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Section 1

Sets

In Exercises $1-10,$ write true or false for each statement. $$3 \in\{2,5,7,9,10\}$$

In Exercises $1-10,$ write true or false for each statement. $$6 \in\{-2,6,9,5\}$$

In Exercises $1-10,$ write true or false for each statement. $$9 \notin\{2,1,5,8\}$$

In Exercises $1-10,$ write true or false for each statement. $$3 \notin\{7,6,5,4\}$$

In Exercises $1-10,$ write true or false for each statement. $$\{2,5,8,9\}=\{2,5,9,8\}$$

In Exercises $1-10,$ write true or false for each statement. $$\{3,7,12,14\}=\{3,7,12,14,0\}$$

In Exercises $1-10,$ write true or false for each statement. $\{\text { all whole numbers greater than } 7 \text { and less than } 10\}=\{8,9\}$

In Exercises $1-10,$ write true or false for each statement. $\{x | x \text { is an odd integer; } 6 \leq x \leq 18\}=\{7,9,11,15,17\}$

In Exercises $1-10,$ write true or false for each statement. $$0 \in \emptyset$$

In Exercises $1-10,$ write true or false for each statement. $$\emptyset \in\{\emptyset\}$$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $$A \multimap U$$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $$E \multimap A$$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $A \multimap E$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $B \multimap C$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $\emptyset \multimap A$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $\{0,2\} \multimap D$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $D \multimap B$

Let $A=\{2,4,6,10,12\}, B=\{2,4,8,10\}, C=\{4,8,12\},$ $D=\{2,10\}, E=\{6\},$ and $U=\{2,4,6,8,10,12,14\} .$ Insert $\subseteq$ or $\underline{\nsubseteq}$ to make the statement true. $A \multimap C$

Repeat Exercises $11-18$ except insert $\subset$ or $\not$ to make the statement true.

What is set-builder notation? Give an example.

Insert a number in each blank to make the statement true, using the sets for Exercises $11-18 .$ There are exactly $-$ subsets of $A$

Insert a number in each blank to make the statement true, using the sets for Exercises $11-18 .$ There are exactly $-$ subsets of $B$ .

Insert a number in each blank to make the statement true, using the sets for Exercises $11-18 .$ There are exactly $-$ subsets of $C$

Insert a number in each blank to make the statement true, using the sets for Exercises $11-18 .$ There are exactly $-$ subsets of $D$

Insert $\cap$ or $\cup$ to make each statement true. $$\{5,7,9,19\}-\{7,9,11,15\}=\{7,9\}$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{8,11,15\}-\{8,11,19,20\}=\{8,11\}$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{2,1,7\}-\{1,5,9\}=\{1,2,5,7,9\}$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{6,12,14,16\}-\{6,14,19\}=\{6,12,14,16,19\}$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{3,5,9,10\}-\emptyset=\emptyset$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{3,5,9,10\}-\emptyset=\{3,5,9,10\}$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{1,2,4\}-\{1,2,4\}=\{1,2,4\}$$

Insert $\cap$ or $\cup$ to make each statement true. $$\{0,10\}-\{10,0\}=\{0,10\}$$

Describe the intersection and union of sets. How do they differ?

Is it possible for two nonempty sets to have the same intersection and union? If so, give an example.

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $X \cap Y$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $X \cup Y$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $X^{\prime}$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $Y^{r}$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $X^{\prime} \cap Y^{\prime}$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $X^{\prime} \cap Z$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $\ln (X \cup Z)$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $X^{\prime} \cap\left(Y^{\prime} \cup Z\right)$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $\left(X \cap Y^{\prime}\right) \cup\left(Z^{\prime} \cap Y^{\prime}\right)$

Let $U=\{1,2,3,4,5,6,7,8,9 |, X=\{2,4,6,8\}, Y=\{2,3,4,5,6\}$ and $Z=\{1,2,3,8,9\} .$ List the members of each set, using set braces. $(X \cap Y) \cup\left(X^{\prime} \cap Z\right)$

In Example $6,$ what set do you get when you calculate $(A \cap B) \cup\left(A \cap B^{\prime}\right) ?$

Explain in words why $(A \cap B) \cup\left(A \cap B^{\prime}\right)=A$

Let $U=$ lall students in this schooll, $M=$ lall students taking this coursel, $N=\{\text { all students taking accountingl, and } P=\{\text { all }$ students taking zoology. Describe each set in words. $M^{\prime}$

Let $U=$ lall students in this schooll, $M=$ lall students taking this coursel, $N=\{\text { all students taking accountingl, and } P=\{\text { all }$ students taking zoology. Describe each set in words. $M \cup N$

Let $U=$ lall students in this schooll, $M=$ lall students taking this coursel, $N=\{\text { all students taking accountingl, and } P=\{\text { all }$ students taking zoology. Describe each set in words. $N \cap P$

Let $U=$ lall students in this schooll, $M=$ lall students taking this coursel, $N=\{\text { all students taking accountingl, and } P=\{\text { all }$ students taking zoology. Describe each set in words. $N^{\prime} \cap P^{\prime}$

Refer to the sets listed for Exercises $11-18 .$ Which pairs of sets are disjoint?

Refer to the sets listed for Exercises $35-44 .$ Which pairs are disjoint?

Refer to Example 8 in the text. Describe each set in Exercises $53-56$ in words; then list the elements of each set. $B^{\prime}$

Refer to Example 8 in the text. Describe each set in Exercises $53-56$ in words; then list the elements of each set. $A \cap B$

Refer to Example 8 in the text. Describe each set in Exercises $53-56$ in words; then list the elements of each set. $(A \cap B)^{\prime}$

Refer to Example 8 in the text. Describe each set in Exercises $53-56$ in words; then list the elements of each set. $(C \cup D)^{\prime}$

Let $A=\{1,2,3,\{3\},\{1,4,7\}\} .$ Answer each of the following as true or false. $$\begin{array}{ll}{\text { a. } 1 \in A} & {\text { b. }\{3\} \in A \qquad \text { c. }\{2\} \in A \qquad \text { d. } 4 \in A} \\ {\text { e. }\{\{3\}\} \subset A} & {\text { f. }\{1,4,7\} \in A \qquad \text { g. }\{1,4,7\} \subseteq A}\end{array}$$

Let $B=\{a, b, c,\{d\},\{e, f\}\} .$ Answer each of the following as true or false. $$\begin{array}{ll}{\text { a. } a \in B} & {\text { b. }\{b, c, d\} \subset B \qquad \text { c. }\{d\} \in B} \\ {\text { d. }\{d\} \subseteq B} & {\text { e. }\{e, f\} \in B \qquad \text { f. }\{a,\{e, f\}\} \subset B} \\ {\text { g. }\{e, f\} \subset B}\end{array}$$

Let $U$ be the smallest possible set that includes all the corporations listed, and $V, F, J,$ and $T$ be the set of top holdings for each mutual fund, respectively. Find each set:$V \cap J$

Let $U$ be the smallest possible set that includes all the corporations listed, and $V, F, J,$ and $T$ be the set of top holdings for each mutual fund, respectively. Find each set:$V \cap(F \cup T)$

Let $U$ be the smallest possible set that includes all the corporations listed, and $V, F, J,$ and $T$ be the set of top holdings for each mutual fund, respectively. Find each set:$(J \cup F)^{\prime}$

Let $U$ be the smallest possible set that includes all the corporations listed, and $V, F, J,$ and $T$ be the set of top holdings for each mutual fund, respectively. Find each set:$J^{\prime} \cap T^{\prime}$

Sales Calls Suppose that Kendra Gallegos has appointments with 9 potential customers. Kendra will be ecstatic if all 9 of these potential customers decide to make a purchase from her. Of course, in sales there are no guarantees. How many different sets of customers may place an order with Kendra? (Hint: Each set of customers is a subset of the original set of 9 customers.)

Let $U$ be the smallest possible set that includes all the symptoms listed, $N$ be the set of symptoms for an underactive thyroid, and 0 be the set of symptoms for an overactive thyroid. Find each set. $O^{\prime}$

Let $U$ be the smallest possible set that includes all the symptoms listed, $N$ be the set of symptoms for an underactive thyroid, and 0 be the set of symptoms for an overactive thyroid. Find each set. $N^{\prime}$

Let $U$ be the smallest possible set that includes all the symptoms listed, $N$ be the set of symptoms for an underactive thyroid, and 0 be the set of symptoms for an overactive thyroid. Find each set. $N \cap O$

Let $U$ be the smallest possible set that includes all the symptoms listed, $N$ be the set of symptoms for an underactive thyroid, and 0 be the set of symptoms for an overactive thyroid. Find each set. $N \cup O$

Let $U$ be the smallest possible set that includes all the symptoms listed, $N$ be the set of symptoms for an underactive thyroid, and 0 be the set of symptoms for an overactive thyroid. Find each set. $N \cap O^{\prime}$

APPLY IT Electoral College U.S. presidential elections are decided by the Electoral College, in which each of the 50 states, plus the District of Columbia, gives all of its votes to a candidate.* Ignoring the number of votes each state has in the Electoral College, but including all possible combinations of states that could be won by either candidate, how many out- comes are possible in the Electoral College if there are two candidates? (Hint: The states that can be won by a candidate form a subset of all the states.)

Musicians A concert featured a cellist, a flutist, a harpist, and a vocalist. Throughout the concert, different subsets of the four musicians performed together, with at least two musicians playing each piece. How many subsets of at least two are possible?

List the elements of the following sets. For exercises $74-76,$ describe each set in words. $F,$ the set of networks that were launched before $1985 .$

List the elements of the following sets. For exercises $74-76,$ describe each set in words. G, the set of networks that feature sports.

List the elements of the following sets. For exercises $74-76,$ describe each set in words. H, the set of networks that have more than 97.6 million viewers.

List the elements of the following sets. For exercises $74-76$ , describe each set in words. $F \cap H$

List the elements of the following sets. For exercises $74-76$ , describe each set in words. $G \cup H$

List the elements of the following sets. For exercises $74-76$ , describe each set in words. $G^{\prime}$

Games In David Gale's game of Subset Takeaway, the object is for each player, at his or her turn, to pick a nonempty proper subset of a given set subject to the condition that no subset chosen earlier by either player can be a subset of the newly chosen set. The winner is the last person who can make a legal move. Consider the set $A=\{1,2,3\} .$ Suppose Joe and Dorothy are playing the game and Dorothy goes first. If she chooses the proper subset $\{1\},$ then Joe cannot choose any subset that includes the element 1. Joe can, however, choose $\{2\}$ or $\{3\}$ or $\{2,3\} .$ Develop a strategy for Joe so that he can always win the game if Dorothy goes first. Source: Scientific American.

a. Describe in words the set $A \cup(B \cap C)^{\prime}$ .b. List all elements in the set $A \cup(B \cap C)^{\prime}$

a. Describe in words the set $(A \cup B)^{\prime} \cap C$ .b. List all elements in the set $(A \cup B)^{\prime} \cap C$ .