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Chapter 2: Problem 144

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

Short Answer

Expert verified
A Venn diagram of proper subsets presents one circle (set A) entirely contained inside another circle (set B). This shows that all elements of set A are also in set B, making A a proper subset of B. The additional elements that set B has and set A does not visibly illustrate the 'proper' part of the proper subset.

Step by step solution

01

Understand Proper Subset

A proper subset is a set in which all the elements of that set are also elements of another set. Furthermore, the second set contains at least one element not found in the first set. For example, if set A = {1, 2} and set B = {1, 2, 3}, then set A is a proper subset of B, because set B contains all elements of set A plus an additional element, 3.
02

Draw the Venn Diagram

Draw two circles, one inside of the other. Label the inner circle A and the outer circle B. Write the elements of set A inside the circle A. Write the unique elements of set B (those not in set A) on the rest of the area in circle B.
03

Interpret the Venn Diagram

The Venn diagram now visually shows the two sets and their relationship to each other. Set A is completely contained within set B, demonstrating that all elements in set A are also in set B. However, there are also elements in set B not found in Set A, indicating that Set A is a proper subset of set B.

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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. \(A \cap U\)

Let $$ \begin{aligned} U &=\\{x \mid x \in \mathbf{N} \text { and } x<9\\} \\ A &=\\{x \mid x \text { is an odd natural number and } x<9\\} \\ B &=\\{x \mid x \text { is an even natural number and } x<9\\} \\ C &=\\{x \mid x \in \mathbf{N} \text { and } 1

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

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

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. \(B \cap U\)
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