Question

Find the complement \(C^{c}\) of the set \(C\) with respect to the space \(\mathcal{C}\) if: (a) \(\mathcal{C}=\\{x: 0

Short answer

The complement sets \(C^{c}\) for each case are: (a) \(C^{c}=\left(0, \frac{5}{8} \right]\), (b) \(C^{c}=\left\{(x, y, z): x^{2}+y^{2}+z^{2}<1\right\}\), (c) \(C^{c}=\left\{(x, y): x^{2}+y^{2} = 2, or |x| + |y| > 2\right\}\)

Step-by-step solution

Identifying complement sets

The first step is to comprehend what a complement set is. For a given set \(C\) and its respective space \(\mathcal{C}\), the complement sets here \(C^{c}\) are all the elements in the space \(\mathcal{C}\) that do not belong to the set \(C\). This understanding will help in determining the elements of the complement set.

Find the complement sets for each case

In case (a), \(C\) is the interval \(\left( \frac{5}{8}, 1 \right)\) so \(C^{c}\) is \(\left(0, \frac{5}{8} \right] \). \nIn case (b), \(C\) includes all points on the surface of a sphere with radius 1. Its complement \(C^{c}\) is all points inside (but not on) the sphere, defined as \(\left\{(x, y, z): x^{2}+y^{2}+z^{2}<1\right\}\). \nIn case (c), \(C\) consists of points (x, y) in a circle with radius \(sqrt(2)\). Its complement \(C^{c}\) are points in the diamond outside the circle and points on the boundary, defined as \(\left\{(x, y): x^{2}+y^{2} = 2, or |x| + |y| > 2\right\}\).

Analyzing the results

After finding out the complement sets in each case, it is noticed that they represent all the elements in the respective space \(\mathcal{C}\) that are not found in the original set \(C\). This aligns with the basic definition of a complement set.

Key concepts

Complement of a Set

In set theory, the complement of a set is a fundamental concept that aids in understanding what elements belong to 'everything else' besides the given set within a certain universe or space. Expressed mathematically as \( C^{c} \), the complement of a set \( C \) with respect to a space \( \mathcal{C} \) includes all elements in \( \mathcal{C} \) that are not found in \( C \).
This can be visualized as the space minus the elements of set \( C \). Essentially, you are identifying what you have when you exclude everything in \( C \).
For example, when you have a real number interval like \( C = \left( \frac{5}{8}, 1 \right) \), the complement in its space \( \mathcal{C} = \{ x : 0 < x < 1 \} \) becomes \( C^{c} = \left(0, \frac{5}{8}\right] \).
This means we select all the numbers in the range \((0, 1)\) except those in \( C \), which gives us numbers less than or equal to \( \frac{5}{8} \).

• Helps frame problems and solutions
• Makes understanding overall space easier
• Key for analyses in mathematical models

Mathematical Spaces

Mathematical spaces can be thought of as the 'playing field' for set operations, providing a universe in which sets and their complements reside. These spaces can take many forms, such as a simple number line or more complex multidimensional realms.
They define the limits within which our mathematical objects exist and operate. In the context of the earlier example, the space \( \mathcal{C} = \left\{x : 0 < x < 1 \right\} \) functions as the boundary for finding the complement of \( C \).
Mathematical spaces may include various geometric shapes or even infinite dimensions, often described by properties such as volume, length, or area.

• Provide structure for set theories
• Enable visualization of relationships
• Crucial for calculus and topology

This concept is crucial for designating where sets start and end, particularly when exploring set complements in scenarios like spheres, circles, or other geometric configurations.

Geometric Visualization of Sets

Visualizing sets geometrically allows for an intuitive comprehension of complex mathematical theories, enabling easier interpretation and solution of problems involving multi-dimensional concepts.
For instance, consider the set \( C \) as points forming the surface of a sphere and its complement \( C^{c} \) as the interior of the sphere. Such visualization turns abstract algebraic mappings into tangible shapes, making them easier to interpret.
The geometrical perspective is vital when dealing with exercises like finding the complement in spaces involving circles or spheres, helping to visually differentiate between the set and its complement.

• Translates abstract concepts to visual forms
• Assists in recognizing spatial relationships
• Facilitates problem-solving and design

By sketching or imagining such representations, students can better grasp the acting forces and interrelations between sets and spaces, ensuring a comprehensive understanding of the problem at hand.