Chapter 1: Problem 17
Suppose that \(|A|=m\) and \(|B|=n .\) Find the following cardinalities. $$ |\\{X \in \mathscr{P}(A):|X| \leq 1\\}| $$
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Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, y): x, y \in \mathbb{R}, x>1\\} $$
Write each of the following sets in set-builder notation. $$ \left\\{\ldots,-\pi,-\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}, 2 \pi, \frac{5 \pi}{2}, \ldots\right\\} $$
Draw Venn diagrams for \(A \cap(B \cup C)\) and \((A \cap B) \cup(A \cap C) .\) Based on your drawings, do you think \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C) ?\)
Draw a Venn diagram for \((A \cap B)-C\).
Sketch the sets \(X=\left\\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leq 1\right\\}\) and \(Y=\left\\{(x, y) \in \mathbb{R}^{2}:-1 \leq y \leq 0\right\\}\) on \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(X \cup Y, X \cap Y, X-Y\) and \(Y-X\).
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