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Chapter 2: Problem 143

Describe the Venn diagram for two disjoint sets. How does this diagram illustrate that the sets have no common elements?

Short Answer

Expert verified
A Venn diagram represents disjoint sets as two non-overlapping circles within a universal set. This absence of overlap demonstrates that the sets have no elements in common.

Step by step solution

01

Understand the concept of disjoint sets

Disjoint sets are those sets that do not possess any common elements. In other words, the intersection of disjoint sets is an empty set. This can be represented as \( A \cap B = \emptyset \) where A and B are two disjoint sets.
02

Draw the Venn diagram for disjoint sets

To represent disjoint sets in a Venn Diagram, draw two circles within a box. The box represents the universal set, and each circle represents one set. The circles should not overlap, as disjoint sets have no elements in common. Hence, each circle contains unique elements pertaining only to their set.
03

Illustrate lack of common elements

As there is no overlap between the two circles in the Venn diagram, it shows that there are no common elements between the sets, which is the characteristic feature of disjoint sets. This lack of overlap visually illustrates the lack of common elements.

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Most popular questions from this chapter

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. \((A \cap B \cap C)^{\prime}\)

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. \(A \cup \varnothing\)

Construct a Venn diagram illustrating the given sets. \(\begin{aligned} A &=\left\\{x_{3}, x_{9}\right\\} \\ B &=\left\\{x_{1}, x_{2}, x_{3}, x_{5}, x_{6}\right\\} \\ C &=\left\\{x_{3}, x_{4}, x_{5}, x_{6}, x_{9}\right\\} \\ U &=\left\\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}\right\\} \end{aligned}\)

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. Even if I'm not sure how mathematicians define irrational and complex numbers, telling me how these sets are related, I can construct a Venn diagram illustrating their relationship.

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. \(C^{\prime} \cap\left(A \cup B^{\prime}\right)\)
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