Section 1
Sets
Reading and Writing Afler reading this section, write out the answers to these questions. Use complete sentences.What is a set?
Reading and Writing Afler reading this section, write out the answers to these questions. Use complete sentences.What is the difference between a finite set and an infinite set?
Reading and Writing Afler reading this section, write out the answers to these questions. Use complete sentences.What is a Venn diagram used for?
Reading and Writing Afler reading this section, write out the answers to these questions. Use complete sentences.What is the difference between the intersection and the union of two sets?
Reading and Writing Afler reading this section, write out the answers to these questions. Use complete sentences.What does it mean to say that set $A$ is a subset of set $B$ ?
Reading and Writing Afler reading this section, write out the answers to these questions. Use complete sentences.Which set is a subset of every set?
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$3 \in A$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$3 \in B$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$11 \notin A$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$3 \notin C$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$C=N$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$A=N$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$A \neq B$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$c \neq N$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$99 \in N$$
Using the sets $A, B, C,$ and $N,$ determine whether each statement is true or false. Explain.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$6 \in C$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cup C$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cup B$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cap C$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cap B$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$B \cup C$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$B \cap C$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cup \varnothing$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$B \cup \varnothing$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cap \varnothing$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$B \cap \varnothing$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cap N$$
Using the sets $A, B, C,$ and $N,$ list the elements in each set. If the set is empty write $\varnothing .$$$A \cup N$$
Use one of the symbols $\in, \notin,=, \neq, \cup,$ or $\cap$ in each blank to make a true statement.$$\begin{array}{ll}A=\{1,3,5,7,9\} & B=\{2,4,6,8\} \\C=\{1,2,3,4,5\} & N=\{1,2,3, \dots\}\end{array}$$
$$A \cap B \text{_____}\varnothing$$
$$A \cap C=\text{_____}\varnothing$$
$$A \text{_____} B=\{1,2,3,4,5,6,7,8,9\}$$
$$A \text{_____}B=\varnothing$$
$$B \text{_____}{C}=\{2,4\}$$
$$B \text{_____}C=\{1,2,3,4,5,6,8\}$$
$$3\text{_____}A \cap B$$
$$3\text{_____}A \cap C$$
$$4\text{_____}B \cap C$$
$$8\text{_____}B \cup C$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$A \subseteq N$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$B \subseteq N$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$\{2,3\} \subseteq C$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$C \subseteq A$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$B \nsubseteq C$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$C \nsubseteq A$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$\varnothing \subseteq B$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$\varnothing \subseteq C$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$A \subseteq \varnothing$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$B \subseteq \varnothing$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$A \cap B \subseteq C$$
Determine whether each statement is true or false. Explain your answer.$$\begin{aligned}&A=\{1,3,5,7,9\} \quad B=\{2,4,6,8\}\\&C=\{1,2,3,4,5\} \quad N=\{1,2,3, \dots\}\end{aligned}$$$$B \cap C \subseteq\{2,4,6,8\}$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$D \cup E$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$D \cap E$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$D \cap F$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$D \cup F$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$E \cup F$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$E \cap F$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$(D \cup E) \cap F$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$(D \cup F) \cap E$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$D \cup(E \cap F)$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$D \cup(F \cap E)$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$(D \cap F) \cup(E \cap F)$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$(D \cap E) \cup(F \cap E)$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$(D \cup E) \cap(D \cup F)$$
Using the sets $D . E,$ and $F$ list the elements in each set. If the set is empty write $\varnothing .$ $$D=\{3,5,7\} \quad E=\{2,4,6,8\} \quad F=\{1,2,3,4,5\}$$$$(D \cup F) \cap(D \cup E)$$
List the elements in each set.$\{x | x \text { is an even natural number less than } 20\}$
List the elements in each set.$\{x | x \text { is a natural number greater than } 6\}$
List the elements in each set.$\{x | x \text { is an odd natural number greater than } 11\}$
List the elements in each set.$\{x | x \text { is an odd natural number less than } 14\}$
List the elements in each set.$\{x | x \text { is an even natural number between } 4 \text { and } 79\}$
List the elements in each set.$\{x | x \text { is an odd natural number between } 12 \text { and } 57\}$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$A \cup B$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$A \cup C$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$A \cap B$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$A \cap C$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(A \cap B) \cup D$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(A \cap C) \cup D$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(A \cup B) \cup D$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(B \cup C) \cap A$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(B \cap C) \cap A$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(B \cap D) \cap C$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(B \cap D) \cup(C \cap D)$$
Using the sets $A, B, C,$ and $D,$ list the elements in each set. If the set is empty write $\varnothing$.$$\begin{array}{ll}A=\{1,2,3,4,5\} & B=\{4,5,6,7,8,9\} \\C=\{1,3,5,7\} & D=\{2,4,6,8\}\end{array}$$$$(A \cup B) \cap(C \cup D)$$
Write each set using set-builder notation. Answers may vary.$$\{3,4,5,6\}$$
Write each set using set-builder notation. Answers may vary.$$\{1,3,5,7\}$$
Write each set using set-builder notation. Answers may vary.$$\{5,7,9,11, \dots\}$$
Write each set using set-builder notation. Answers may vary.$$\{4,5,6,7, \dots\}$$
Write each set using set-builder notation. Answers may vary.$$\{6,8,10,12, \dots, 82\}$$
Write each set using set-builder notation. Answers may vary.$$\{9,11,13,15, \dots, 51\}$$
In a class of 30 students all of them are either female or smokers, while only 5 are female and smokers. If there are 12 male smokers in the class, then how many female nonsmokers are in the class?
All of the 500 students at Tickfaw College take either math or English, while 300 of them take math and English. If only 50 take math but not English, then how many take English but not math?
Discussion. If $A$ and $B$ are finite sets, could $A \cup B$ be infinite? Explain.
Cooperative learning. Work with a small group to answer the following questions. If $A \subseteq B$ and $B \subseteq A$, then what can you conclude about $A$ and $B ?$ If $(A \cup B) \subseteq(A \cap B)$, then what can you conclude about $A$ and $B$ ?
Discussion. What is wrong with each statement? Explain.a) $3 \subseteq\{1,2,3\}$b) $\{3\} \in\{1,2,3\}$c) $\varnothing=\{\varnothing\}$
Exploration. There are only two possible subsets of $\{1\}$, namely, $\varnothing$ and $\{1\}.$a) List all possible subsets of $\{1,2\} .$ How many are there?b) List all possible subsets of $\{1,2,3\} .$ How many are there?c) Guess how many subsets there are of $\{1,2,3,4\} .$ Verif your guess by listing all the possible subsets.d) How many subsets are there for $\{1,2,3, \dots, n\} ?$