Section 1
Sets and Set Operations
List the elements in each of the sets. The set $F$ consisting of the four seasons.
List the elements in each of the sets. The set $A$ consisting of the authors of this book.
List the elements in each of the sets. The set $I$ of all positive integers no greater than $6 .$
List the elements in each of the sets. The set $N$ of all negative integers greater than $-3$.
List the elements in each of the sets. $A=\{n \mid n$ is a positive integer and $0 \leq n \leq 3\}$.
List the elements in each of the sets. $A=\{n \mid n$ is a positive integer and $0<n<8\}$.
List the elements in each of the sets. $B=\{n \mid n$ is an even positive integer and $0 \leq n \leq 8\}$.
List the elements in each of the sets. $B=\{n \mid n$ is an odd positive integer and $0 \leq n \leq 8\}$.
List the elements in each of the sets. The set of all outcomes of tossing a pair of (a) distinguishable coins (b) indistinguishable coins.
List the elements in each of the sets. The set of outcomes of tossing three (a) distinguishable coins(b) indistinguishable coins.
List the elements in each of the sets. The set of all outcomes of rolling two distinguishable dice such that the numbers add to 6 .
List the elements in each of the sets. The set of all outcomes of rolling two distinguishable dice such that the numbers add to 8 .
List the elements in each of the sets. The set of all outcomes of rolling two indistinguishable dice such that the numbers add to 6 .
List the elements in each of the sets. The set of all outcomes of rolling two indistinguishable dice such that the numbers add to 8 .
List the elements in each of the sets. The set of all outcomes of rolling two distinguishable dice such that the numbers add to 13 .
List the elements in each of the sets. The set of all outcomes of rolling two distinguishable dice such that the numbers add to 1 .
Draw a Venn diagram that illustrates the relationships among the given sets. $$\begin{aligned}&S=\{\text { eBay, Googletm, Amazon, OHaganBooks, Hotmail }\}, \\&A=\{\text { Amazon, OHaganBooks }\}, \quad B=\{\text { eBay, Amazon }\} \\&C=\{\text { Amazon, Hotmail }\}\end{aligned}$$
Draw a Venn diagram that illustrates the relationships among the given sets. $S=\{$ Apple, Dell, Gateway, Pomegranate, Compaq $\}, A=$ \{Gateway, Pomegranate, Compaq\}, $B=\{$ Dell, Gateway,Pomegranate, Compaq $\}, C=\{$ Apple, Dell, Compaq\}
Draw a Venn diagram that illustrates the relationships among the given sets. $S=\left\{\right.$ eBay, Google $^{\mathrm{TM}}$ Amazon, OHaganBooks, Hotmail\}, $A=\{$ Amazon, Hotmail $\}, B=\left\{\right.$ eBay, Google $^{\mathrm{TM}}$ Amazon, Hotmail $\}, C=\{$ Amazon, Hotmail $\}$
Draw a Venn diagram that illustrates the relationships among the given sets. $S=\{$ Apple, Dell, Gateway, Pomegranate, Compaq $\}, A=$ $\{$ Apple, Dell, Pomegranate, Compaq $\}, B=\{$ Pomegranate $\}$, $C=\{$ Pomegranate $\}$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cup B$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cup C$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cup \emptyset$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$B \cup \emptyset$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cup(B \cup C)$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$(A \cup B) \cup C$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$C \cap B$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$C \cap A$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cap \emptyset$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$\emptyset \cap B$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$(A \cap B) \cap C$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cap(B \cap C)$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$(A \cap B) \cup C$$
Let $A=\{$ June, Janet, Jill, Justin, Jeffrey, Jello\}, $B=\{$ Janet, Jello, Justin\}, and $C=\{$ Sally, Solly, Molly, Jolly, Jello\}. Find each set. $$A \cup(B \cap C)$$
A=\{$ small, medium, large $\}, B=\{$ blue, green $\}$, and $C=\{$ triangle, square $\}$. List the elements of $A \times C$.
A=\{$ small, medium, large $\}, B=\{$ blue, green $\}$, and $C=\{$ triangle, square $\}$. List the elements of $B \times C$.
A=\{$ small, medium, large $\}, B=\{$ blue, green $\}$, and $C=\{$ triangle, square $\}$. List the elements of $A \times B$
A={ small, medium, large }, B={ blue, green }, and C={triangle, square}. The elements of $A \times B \times C$ are the ordered triples $(a, b, c)$ with $a \in A, b \in B$, and $c \in C .$ List all the elements of $A \times B \times C .$
A={ small, medium, large }, B={ blue, green }, and C={triangle, square}. Represent $B \times C$ as cells in a spreadsheet.
A={ small, medium, large }, B={ blue, green }, and C={triangle, square}. Represent $A \times C$ as cells in a spreadsheet.
A={ small, medium, large }, B={ blue, green }, and C={triangle, square}. Represent $A \times B$ as cells in a spreadsheet.
A={ small, medium, large }, B={ blue, green }, and C={triangle, square}. Represent $A \times A$ as cells in a spreadsheet.
Let $A=\{H, T\}$ be the set of outcomes when a coin is tossed, and let $B=\{1,2,3,4,5,6\}$ be the set of outcomes when a die is rolled. Write each set in terms of A and/or $B$ and list its elements. The set of outcomes when a die is rolled and then a cointossed.
Let $A=\{H, T\}$ be the set of outcomes when a coin is tossed, and let $B=\{1,2,3,4,5,6\}$ be the set of outcomes when a die is rolled. Write each set in terms of A and/or $B$ and list its elements. The set of outcomes when a coin is tossed twice.
Let $A=\{H, T\}$ be the set of outcomes when a coin is tossed, and let $B=\{1,2,3,4,5,6\}$ be the set of outcomes when a die is rolled. Write each set in terms of A and/or $B$ and list its elements. The set of outcomes when a coin is tossed three times.
Let $A=\{H, T\}$ be the set of outcomes when a coin is tossed, and let $B=\{1,2,3,4,5,6\}$ be the set of outcomes when a die is rolled. Write each set in terms of A and/or $B$ and list its elements. The set of outcomes when a coin is tossed twice and then a die is rolled.
Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$E^{\prime}$$
Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$F^{\prime}$$
Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$(E \cup F)$$
Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$(E \cap F)^{\prime}$$
Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$E^{\prime} \cup F^{\prime}$$
Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$E^{\prime} \cap F^{\prime}$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \quad \text { De Morgan's Law }$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \quad \text { De Morgan's Law }$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$\begin{aligned}&(A \cap B) \cap C=A \cap(B \cap C)\\&\text { Associative Law }\end{aligned}$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$(A \cup B) \cup C=A \cup(B \cup C) \quad \text { Associative Law }$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$A \cup(B \cap C)=(A \cup B) \cap(A \cup C) \quad \text { Distributive Law }$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \quad \text { Distributive Law }$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$S^{\prime}=\emptyset$$
Use Venn diagrams to illustrate the following identities for subsets $A, B$, and $\operatorname{Cof} S .$ $$\emptyset^{\prime}=S$$
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who owe her money and have done at least $\$ 10,000$ worth of business with her.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who owe her money or have done at least $\$ 10,000$ worth of business with her.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who have done at least $\$ 10,000$ worth of business with her or have employed her in the last year.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who have done at least $\$ 10,000$ worth of business with her and have employed her in the last year.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who do not owe her money and have employed her in the last year.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who do not owe her money or have employed her in the last year.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who owe her money, have not done at least $\$ 10,000$ worth of business with her, and have not employed her in the last year.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. The clients who either do not owe her money, have done at least $\$ 10,000$ worth of business with her, or have employed her in the last year.
A subset of clients is described that the consultant could find using her database. Write each subset in terms of $A, B$, and $C$ and list the clients in that subset. You are given data on revenues from sales of sail boats, motor boats, and yachts for each of the years 2003 through $2006 .$ How would you represent these data in a spreadsheet? The cells in your spreadsheet represent elements of which set?
Spending in most categories of health care in the United States increased dramatically in the last 30 years of the $1900 \mathrm{~s} .{ }^{1}$ You are given data showing total spending on prescription drugs, nursing homes, hospital care, and professional services for each of the last three decades of the $1900 \mathrm{~s}$. How would you represent these data in a spreadsheet? The cells in your spreadsheet represent elements of which set?
You sell iPods ${ }^{\circledast}$ and $j$ Pods. Let $I$ be the set of all iPods you sold last year, and let $J$ be the set of all jPods you sold last year. What set represents the collection of all iPods and jPods you sold combined?
You sell two models of music players: the yoVaina Grandote and the yoVaina Minúsculito, and each comes in three colors:Infraroja, Ultravioleta, and Radiografia. Let $M$ be the set of models and let $C$ be the set of colors. What set represents the different choices a customer can make?
You are searching online for techno music that is neither European nor Dutch. In set notation, which set of music files are you searching for?(A) Techno $\cap$ (European $\cap$ Dutch)'(B) Techno $\cap$ (European $\cup$ Dutch)'(C) Techno $\cup$ (European $\cap$ Dutch) $^{\prime}$(D) Techno $\cup(\text { European } \cup \text { Dutch })^{\prime}$
You would like to see either a World War II movie, or one that that is based on a comic book character but does not feature aliens. Which set of movies are you interested in seeing?(A) WWII $\cap$ (Comix $\cap$ Aliens')(B) WWII $\cap$ (Comix $\cup$ Aliens')(C) WWII $\cup$ (Comix $\cap$ Aliens')(D) WWII $\cup$ (Comix $\cup$ Aliens' $^{\prime}$ )
Explain, illustrating by means of an example, why $(A \cap B) \cup C \neq A \cap(B \cup C)$
Explain, making reference to operations on sets, why the statement "He plays soccer or rugby and cricket" is ambiguous.
Explain the meaning of a universal set, and give two different universal sets that could be used in a discussion about sets of positive integers.
Is the set of outcomes when two indistinguishable dice are rolled (Example 1) a Cartesian product of two sets? If so, which two sets; if not, why not?
Design a database scenario that leads to the following statement: To keep the factory operating at maximum capacity, the plant manager should select the suppliers in $A \cap\left(B \cup C^{\prime}\right)$.
Design a database scenario that leads to the following statement: To keep her customers happy, the bookstore owner should stock periodicals in $A \cup\left(B \cap C^{\prime}\right)$.
Rewrite in set notation: She prefers movies that are not violent, are shorter than two hours, and have neither a tragic ending nor an unexpected ending.
Rewrite in set notation: He will cater for any event as long as there are no more than 1,000 people, it lasts for at least three hours, and it is within a 50 mile radius of Toronto.