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Chapter 2: Problem 145

Describe the Venn diagram for two equal sets. How does this diagram illustrate that the sets are equal?

Short Answer

Expert verified
A Venn diagram of two equal sets is visually depicted as two completely overlapping circles, with no unique sections. This visually confirms that the two sets contain exactly the same elements, hence they are equal.

Step by step solution

01

Define equal sets

Two sets are considered to be equal if they contain exactly the same elements and each element of a set is an element of the other.
02

Drawing a Venn Diagram for two equal sets

A Venn diagram for two equal sets consists of two overlapping circles within a rectangle. The rectangle represents the universal set, and each circle represents a set. Since the sets are equal, every element in one set is also in the other set. Thus, the two circles overlap completely, forming a single area that represents both equal sets.
03

Understanding the implication of the Venn diagram

The single shared area in the Venn diagram demonstrates that the two sets are equal. Each element of one is an element of the other. There are no elements in one set that are not in the other set, so there are no unique areas in each circle. The diagram visually proves the equivalence of the two sets.

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Most popular questions from this chapter

Describe what is meant by the intersection of two sets. Give an example.

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. \(B \cap C\)

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ 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. \(A^{\prime} \cup B^{\prime}\)

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