Section 1
Sets
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$A \cup B$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$B \cap C$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$A-B$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$B-A$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$\bar{A}$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$U-C$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$\bar{U}$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$A \cup \varnothing$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$B \cap \varnothing$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$A \cup U$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$B \cap U$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$A \cap(B \cup C)$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$\bar{B} \cap(C-A)$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$(A \cap B)-C$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$\overline{A \cap B} \cup C$$
Let the universe be the set $U=\{1,2,3, \ldots, 10\}$ Let $A=\{1,4,7,10\}, B=\{1,2,3,4,5\},$ and $C=\{2,4,6,8\} .$ List the elements of each set.$$(A \cup B)-(C-B)$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.Describe $\bar{Y}$ in words.
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$\bar{X}$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$\bar{Y}$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$X \cap Y$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$X \cup Y$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$\bar{X} \cap Y$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$\bar{X} \cup Y$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$X \cap \bar{Y}$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$X \cup \bar{Y}$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$\bar{X} \cap \bar{Y}$$
Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$.$$\bar{X} \cup \bar{Y}$$
What is the cardinality of $\varnothing ?$
What is the cardinality of $\{\varnothing\} ?$
What is the cardinality of $\{a, b, a, c\} ?$
What is the cardinality of $\{\{a\},\{a, b\},\{a, c\}, a, b\} ?$
Show, as in Examples 1.1.2 and 1.1.3, that $A=B$.$$A=\{3,2,1\}, B=\{1,2,3\}$$
Show, as in Examples 1.1.2 and 1.1.3, that $A=B$.$C=\{1,2,3\}, D=\{2,3,4\}, A=\{2,3\}, B=C \cap D$
Show, as in Examples 1.1.2 and 1.1.3, that $A=B$.$A=\{1,2,3\}, B=\left\{n \mid n \in \mathbf{Z}^{+}\right.$ and $\left.n^{2}<10\right\}$
Show, as in Examples 1.1.2 and 1.1.3, that $A=B$.$A=\left\{x \mid x^{2}-4 x+4=1\right\}, B=\{1,3\}$
Show, as in Example 1.1.4, that $A \neq B$.$A=\{1,2,3\}, B=\varnothing$
Show, as in Example 1.1.4, that $A \neq B$.$A=\{1,2\}, B=\left\{x \mid x^{3}-2 x^{2}-x+2=0\right\}$
Show, as in Example 1.1.4, that $A \neq B$.$A=\{1,3,5\}, B=\left\{n \mid n \in \mathbf{Z}^{+}\right.$ and $\left.n^{2}-1 \leq n\right\}$
Show, as in Example 1.1.4, that $A \neq B$.$B=\{1,2,3,4\}, C=\{2,4,6,8\}, A=B \cap C$
Determine whether each pair of sets is equal.$$\{1,2,2,3\},\{1,2,3\}$$
Determine whether each pair of sets is equal.$$\{1,1,3\},\{3,3,1\}$$
Determine whether each pair of sets is equal.$$\left\{x \mid x^{2}+x=2\right\},\{1,-1\}$$
Determine whether each pair of sets is equal.$$\{x \mid x \in \mathbf{R} \text { and } 0<x \leq 2\},\{1,2\}$$
Show, as in Examples 1.1 .5 and $1.1 .6,$ that $A \subseteq B$.$A=\{1,2\}, B=\{3,2,1\}$
Show, as in Examples 1.1 .5 and $1.1 .6,$ that $A \subseteq B$.$$A=\{1,2\}, B=\left\{x \mid x^{3}-6 x^{2}+11 x=6\right\}$$
Show, as in Examples 1.1 .5 and $1.1 .6,$ that $A \subseteq B$.$A=\{1\} \times\{1,2\}, B=\{1\} \times\{1,2,3\}$
Show, as in Examples 1.1 .5 and $1.1 .6,$ that $A \subseteq B$.$A=\left\{2 n \mid n \in \mathbf{Z}^{+}\right\}, B=\left\{n \mid n \in \mathbf{Z}^{+}\right\}$
Show, as in Example 1.1.9, that A is not a subset of $B$.$A=\{1,2,3\}, B=\{1,2\}$
Show, as in Example 1.1.9, that A is not a subset of $B$.$A=\left\{x \mid x^{3}-2 x^{2}-x+2=0\right\}, B=\{1,2\}$
Show, as in Example 1.1.9, that A is not a subset of $B$.$A=\{1,2,3,4\}, C=\{5,6,7,8\}, B=\{n \mid n \in A$ and $n+m=8$ for some $m \in C\}$
Show, as in Example 1.1.9, that A is not a subset of $B$.$A=\{1,2,3\}, B=\varnothing$
Draw a Venn diagram and shade the given set.$A \cap \bar{B}$
Draw a Venn diagram and shade the given set.$\bar{A}-B$
Draw a Venn diagram and shade the given set.$$B \cup(B-A)$$
Draw a Venn diagram and shade the given set.$$(A \cup B)-B$$
Draw a Venn diagram and shade the given set.$$B \cap \overline{(C \cup A)}$$
Draw a Venn diagram and shade the given set.$$(\bar{A} \cup B) \cap(\bar{C}-A)$$
Draw a Venn diagram and shade the given set.$$((C \cap A)-\overline{(B-A)}) \cap C$$
Draw a Venn diagram and shade the given set.$$(B-\bar{C}) \cup((B-\bar{A}) \cap(C \cup B))$$
A television commercial for a popular beverage showed the following Venn diagram. What does the shaded area represent?
Refer to a group of 191 students, of which 10 are taking French, business, and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French; 76 are taking business; and 63 are taking music.How many are taking French and music but not business?
Refer to a group of 191 students, of which 10 are taking French, business, and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French; 76 are taking business; and 63 are taking music.How many are taking business and neither French nor music?
Refer to a group of 191 students, of which 10 are taking French, business, and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French; 76 are taking business; and 63 are taking music.How many are taking French or business (or both)?
Refer to a group of 191 students, of which 10 are taking French, business, and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French; 76 are taking business; and 63 are taking music.How many are taking music or French (or both) but not business?
Refer to a group of 191 students, of which 10 are taking French, business, and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French; 76 are taking business; and 63 are taking music.How many are taking none of the three subjects?
A television poll of 151 persons found that 68 watched "Law and Disorder"; 61 watched "25"; 52 watched "The Tenors"; 16 watched both "Law and Disorder" and "25"; 25 watched both "Law and Disorder" and "The Tenors"; 19 watched both "25" and "The Tenors"; and 26 watched none of these shows. How many persons watched all three shows?
In a group of students, each student is taking a mathematics course or a computer science course or both. One-fifth of those taking a mathematics course are also taking a computer science course, and one-eighth of those taking a computer science course are also taking a mathematics course. Are more than one-third of the students taking a mathematics course?
Let $X=\{1,2\}$ and $Y=\{a, b, c\} .$ List the elements in each set.$$X \times Y$$
Let $X=\{1,2\}$ and $Y=\{a, b, c\} .$ List the elements in each set.$$Y \times X$$
Let $X=\{1,2\}$ and $Y=\{a, b, c\} .$ List the elements in each set.$$X \times X$$
Let $X=\{1,2\}$ and $Y=\{a, b, c\} .$ List the elements in each set.$$Y \times Y$$
Let $X=\{1,2\}, Y=\{a\},$ and $Z=\{\alpha, \beta\} .$ List the elements of each set.$$X \times Y \times Z$$
Let $X=\{1,2\}, Y=\{a\},$ and $Z=\{\alpha, \beta\} .$ List the elements of each set.$$X \times X \times X$$
Let $X=\{1,2\}, Y=\{a\},$ and $Z=\{\alpha, \beta\} .$ List the elements of each set.$$Y \times X \times Y \times Z$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{R} \times \mathbf{R}$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{Z} \times \mathbf{R}$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{R} \times \mathbf{Z}^{\text {nonneg }}$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{Z} \times \mathbf{Z}$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{R} \times \mathbf{R} \times \mathbf{R}$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{R} \times \mathbf{R} \times \mathbf{Z}$$
Give a geometric description of each set in words. Consider the elements of the sets to be coordinates. For example, $\mathbf{R} \times \mathbf{Z}$ is the set $\{(x, n) \mid x \in \mathbf{R}$ and $n \in \mathbf{Z}\} .$ Interpreting the ordered pairs $(x, n)$ as coordinates in the plane, the graph of allsuch ordered pairs is the set of all parallel horizontal lines spaced one unit apart, one of which passes through (0,0).$$\mathbf{R} \times \mathbf{Z} \times \mathbf{Z}$$
List all partitions of the set.$$\{1\}$$
List all partitions of the set.$$\{1,2\}$$
List all partitions of the set.$$\{a, b, c\}$$
List all partitions of the set.$$\{a, b, c, d\}$$
Answer true or false.$$\{x\} \subseteq\{x\}$$
Answer true or false.$$\{x\} \in\{x\}$$
Answer true or false.$$\{x\} \in\{x,\{x\}\}$$
Answer true or false.$$\{x\} \subseteq\{x,\{x\}\}$$
Answer true or false.$$\{2\} \subseteq \mathcal{P}(\{1,2\})$$
Answer true or false.$$\{2\} \in \mathcal{P}(\{1,2\})$$
List the members of $\mathcal{P}(\{a, b\})$. Which are proper subsets of $\{a, b\} ?$
List the members of $\mathcal{P}(\{a, b, c, d\}) .$ Which are proper subsets of $\{a, b, c, d\} ?$
If $X$ has 10 members, how many members does $\mathcal{P}(X)$ have? How many proper subsets does $X$ have?
If $X$ has $n$ members, how many proper subsets does $X$ have?
What relation must hold between sets $A$ and $B$ in order for the given condition to be true?$$A \cap B=A$$
What relation must hold between sets $A$ and $B$ in order for the given condition to be true?$$A \cup B=A$$
What relation must hold between sets $A$ and $B$ in order for the given condition to be true?$$\bar{A} \cap U=\varnothing$$
What relation must hold between sets $A$ and $B$ in order for the given condition to be true?$$\overline{A \cap B}=\bar{B}$$
If $A=\{1,2,3\}$ and $B=\{2,3,4,5\},$ find $A \Delta B$.
Describe the symmetric difference of sets $A$ and $B$ in words.
Given a universe $U,$ describe $A \triangle A, A \triangle \bar{A}, U \Delta A,$ and $\varnothing \Delta A$.
Let $C$ be a circle and let $\mathcal{D}$ be the set of all diameters of $C$. Whatis $\cap \mathcal{D} ?$ (Here, by "diameter" we mean a line segment through the center of the circle with its endpoints on the circumference of the circle.)
Let $P$ denote the set of integers greater than $1 .$ For $i \geq 2,$ define $$X_{i}=\{i k \mid k \in P\}$$ Describe $P-\bigcup_{i=2}^{\infty} X_{i}$.