Find the complement C^c of the set C with respect to the space 𝒞 if (a) 𝒞={x: 0<x<1}, C={x: (5)/

Name: Problem 2, Chapter 1: Introduction to Mathematical Statistics
Uploaded: 2021-07-10T04:51:49-08:00
Duration: 5 min 17 s
Channel: Ahmad Reda
Description: Find the complement C^c of the set C with respect to the space 𝒞 if (a) 𝒞={x: 0<x<1}, C={x: (5)/(8)<x<1}. (b) 𝒞={(x, y, z): x^2+y^2+z^2 ≤1}, C={(x, y, z): x^2+y^2+z^2=1}. (c) 𝒞={(x, y):|x|+|y| ≤2}, C={(x, y): x^2+y^2<2}.

step 1 In problem 2 we want to find the complement of the event C with respect to the space C. For part

Question

Find the complement $C^{c}$ of the set $C$ with respect to the space $\mathcal{C}$ if (a) $\mathcal{C}=\{x: 0<x<1\}, C=\left\{x: \frac{5}{8}<x<1\right\}$. (b) $\mathcal{C}=\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1\right\}, C=\left\{(x, y, z): x^{2}+y^{2}+z^{2}=1\right\}$. (c) $\mathcal{C}=\{(x, y):|x|+|y| \leq 2\}, C=\left\{(x, y): x^{2}+y^{2}<2\right\}$.

   Find the complement $C^{c}$ of the set $C$ with respect to the space $\mathcal{C}$ if
(a) $\mathcal{C}=\{x: 0<x<1\}, C=\left\{x: \frac{5}{8}<x<1\right\}$.
(b) $\mathcal{C}=\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1\right\}, C=\left\{(x, y, z): x^{2}+y^{2}+z^{2}=1\right\}$.
(c) $\mathcal{C}=\{(x, y):|x|+|y| \leq 2\}, C=\left\{(x, y): x^{2}+y^{2}<2\right\}$.

Introduction to Mathematical Statistics

Robert V. Hogg,… 8th Edition

Chapter 1, Problem 2

Instant Answer

Solved by Expert Ahmad Reda

07/10/2021

Step 1

The complement of $C$ with respect to $\mathcal{C}$ is the set of all elements in $\mathcal{C}$ that are not in $C$. This is the set of all $x$ such that $0<x\leq\frac{5}{8}$. Show more…

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Find the complement $C^{c}$ of the set $C$ with respect to the space $\mathcal{C}$ if (a) $\mathcal{C}=\{x: 0<x<1\}, C=\left\{x: \frac{5}{8}<x<1\right\}$. (b) $\mathcal{C}=\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1\right\}, C=\left\{(x, y, z): x^{2}+y^{2}+z^{2}=1\right\}$. (c) $\mathcal{C}=\{(x, y):|x|+|y| \leq 2\}, C=\left\{(x, y): x^{2}+y^{2}<2\right\}$.

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Key Concepts

Set Operations

Set operations such as union, intersection, and complement are essential tools in mathematics. These operations allow one to combine, compare, and contrast different sets, and they are fundamental to various fields of study including algebra, topology, and analysis.

Relative Complement

The relative complement of a set C with respect to a space ? is defined as the set of all elements in ? that are not in C. This concept is a cornerstone of set theory and helps in understanding how a subset relates to its larger space by focusing on what remains when the subset is removed from the universal set being considered.

Universal Set

In the context of complements, the universal set (here denoted as ?) is the complete collection of elements under consideration. It sets the boundary for taking complements, ensuring that the complement of a set C includes only those elements that are defined within the space ? and not beyond it.

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