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Chapter 2: Problem 94

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 30 elements, set \(B\) contains 18 elements, and 5 elements are common to sets \(A\) and \(B\). How many elements are in \(A \cup B\) ?

Short Answer

Expert verified
The set \(A \cup B\) contains 43 elements.

Step by step solution

01

Identify the given values

In this problem, we are given that set \(A\) contains 30 elements, set \(B\) contains 18 elements, and 5 elements are common to sets \(A\) and \(B\). We write these as \( |A| = 30 \), \( |B| = 18 \), and \( |A \cap B| = 5 \).
02

Apply the formula for the cardinal number of the union of two sets

The formula for the number of elements in the union of two sets is defined as \( |A \cup B| = |A| + |B| - |A \cap B| \). Substituting our given values into this formula, we get \( |A \cup B| = 30 + 18 - 5 \).
03

Compute the cardinal number of the union of A and B

After performing the calculations, we find that the set \(A \cup B\) contains \( 30 + 18 - 5 = 43 \) elements.

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Most popular questions from this chapter

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 &=\\{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

Use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible to meet. \(n(U)=42, n(A)=26, n(B)=22, n(C)=25\) \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=9\) \(n(A \cap B \cap C)=5\)

A survey of 120 college students was taken at registration. Of those surveyed, 75 students registered for a math course, 65 for an English course, and 40 for both math and English. Of those surveyed, a. How many registered only for a math course? b. How many registered only for an English course? c. How many registered for a math course or an English course? d. How many did not register for either a math course or an English course?

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