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 complements of the set \(C\) with respect to the space \(\mathcal{C}\) for the parts (a), (b), and (c) of the exercise are respectively \(C^{c}=\{x: 0<x\leq\frac{5}{8}\}\), \(C^{c}=\left\{(x, y, z): x^{2}+y^{2}+z^{2}<1\right\}\), and \(C^{c}=\left\{(x, y): x^2+y^2\geq2 \text{ or } |x|+|y|=2\right\}\).

Step-by-step solution

Find the complement of set \(C\) in \(\mathcal{C}\) for part (a))

For part (a), we have the space \(\mathcal{C}=\{x: 0<x<1\}\) and the set \(C=\{x: \frac{5}{8}<x<1\}\). The elements in \(\mathcal{C}\) that are not in \(C\) are the ones for which \(0<x\leq\frac{5}{8}\). Therefore, the complement of \(C\) with respect to \(\mathcal{C}\) is \(C^{c}=\{x: 0<x\leq\frac{5}{8}\}\).

Find the complement of set \(C\) in \(\mathcal{C}\) for part (b))

In part (b), \(\mathcal{C}=\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1\right\}\) and \(C=\left\{(x, y, z): x^{2}+y^{2}+z^{2}=1\right\}\). Here, the elements in \(\mathcal{C}\) that are not in \(C\) are the ones that satisfy the inequality \(x^{2}+y^{2}+z^{2}<1\). Therefore, the complement of \(C\) in \(\mathcal{C}\) is \(C^{c}=\left\{(x, y, z): x^{2}+y^{2}+z^{2}<1\right\}\).

Find the complement of set \(C\) in \(\mathcal{C}\) for part (c))

For part (c), we have \(\mathcal{C}=\{(x, y):|x|+|y| \leq 2\}\) and \(C=\left\{(x, y):x^{2}+y^{2}<2\right\}\). The elements in \(\mathcal{C}\) that are not in \(C\) are the ones where either \(|x|+|y|=2\) or \(x^{2}+y^{2}\geq2\). Therefore, the complement of \(C\) in \(\mathcal{C}\) is \(C^{c}=\left\{(x, y): x^2+y^2\geq2 \text{ or } |x|+|y|=2\right\}\).

Key concepts

Mathematical Sets

At the heart of many mathematical disciplines lies the concept of a set. A set is a collection of distinct objects, considered as an object in its own right. Objects can be anything: numbers, symbols, points in space, etc. Typically, sets are denoted by capital letters and the objects within them are called members or elements.

To visualize a set, imagine a container holding certain items. Those items inside the container belong to the set, and anything outside the container does not. In mathematics, the notion of sets is used to formalize and structure the way objects relate to one another. For instance, the set \( \{x: 0
When we describe sets, we often do so by specifying a property that all members share. For example, \( C = \{x: \frac{5}{8}

Set Operations

Set operations are processes that can be applied to sets, allowing us to combine, relate, and compare them. The basic set operations include union, intersection, difference, and complement. Each operation is a way to construct a new set from one or more other sets.

For example, the union of two sets \( A \) and \( B \) includes all elements that are in \( A \) or in \( B \) or in both. The intersection of \( A \) and \( B \) includes only elements that are in both \( A \) and \( B \) simultaneously. The difference \( A - B \) contains elements that are in \( A \) but not in \( B \).

In contrast, the complement of a set \( A \) within a universal set \( U \) consists of all elements that are in \( U \) but not in \( A \). This operation is key when determining what lies outside the subset or how one context (the universal set) can restrict the scope of another (the subset). Understanding these operations is vital because they allow us to manipulate sets for analysis and problem-solving, like in our exercise.

Complement of a Set

The complement of a set \( A \) typically denoted \( A^{c} \), is one of the fundamental operations in set theory. It refers to all elements not in \( A \) but within a larger set that we consider as the universal set \( U \). The construction of a complement depends entirely on the context provided by the universal set; without \( U \) being specified, the complement of \( A \) would not be well-defined.

In the given exercise, for different scenarios (a, b, and c), we analyze the set \( C \) and find its complement with respect to the universal space \( \mathcal{C} \). The complement contains all points in the space that are not in the set \( C \), which for scenario (a) are real numbers between 0 and \( \frac{5}{8} \) inclusive. For scenario (b), it includes points within the sphere of radius 1. And for scenario (c), it involves points where the sum of their absolute values equals 2 or where the squared sum is greater than or equal to 2.

Identifying the complement is crucial for understanding which elements of the universal set are excluded from the subset in question. This concept is widely used in probability, logic, and across various mathematical disciplines.

Set Operations

Set operations are processes that can be applied to sets, allowing us to combine, relate, and compare them. The basic set operations include union, intersection, difference, and complement. Each operation is a way to construct a new set from one or more other sets.

For example, the union of two sets \( A \) and \( B \) includes all elements that are in \( A \) or in \( B \) or in both. The intersection of \( A \) and \( B \) includes only elements that are in both \( A \) and \( B \) simultaneously. The difference \( A - B \) contains elements that are in \( A \) but not in \( B \).

In contrast, the complement of a set \( A \) within a universal set \( U \) consists of all elements that are in \( U \) but not in \( A \). This operation is key when determining what lies outside the subset or how one context (the universal set) can restrict the scope of another (the subset). Understanding these operations is vital because they allow us to manipulate sets for analysis and problem-solving, like in our exercise.

Complement of a Set

The complement of a set \( A \) typically denoted \( A^{c} \), is one of the fundamental operations in set theory. It refers to all elements not in \( A \) but within a larger set that we consider as the universal set \( U \). The construction of a complement depends entirely on the context provided by the universal set; without \( U \) being specified, the complement of \( A \) would not be well-defined.

In the given exercise, for different scenarios (a, b, and c), we analyze the set \( C \) and find its complement with respect to the universal space \( \mathcal{C} \). The complement contains all points in the space that are not in the set \( C \), which for scenario (a) are real numbers between 0 and \( \frac{5}{8} \) inclusive. For scenario (b), it includes points within the sphere of radius 1. And for scenario (c), it involves points where the sum of their absolute values equals 2 or where the squared sum is greater than or equal to 2.

Identifying the complement is crucial for understanding which elements of the universal set are excluded from the subset in question. This concept is widely used in probability, logic, and across various mathematical disciplines.