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Chapter 2: Problem 147

Describe what is meant by the complement of a set.

Short Answer

Expert verified
The complement of a set A, denoted as \(A'\) or \(A^c\), is the set of all elements in the universal set U that are not in set A.

Step by step solution

01

- Define Complement of a Set

The complement of a set, denoted as \(A'\) or \(A^c\), refers to the set of all elements in the universal set that are not in the given set. If the universal set is denoted as \(U\) and the set in question is represented as \(A\), then the complement of \(A\) in \(U\) is given by elements that are in \(U\) but not in \(A\).
02

- Illustrate with an Example

For example, let \(U = \{1, 2, 3, 4, 5, 6\}\) be the universal set and \(A = \{1, 2, 3\}\) be a subset of \(U\). The complement of \(A\) in \(U\), denoted as \(A'\) or \(A^c\), would then be \(\{4, 5, 6\}\): the set of all elements in \(U\) that are not in \(A\).

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Most popular questions from this chapter

Let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((A \cup B) \cap(A \cup C)\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A \cap B\)

Describe the Venn diagram for proper subsets. How does this diagram illustrate that the elements of one set are also in the second set?

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cup C^{\prime}\)

Let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((B \cup C)^{\prime} \cap A\)
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