91Ó°ÊÓ

In Exercises 61-68, calculate the number of distinct subsets and the number of distinct proper subsets for each set. \(\\{x \mid x \in \mathbf{N}\) and \(2 \leq x \leq 6\\}\)

In Exercises 69-82, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The set \(\\{1,2,3, \ldots, 1000\\}\) has \(2^{1000}\) proper subsets.

a. Let \(A=\\{\mathrm{c}\\}, B=\\{\mathrm{a}, \mathrm{b}\\}, C=\\{\mathrm{b}, \mathrm{d}\\}\), and \(U=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\}\). Find \(A \cup\left(B^{\prime} \cap C^{\prime}\right)\) and \(\left(A \cup B^{\prime}\right) \cap\left(A \cup C^{\prime}\right)\). b. Let \(A=\\{1,3,7,8\\}, B=\\{2,3,6,7\\}, C=\\{4,6,7,8\\}\), and \(U=\\{1,2,3, \ldots, 8\\}\). Find \(A \cup\left(B^{\prime} \cap C^{r}\right)\) and \(\left(A \cup B^{r}\right) \cap\left(A \cup C^{\prime}\right)\). c. Based on your results in parts (a) and (b), use inductive reasoning to write a conjecture that relates \(A \cup\left(B^{\prime} \cap C^{\prime}\right)\) and \(\left(A \cup B^{\prime}\right) \cap\left(A \cup C^{\prime}\right)\). d. Use deductive reasoning to determine whether your conjecture in part (c) is a theorem.

In Exercises 67-80, find the cardinal number for each set. \(B=\\{2,4,6, \ldots, 30\\}\)

In Exercises 67-80, find the cardinal number for each set. \(B=\\{1,3,5, \ldots, 21\\}\)

a. Let \(A=\\{3\\}, B=\\{1,2\\}, C=\\{2,4\\}\), and \(U=\\{1,2,3,4,5,6\\}\). Find \((A \cup B)^{\prime} \cap C\) and \(A^{\prime} \cap\left(B^{\prime} \cap C\right)\). b. Let \(A=\\{\mathrm{d}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\}, B=\\{\mathrm{a}, \mathrm{c}, \mathrm{f}, \mathrm{h}\\}, C=\\{\mathrm{c}, \mathrm{e}, \mathrm{g}, \mathrm{h}\\}\), and \(U=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \ldots, \mathrm{h}\\}\). Find \((A \cup B)^{\prime} \cap C\) and \(A^{\prime} \cap\left(B^{\prime} \cap C\right)\) c. Based on your results in parts (a) and (b), use inductive reasoning to write a conjecture that relates \((A \cup B)^{\prime} \cap C\) and \(A^{\prime} \cap\left(B^{\prime} \cap C\right)\). d. Use deductive reasoning to determine whether your conjecture in part (c) is a theorem.

In Exercises 67-80, find the cardinal number for each set. \(C=\\{x \mid x\) is a day of the week that begins with the letter A \(\\}\)

In Exercises 69-82, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\\{x \mid x \in \mathbf{N}\) and \(30

In Exercises 67-80, find the cardinal number for each set. \(C=\\{x \mid x\) is a month of the year that begins with the letter W \(\\}\)

In Exercises 69-82, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\\{x \mid x \in \mathbf{N}\) and \(20 \leq x \leq 60\\} \nsubseteq\\{x \mid x \in \mathbf{N}\) and \(20