91Ó°ÊÓ

Write the following sets by listing their elements between braces. $$ \mathscr{P}(\\{\\{a, b\\},\\{c\\}\\}) $$

Write the following sets by listing their elements between braces. $$ \mathscr{P}(\\{1,2,3,4\\}) $$

Sketch the set \(X=[1,3] \times[1,2]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(\bar{X}\) and \(\bar{X} \cap([0,2] \times[0,3])\)

Suppose \(A=\\{0,1\\}\) and \(B=\\{1,2\\} .\) Find: (a) \((A \times B) \cap(B \times B)\) (b) \((A \times B) \cup(B \times B)\) (c) \((A \times B)-(B \times B)\) (d) \((A \cap B) \times A\) (e) \((A \times B) \cap B\) (f) \(\mathscr{P}(A) \cap \mathscr{P}(B)\) \((\mathbf{g}) \mathscr{P}(A)-\mathscr{P}(B)\) (h) \(\mathscr{P}(A \cap B)\) (i) \(\mathscr{P}(A \times B)\)

Write each of the following sets by listing their elements between braces. $$ \\{x \in \mathbb{Z}:-2 \leq x<7\\} $$

Draw Venn diagrams for \(A \cup(B \cap C)\) and \((A \cup B) \cap(A \cup C)\). Based on your drawings, do you think \(A \cup(B \cap C)=(A \cup B) \cap(A \cup C) ?\)

Sketch the sets \(X=[1,3] \times[1,3]\) and \(Y=[2,4] \times[2,4]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(X \cup Y, X \cap Y, X-Y\) and \(Y-X .\) (Hint: \(X\) and \(Y\) are Cartesian products of intervals. You may wish to review how you drew sets like \([1,3] \times[1,3]\) in the exercises for Section 1.2.)

Draw Venn diagrams for \(A \cap(B \cup C)\) and \((A \cap B) \cup(A \cap C) .\) Based on your drawings, do you think \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C) ?\)

List all the subsets of the following sets. $$ \\{\mathbb{R},\\{\mathbb{Q}, \mathbb{N}\\}\\} $$

Suppose sets \(A\) and \(B\) are in a universal set \(U .\) Draw Venn diagrams for \(\overline{A \cup B}\) and \(\bar{A} \cap \bar{B}\). Based on your drawings, do you think it's true that \(\overline{A \cup B}=\bar{A} \cap \bar{B}\) ?