intersections: Allows common elements of sets to form a new set.
• Complements: Result from excluding a particular set's elements from a universal set.

Understanding these properties helps in analyzing and manipulating sets to address various mathematical problems.

Properties of the X-axis

The X-axis is a fundamental component of the Cartesian coordinate system. It's the line where the value of \( y \) is zero, and it's used primarily to determine the position of a point on a two-dimensional plane.

There are several key properties and aspects of the X-axis to remember:

• Direction: It runs horizontally from left to right.
• Origin: The X-axis intersects with the Y-axis at the origin, which is the point (0,0).
• Symmetry: If a point (x,y) lies on the X-axis, then y = 0. Therefore, points below or above this axis are equidistant from it.

In the exercise, sets \( S_1 \) and \( S_2 \) are defined based on their position relative to this axis. Positivity on the X-axis signifies being to the right of the origin, while non-positive numbers lie to the left or directly on the origin. These properties are essential to understanding how points are positioned and categorized within a coordinate plane, aiding in the visualization and solving of many geometrical and algebraic problems.