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Chapter 2: Problem 61
In Exercises 61-68, calculate the number of distinct subsets and the number of distinct proper subsets for each set. \(\\{2,4,6,8\\}\)
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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. \(A \cap(B \cup C)\)
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. \(n(U)=42, n(A)=26, n(B)=22, n(C)=25\) \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=9\) \(n(A \cap B \cap C)=5\)
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. \((A \cap B) \cup(A \cap C)\)
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. \(B \cap U\)
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
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