Sets and Operations on Sets

Section 5.1 Sets and Operations on Sets

Beginning Activity Beginning Activity 1: Set Operations
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Before beginning this section, it would be a good idea to review sets and set notation, including the roster method and set builder notation, in Section 2.3.

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In Section 2.1, we used logical operators (conjunction, disjunction, negation) to form new statements from existing statements. In a similar manner, there are several ways to create new sets from sets that have already been defined. In fact, we will form these new sets using the logical operators of conjunction (and), disjunction (or), and negation (not). For example, if the universal set is the set of natural numbers

N

and

A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7, 9},

The set consisting of all natural numbers that are in $A$ and are in $B$ is the set ${1, 3, 5};$
The set consisting of all natural numbers that are in $A$ or are in $B$ is the set ${1, 2, 3, 4, 5, 6, 7, 9};$ and
The set consisting of all natural numbers that are in $A$ and are not in $B$ is the set ${2, 4, 6} .$

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These sets are examples of some of the most common set operations, which are given in the following definitions.

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Definition.

Let $A$ and $B$ be subsets of some universal set $U .$ The intersection of $A$ and $B,$ written $A \cap B$ and read “ $A$ intersect $B,$ ” is the set of all elements that are in both $A$ and $B .$ That is,

A \cap B = {x \in U ∣ x \in A and x \in B} .

The union of $A$ and $B,$ written $A \cup B$ and read “ $A$ union $B,$ ” is the set of all elements that are in $A$ or in $B .$ That is,

A \cup B = {x \in U ∣ x \in A or x \in B} .

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Definition.

Let $A$ and $B$ be subsets of some universal set $U .$ The set difference of $A$ and $B,$ or relative complement of $B$ with respect to $A,$ written $A - B$ and read “ $A$ minus $B$ ” or “the complement of $B$ with respect to $A,$ ” is the set of all elements in $A$ that are not in $B .$ That is,

A - B = {x \in U ∣ x \in A and x \notin B} .

The complement of the set $A,$ written $A^{c}$ and read “the complement of $A,$ ” is the set of all elements of $U$ that are not in $A .$ That is,

A^{c} = {x \in U ∣ x \notin A} .

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For the rest of this beginning activity, the universal set is

U = {0, 1, 2, 3, \dots, 10},

and we will use the following subsets of

U :

A = {0, 1, 2, 3, 9} and B = {2, 3, 4, 5, 6} .

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So in this case,

A \cap B = {x \in U ∣ x \in A and x \in B} = {2, 3} .

Use the roster method to specify each of the following subsets of

U .

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1.

$A \cup B$

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2.

$A^{c}$

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3.

$B^{c}$

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We can now use these sets to form even more sets. For example,

A \cap B^{c} = {0, 1, 2, 3, 9} \cap {0, 1, 7, 8, 9, 10} = {0, 1, 9} .

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Use the roster method to specify each of the following subsets of

U .

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4.

$A \cup B^{c}$

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5.

$A^{c} \cap B^{c}$

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6.

$A^{c} \cup B^{c}$

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7.

$(A \cap B)^{c}$

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Beginning Activity Beginning Activity 2: Venn Diagrams for Two Sets
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In Beginning Activity 1, we worked with verbal and symbolic definitions of set operations. However, it is also helpful to have a visual representation of sets. Venn diagrams are used to represent sets by circles (or some other closed geometric shape) drawn inside a rectangle. The points inside the rectangle represent the universal set

U,

and the elements of a set are represented by the points inside the circle that represents the set. For example, Figure 5.1 is a Venn diagram showing two sets.

Venn Diagram with two circles A and B inside the universal set U. A overlaps B creating 3 inner-circle regions. The leftmost is labelled 1, the center 2, and the right 3. The area outside the circles but in U is labelled 4. — 🔗
Figure 5.1. Venn Diagram for Two Sets

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In Figure 5.1, the elements of

A

are represented by the points inside the left circle, and the elements of

B

are represented by the points inside the right circle. The four distinct regions in the diagram are numbered for reference purposes only. (The numbers do not represent elements in a set.) The following table describes the four regions in the diagram.

Region	Elements of $U$	Set
1	In $A$ and not in $B$	$A - B$
2	In $A$ and in $B$	$A \cap B$
3	In $B$ and not in $A$	$B - A$
4	Not in $A$ and not in $B$	$A^{c} \cap B^{c}$

We can use these regions to represent other sets. For example, the set

A \cup B

is represented by regions 1, 2, and 3 or the shaded region in Figure 5.2.

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Let

A

and

B

be subsets of a universal set

U .

For each of the following, draw a Venn diagram for two sets and shade the region that represent the specified set. In addition, describe the set using set builder notation.

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1.

$A^{c}$

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2.

$B^{c}$

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3.

$A^{c} \cup B$

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4.

$A^{c} \cup B^{c}$

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5.

${(A \cap B)}^{c}$

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6.

$(A \cup B) - (A \cap B)$

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Subsection Set Equality, Subsets, and Proper Subsets

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In Section 2.3, we introduced some basic definitions used in set theory, what it means to say that two sets are equal and what it means to say that one set is a subset of another set. See Definition. We need one more definition.

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Definition.

Let $A$ and $B$ be two sets contained in some universal set $U .$ The set $A$ is a proper subset of $B$ provided that $A \subseteq B$ and $A \neq B .$ When $A$ is a proper subset of $B,$ we write $A \subset B .$

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One reason for the definition of proper subset is that each set is a subset of itself. That is, If

A

is a set, then

A \subseteq A .

However, sometimes we need to indicate that a set

X

is a subset of

Y

but

X \neq Y .

For example, if

X = {1, 2} and Y = {0, 1, 2, 3},

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then

X \subset Y .

We know that

X \subseteq Y

since each element of

X

is an element of

Y,

but

X \neq Y

since

0 \in Y

and

0 \notin X .

(Also,

3 \in Y

and

3 \notin X .

) Notice that the notations

A \subset B

and

A \subseteq B

are used in a manner similar to inequality notation for numbers (

a < b

and

a \leq b

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It is often very important to be able to describe precisely what it means to say that one set is not a subset of the other. In the preceding example,

Y

is not a subset of

X

since there exists an element of

Y

(namely, 0) that is not in

X .

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In general, the subset relation is described with the use of a universal quantifier since

A \subseteq B

means that for each element

x

U,

x \in A,

then

x \in B .

So when we negate this, we use an existential quantifier as follows:

\begin{aligned} A \subseteq B & means & (\forall x \in U) [(x \in A) \to (x \in B)] \\ A ⊈ B & means & \neg (\forall x \in U) [(x \in A) \to (x \in B)] \\ (\exists x \in U) \neg [(x \in A) \to (x \in B)] \\ (\exists x \in U) [(x \in A) \land (x \notin B)] . \end{aligned}

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So we see that

A ⊈ B

means that there exists an

x

U

such that

x \in A

and

x \notin B .

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Notice that if

A = \emptyset,

then the conditional statement, “For each

x \in U,

x \in \emptyset,

then

x \in B

” must be true since the hypothesis will always be false. Another way to look at this is to consider the following statement:

$\emptyset ⊈ B$ means that there exists an $x \in \emptyset$ such that $x \notin B .$

However, this statement must be false since there does not exist an

x

\emptyset .

Since this is false, we must conclude that

\emptyset \subseteq B .

Although the facts that

\emptyset \subseteq B

and

B \subseteq B

may not seem very important, we will use these facts later, and hence we summarize them in Theorem 5.3.

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Theorem 5.3.

For any set $B,$ $\emptyset \subseteq B$ and $B \subseteq B .$

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In Section 2.3, we also defined two sets to be equal when they have precisely the same elements. For example,

{x \in R | x^{2} = 4} = {- 2, 2} .

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If the two sets

A

and

B

are equal, then it must be true that every element of

A

is an element of

B,

that is,

A \subseteq B,

and it must be true that every element of

B

is an element of

A,

that is,

B \subseteq A .

Conversely, if

A \subseteq B

and

B \subseteq A,

then

A

and

B

must have precisely the same elements. This gives us the following test for set equality:

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Theorem 5.4.

Let $A$ and $B$ be subsets of some universal set $U .$ Then $A = B$ if and only if $A \subseteq B$ and $B \subseteq A .$

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Progress Check 5.5. Using Set Notation.

Let the universal set be $U = {1, 2, 3, 4, 5, 6},$ and let

A = {1, 2, 4}, B = {1, 2, 3, 5}, C = {x \in U | x^{2} \leq 2} .

In each of the following, fill in the blank with one or more of the symbols $\subset, \subseteq, =, \neq, \in, or \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

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(a)

$A$ $B$

Solution.

$⊄, ⊈\neq$

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(b)

5 $B$

Solution.

$\in$

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(c)

$A$ $C$

Solution.

$\neq$

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(d)

${1, 2}$ $A$

Solution.

$\subset, \subseteq, \neq$

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(e)

6 $A$

Solution.

$\notin$

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(f)

$\emptyset$ $A$

Solution.

$\subset, \subseteq, \neq$

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(g)

${5}$ $B$

Solution.

$\subset, \subseteq, \neq$

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(h)

${1, 2}$ $C$

Solution.

$\neq$

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(i)

${4, 2, 1}$ $A$

Solution.

$\subseteq, =$

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(j)

$B$ $\emptyset$

Solution.

$\neq$

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Subsection More about Venn Diagrams

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In Beginning Activity 2, we learned how to use Venn diagrams as a visual representation for sets, set operations, and set relationships. In that activity, we restricted ourselves to using two sets. We can, of course, include more than two sets in a Venn diagram. Figure 5.6 shows a general Venn diagram for three sets (including a shaded region that corresponds to

A \cap C

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In this diagram, there are eight distinct regions, and each region has a unique reference number. For example, the set

A

is represented by the combination of regions 1, 2, 4, and 5, whereas the set

C

is represented by the combination of regions 4, 5, 6, and 7. This means that the set

A \cap C

is represented by the combination of regions 4 and 5. This is shown as the shaded region in Figure 5.6.

A Venn diagram for three sets, A, B, and C. The intersection of A and C is shaded. — 🔗
Figure 5.6. Venn Diagram for $A \cap C$

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Finally, Venn diagrams can also be used to illustrate special relationships between sets. For example, if

A \subseteq B,

then the circle representing

A

should be completely contained in the circle for

B .

So if

A \subseteq B,

and we know nothing about any relationship between the set

C

and the sets

A

and

B,

we could use the Venn diagram shown in Figure 5.7.

Venn Diagram with three circles A, B, and C. B and C overlap and A is completely inside B while overlapping C. — 🔗
Figure 5.7. Venn Diagram Showing $A \subseteq B$

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Progress Check 5.8. Using Venn Diagrams.

Let $A,$ $B,$ and $C$ be subsets of a universal set $U .$

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(a)

For each of the following, draw a Venn diagram for three sets and shade the region(s) that represent the specified set.

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(i)

$(A \cap B) \cap C$

Solution.

For the set

(A \cap B) \cap C,

region 5 is shaded.

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(ii)

$(A \cap B) \cup C$

Solution.

For the set

(A \cap B) \cup C,

the regions 2, 4, 5, 6, 7 are shaded.

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(iii)

$(A^{c} \cup B)$

Solution.

For the set

(A^{c} \cup B),

the regions 2, 3, 5, 6, 7, 8 are shaded.

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(iv)

$A^{c} \cap (B \cup C)$

Solution.

For the set

(A^{c} \cap (B \cup C)),

the regions 3, 6, 7 are shaded.

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(b)

Draw the most general Venn diagram showing $B \subseteq (A \cup C) .$

Solution.

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(c)

Draw the most general Venn diagram showing $A \subseteq (B^{c} \cup C) .$

Solution.

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Subsection The Power Set of a Set

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The symbol

\in

is used to describe a relationship between an element of the universal set and a subset of the universal set, and the symbol

\subseteq

is used to describe a relationship between two subsets of the universal set. For example, the number 5 is an integer, and so it is appropriate to write

5 \in Z .

It is not appropriate, however, to write

5 \subseteq Z

since 5 is not a set. It is important to distinguish between 5 and

{5} .

The difference is that 5 is an integer and

{5}

is a set consisting of one element. Consequently, it is appropriate to write

{5} \subseteq Z,

but it is not appropriate to write

{5} \in Z .

The distinction between these two symbols

(5 and {5})

is important when we discuss what is called the power set of a given set.

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Definition.

If $A$ is a subset of a universal set $U,$ then the set whose members are all the subsets of $A$ is called the power set of $A .$ We denote the power set of $A$ by $P (A)$ . Symbolically, we write

P (A) = {X \subseteq U ∣ X \subseteq A} .

That is, $X \in P (A)$ if and only if $X \subseteq A .$

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When dealing with the power set of

A,

we must always remember that

\emptyset \subseteq A

and

A \subseteq A .

For example, if

A = {a, b},

then the subsets of

A

are

\begin{matrix} (37) & \emptyset, {a}, {b}, {a, b} . \end{matrix}

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We can write this as

P (A) = {\emptyset, {a}, {b}, {a, b}} .

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Now let

B = {a, b, c} .

Notice that

B = A \cup {c} .

We can determine the subsets of

B

by starting with the subsets of

A

in (37). We can form the other subsets of

B

by taking the union of each set in (37) with the set

{c} .

This gives us the following subsets of

B .

\begin{matrix} (38) & {c}, {a, c}, {b, c}, {a, b, c} . \end{matrix}

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So the subsets of

B

are those sets in (37) combined with those sets in (38). That is, the subsets of

B

are

\begin{matrix} (39) & \emptyset, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}, \end{matrix}

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which means that

P (B) = {\emptyset, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}} .

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Notice that we could write

{a, c} \subseteq B or that {a, c} \in P (B) .

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Also, notice that

A

has two elements and

A

has four subsets, and

B

has three elements and

B

has eight subsets. Now, let

n

be a nonnegative integer. The following result can be proved using mathematical induction. (See Activity 29.)

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Theorem 5.9.

Let $n$ be a nonnegative integer and let $T$ be a subset of some universal set. If the set $T$ has $n$ elements, then the set $T$ has $2^{n}$ subsets. That is, $P (T)$ has $2^{n}$ elements.

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Subsection The Cardinality of a Finite Set

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In our discussion of the power set, we were concerned with the number of elements in a set. In fact, the number of elements in a finite set is a distinguishing characteristic of the set, so we give it the following name.

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Definition.

The number of elements in a finite set $A$ is called the cardinality of $A$ and is denoted by $card (A) .$

For example,

card (\emptyset) = 0;

card ({a, b}) = 2;

card (P ({a, b})) = 4 .

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Theoretical Note.

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There is a mathematical way to distinguish between finite and infinite sets, and there is a way to define the cardinality of an infinite set. We will not concern ourselves with this at this time. More about the cardinality of finite and infinite sets is discussed in Chapter 9.

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Subsection Standard Number Systems

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We can use set notation to specify and help describe our standard number systems. The starting point is the set of natural numbers, for which we use the roster method.

N = {1, 2, 3, 4, \dots}

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The integers consist of the natural numbers, the negatives of the natural numbers, and zero. If we let

N^{-} = {\dots, - 4, - 3, - 2, - 1},

then we can use set union and write

Z = N^{-} \cup {0} \cup N .

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So we see that

N \subseteq Z,

and in fact,

N \subset Z .

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We need to use set builder notation for the set

Q

of all rational numbers, which consists of quotients of integers.

Q = {\frac{m}{n} | m, n \in Z and n \neq 0}

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Since any integer

n

can be written as

n = \frac{n}{1},

we see that

Z \subseteq Q .

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We do not yet have the tools to give a complete description of the real numbers. We will simply say that the real numbers consist of the rational numbers and the irrational numbers. In effect, the irrational numbers are the complement of the set of rational numbers

Q

R .

So we can use the notation

Q^{c} = {x \in R ∣ x \notin Q}

and write

R = Q \cup Q^{c} and Q \cap Q^{c} = \emptyset .

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A number system that we have not yet discussed is the set of complex numbers. The complex numbers,

C,

consist of all numbers of the form

a + b i,

where

a, b \in R

and

i = \sqrt{- 1}

(or

i^{2} = - 1

). That is,

C = {a + b i | a, b \in R and i = \sqrt{- 1}} .

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We can add and multiply complex numbers as follows: If

a, b, c, d \in R,

then

\begin{aligned} (a + b i) + (c + d i) & = (a + c) + (b + d) i, and \\ (a + b i) (c + d i) & = a c + a d i + b c i + b d i^{2} \\ = (a c - b d) + (a d + b c) i . \end{aligned}

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Exercises Exercises

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1.

Assume the universal set is the set of real numbers. Let

\begin{aligned} A & = {- 3, - 2, 2, 3}, & B & = {x \in R | x^{2} = 4 or x^{2} = 9}, \\ C & = {x \in R | x^{2} + 2 = 0}, & D & = {x \in R | x > 0} . \end{aligned}

Respond to each of the following questions. In each case, explain your answer.

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(a)

Is the set $A$ equal to the set $B ?$

Interval Notation	Set Notation	Name
$(a, b) =$	${x \in R \| a < x < b}$	Open interval from $a$ to $b$
$[a, b] =$	${x \in R \| a \leq x \leq b}$	Closed interval from $a$ to $b$
$[a, b) =$	${x \in R \| a \leq x < b}$	Half-open interval
$(a, b] =$	${x \in R \| a < x \leq b}$	Half-open interval
$(a, + \infty) =$	${x \in R \| x > a}$	Open ray
$(- \infty, b) =$	${x \in R \| x < b}$	Open ray
$[a, + \infty) =$	${x \in R \| x \geq a}$	Closed ray
$(- \infty, b] =$	${x \in R \| x \leq b}$	Closed ray

Section 5.1 Sets and Operations on Sets

Beginning Activity Beginning Activity 1: Set Operations A4US

Definition.

Definition.

1.

2.

3.

4.

5.

6.

7.

Beginning Activity Beginning Activity 2: Venn Diagrams for Two Sets A4US

1.

2.

3.

4.

5.

6.

Subsection Set Equality, Subsets, and Proper Subsets

Definition.

Theorem 5.3.

Theorem 5.4.

Progress Check 5.5. Using Set Notation.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Subsection More about Venn Diagrams

Progress Check 5.8. Using Venn Diagrams.

(a)

(i)

(ii)

(iii)

(iv)

(b)

(c)

Subsection The Power Set of a Set

Definition.

Theorem 5.9.

Subsection The Cardinality of a Finite Set

Definition.

Theoretical Note.

Subsection Standard Number Systems

Exercises Exercises

1.

(a)

(b)

(c)

(d)

(e)

2.

(a)

(b)

3.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

4.

5.

(a)

(b)

(c)

(d)

Beginning Activity Beginning Activity 1: Set Operations
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Beginning Activity Beginning Activity 2: Venn Diagrams for Two Sets
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