Open Sentences and Sets

Section 2.3 Open Sentences and Sets

Beginning Activity Beginning Activity 1: Sets and Set Notation
A4 US

The theory of sets is fundamental to mathematics in the sense that many areas of mathematics use set theory and its language and notation. This language and notation must be understood if we are to communicate effectively in mathematics. At this point, we will give a very brief introduction to some of the terminology used in set theory.

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A set is a well-defined collection of objects that can be thought of as a single entity itself. For example, we can think of the set of integers that are greater than 4. Even though we cannot write down all the integers that are in this set, it is still a perfectly well-defined set. This means that if we are given a specific integer, we can tell whether or not it is in the set of integers greater than 4.

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The most basic way of specifying the elements of a set is to list the elements of that set. This works well when the set contains only a small number of objects. The usual practice is to list these elements between braces. For example, if the set

C

consists of the integer solutions of the equation

x^{2} = 9,

we would write

C = {- 3, 3} .

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For larger sets, it is sometimes inconvenient to list all of the elements of the set. In this case, we often list several of them and then write a series of three dots (

\dots

) to indicate that the pattern continues. For example,

D = {1, 3, 5, 7, \dots, 49}

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is the set of all odd natural numbers from 1 to 49, inclusive.

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For some sets, it is not possible to list all of the elements of a set; we then list several of the elements in the set and again use a series of three dots (

\dots

) to indicate that the pattern continues. For example, if

F

is the set of all even natural numbers, we could write

F = {2, 4, 6, \dots} .

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We can also use the three dots before listing specific elements to indicate the pattern prior to those elements. For example, if

E

is the set of all even integers, we could write

E = {\dots - 6, - 4, - 2, 0, 2, 4, 6, \dots} .

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Listing the elements of a set inside braces is called the roster method of specifying the elements of the set. We will learn other ways of specifying the elements of a set later in this section.

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1.

Use the roster method to specify the elements of each of the following sets:

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(a)

The set of real numbers that are solutions of the equation $x^{2} - 5 x = 0 .$

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(b)

The set of natural numbers that are less than or equal to 10.

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(c)

The set of integers that are greater than $- 2 .$

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2.

Each of the following sets is defined using the roster method. For each set, determine four elements of the set other than the ones listed using the roster method.

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(a)

$A = {1, 4, 7, 10, \dots}$

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(b)

$B = {2, 4, 8, 16, \dots}$

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(c)

$C = {\dots, - 8, - 6, - 4, - 2, 0}$

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(d)

$D = {\dots, - 9, - 6, - 3, 0, 3, 6, 9, \dots}$

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Beginning Activity Beginning Activity 2: Variables
A4 US

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Not all mathematical sentences are statements. For example, an equation such as

x^{2} - 5 = 0

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is not a statement. In this sentence, the symbol

x

is a variable. It represents a number that may be chosen from some specified set of numbers. The sentence (equation) becomes true or false when a specific number is substituted for

x .

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1.

Does the equation $x^{2} - 25 = 0$ become a true statement...

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(a)

if $- 5$ is substituted for $x ?$

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(b)

if $\sqrt{5}$ is substituted for $x ?$

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Definition.

A variable is a symbol representing an unspecified object that can be chosen from a given set $U .$ The set $U$ is called the universal set for the variable. It is the set of specified objects from which objects may be chosen to substitute for the variable. A constant is a specific member of the universal set.

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Some sets that we will use frequently are the usual number systems. Recall that we use the symbol

R

to stand for the set of all real numbers, the symbol

Q

to stand for the set of all rational numbers, the symbol

Z

to stand for the set of all integers, and the symbol

N

to stand for the set of all natural numbers.

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2.

What real numbers will make the sentence “ $y^{2} - 2 y - 15 = 0$ ” a true statement when substituted for $y ?$

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3.

What natural numbers will make the sentence “ $y^{2} - 2 y - 15 = 0$ ” a true statement when substituted for $y ?$

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4.

What real numbers will make the sentence “ $\sqrt{x}$ is a real number” a true statement when substituted for $x ?$

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5.

What real numbers will make the sentence “ $\sin^{2} x + \cos^{2} x = 1$ ” a true statement when substituted for $x ?$

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6.

What natural numbers will make the sentence “ $\sqrt{n}$ is a natural number” a true statement when substituted for $n ?$

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7.

What real numbers will make the sentence

\int_{0}^{y} t^{2} d t > 9

a true statement when substituted for $y ?$

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Subsection Some Set Notation

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In Beginning Activity 1, we indicated that a set is a well-defined collection of objects that can be thought of as an entity itself.

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List 2.13. Elements of Sets

If $A$ is a set and $y$ is one of the objects in the set $A,$ we write $y \in A$ and read this as “ $y$ is an element of $A$ ” or “ $y$ is a member of $A .$ ” For example, if $B$ is the set of all integers greater than 4, then we could write $5 \in B$ and $10 \in B .$
If an object $z$ is not an element in the set $A,$ we write $z \notin A$ and read this as “ $z$ is not an element of $A .$ ” For example, if $B$ is the set of all integers greater than 4, then we could write $- 2 \notin B$ and $4 \notin B .$

When working with a mathematical object, such as set, we need to define when two of these objects are equal. We are also often interested in whether or not one set is contained in another set.

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Definition.

Two sets, $A$ and $B,$ are equal when they have precisely the same elements. In this case, we write $A = B$ . When the sets $A$ and $B$ are not equal, we write $A \neq B .$

The set $A$ is a subset of a set $B$ provided that each element of $A$ is an element of $B .$ In this case, we write $A \subseteq B$ and also say that $A$ is contained in $B .$ When $A$ is not a subset of $B,$ we write $A ⊈ B .$

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Using these definitions, we see that for any set

A,

A = A

and since it is true that for each

x \in U,

x \in A,

then

x \in A,

we also see that

A \subseteq A .

That is, any set is equal to itself and any set is a subset of itself. For some specific examples, we see that:

${1, 3, 5} = {3, 5, 1}$
${4, 8, 12} = {4, 4, 8, 12, 12}$
${5, 10} = {5, 10, 5}$
${5, 10} \neq {5, 10, 15} but {5, 10} \subseteq {5, 10, 15} and {5, 10, 15} ⊈ {5, 10}$

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In each of the first three examples, the two sets have exactly the same elements even though the elements may be repeated or written in a different order.

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Progress Check 2.14. Set Notation.

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(a)

Let $A = {- 4, - 2, 0, 2, 4, 6, 8, \dots} .$ Use correct set notation to indicate which of the following integers are in the set $A$ and which are not in the set $A .$ For example, we could write $6 \in A$ and $5 \notin A .$

10 22 13 - 3 0 - 12

Solution.

$10 \in A,$ $22 \in A,$ $13 \notin A,$ $- 3 \notin A,$ $0 \in A,$ $- 12 \notin A$

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(b)

Use correct set notation (using $=$ or $\subseteq$ ) to indicate which of the following sets are equal and which are subsets of one of the other sets.

\begin{aligned} A & = {3, 6, 9} & B & = {6, 9, 3, 6} \\ C & = {3, 6, 9, \dots} & D & = {3, 6, 7, 9} \\ E & = {9, 12, 15, \dots} & F & = {9, 7, 6, 2} \end{aligned}

Solution.

$A = B,$ $A \subseteq B,$ $B \subseteq A,$ $A \subseteq C,$ $A \subseteq D,$ $B \subseteq C,$ $B \subseteq D$

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Subsection Variables and Open Sentences

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As we have seen in the beginning activities, not all mathematical sentences are statements. This is often true if the sentence contains a variable. The following terminology is useful in working with sentences and statements.

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Definition.

An open sentence is a sentence $P (x_{1}, x_{2}, \dots, x_{n})$ involving variables $x_{1}, x_{2}, \dots, x_{n}$ with the property that when specific values from the universal set are assigned to $x_{1}, x_{2}, \dots, x_{n},$ then the resulting sentence is either true or false. That is, the resulting sentence is a statement. An open sentence is also called a predicate or a propositional function.

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Notation.

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One reason an open sentence is sometimes called a propositional function is the fact that we use function notation

P (x_{1}, x_{2}, \dots, x_{n})

for an open sentence in

n

variables. When there is only one variable, such as

x,

we write

P (x),

which is read “

P

x .

” In this notation,

x

represents an arbitrary element of the universal set, and

P (x)

represents a sentence. When we substitute a specific element of the universal set for

x,

the resulting sentence becomes a statement. This is illustrated in the next example.

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Example 2.15. Open Sentences.

If the universal set is $R,$ then the sentence “ $x^{2} - 3 x - 10 = 0$ ” is an open sentence involving the one variable $x .$

If we substitute $x = 2,$ we obtain the false statement “ $2^{2} - 3 \cdot 2 - 10 = 0 .$ ”

If we substitute $x = 5,$ we obtain the true statement “ $5^{2} - 3 \cdot 5 - 10 = 0 .$ ”

In this example, we can let

P (x)

be the predicate “

x^{2} - 3 x - 10 = 0

” and then say that

P (2)

is false and

P (5)

is true.

Using similar notation, we can let $Q (x, y)$ be the predicate “ $x + 2 y = 7 .$ ” This predicate involves two variables. Then,

$Q (1, 1)$ is false since “ $1 + 2 \cdot 1 = 7$ ” is false; and

$Q (3, 2)$ is true since “ $3 + 2 \cdot 2 = 7$ ” is true.

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Progress Check 2.16.

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(a)

Assume the universal set for all variables is $Z$ and let $P (x)$ be the predicate “ $x^{2} \leq 4 .$ ”

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(i)

Find two values of $x$ for which $P (x)$ is false.

Solution.

Two values of $x$ for which $P (x)$ is false are $x = 3$ and $x = - 4 .$

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(ii)

Find two values of $x$ for which $P (x)$ is true.

Solution.

The set of all $x$ for which $P (x)$ is true is the set ${- 2, - 1, 0, 1, 2} .$

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(iii)

Use the roster method to specify the set of all $x$ for which $P (x)$ is true.

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(b)

Assume the universal set for all variables is $Z,$ and let $R (x, y, z)$ be the predicate “ $x^{2} + y^{2} = z^{2} .$ ”

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(i)

Find two different examples for which $R (x, y, z)$ is false.

Solution.

Two examples for which $R (x, y, z)$ is false are: $x = 1,$ $y = 1,$ $z = 1$ and $x = 3,$ $y = - 1,$ $z = 5 .$

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(ii)

Find two different examples for which $R (x, y, z)$ is true.

Solution.

Two examples for which $R (x, y, z)$ is true are: $x = 3,$ $y = 4,$ $z = 5$ and $x = 5,$ $y = 12,$ $z = 13 .$

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Without using the term, Example 2.15 and Progress Check 2.16 (and Beginning Activity 2) dealt with a concept called the truth set of a predicate.

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Definition.

The truth set of an open sentence with one variable is the collection of objects in the universal set that can be substituted for the variable to make the predicate a true statement.

One part of elementary mathematics consists of learning how to solve equations. In more formal terms, the process of solving an equation is a way to determine the truth set for the equation, which is an open sentence. In this case, we often call the truth set the solution set. Following are three examples of truth sets.

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If the universal set is $R,$ then the truth set of the equation $3 x - 8 = 10$ is the set ${6} .$
If the universal set is $R,$ then the truth set of the equation “ $x^{2} - 3 x - 10 = 0$ ” is ${- 2, 5} .$
If the universal set is $N,$ then the truth set of the open sentence “ $\sqrt{n} \in N$ ” is ${1, 4, 9, 16, \dots} .$

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Subsection Set Builder Notation

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Sometimes it is not possible to list all the elements of a set. For example, if the universal set is

R,

we cannot list all the elements of the truth set of “

x^{2} < 4 .

” In this case, it is sometimes convenient to use the so-called set builder notation in which the set is defined by stating a rule that all elements of the set must satisfy. If

P (x)

is a predicate in the variable

x,

then the notation

{x \in U ∣ P (x)}

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stands for the set of all elements

x

in the universal set

U

for which

P (x)

is true. If it is clear what set is being used for the universal set, this notation is sometimes shortened to

{x ∣ P (x)} .

This is usually read as “the set of all

x

such that

P (x) .

” The vertical bar stands for the phrase “such that.” Some writers will use a colon (:) instead of the vertical bar.

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For a non-mathematical example,

P

could be the property that a college student is a mathematics major. Then

{x ∣ P (x)}

denotes the set of all college students who are mathematics majors. This could be written as

{x ∣ x is a  college student who is a mathematics major} .

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Example 2.17. Truth Sets.

Assume the universal set is $R$ and $P (x)$ is “ $x^{2} < 4 .$ ” We can describe the truth set of $P (x)$ as the set of all real numbers whose square is less than 4. We can also use set builder notation to write the truth set of $P (x)$ as

{x \in R ∣ x^{2} < 4} .

However, if we solve the inequality $x^{2} < 4,$ we obtain $- 2 < x < 2 .$ So we could also write the truth set as

{x \in R ∣ - 2 < x < 2} .

We could read this as the set of all real numbers that are greater than $- 2$ and less than 2. We can also write

{x \in R ∣ x^{2} < 4} = {x \in R ∣ - 2 < x < 2} .

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Progress Check 2.18. Working with Truth Sets.

Let $P (x)$ be the predicate “ $x^{2} \leq 9 .$ ”

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(a)

If the universal set is $R,$ describe the truth set of $P (x)$ using English and write the truth set of $P (x)$ using set builder notation.

Solution.

The truth set is the set of all real numbers whose square is less than or equal to 9. The truth set is ${x \in R | x^{2} \leq 9} = {x \in R | - 3 \leq x \leq 3} .$

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(b)

If the universal set is $Z,$ then what is the truth set of $P (x) ?$ Describe this set using English and then use the roster method to specify all the elements of this truth set.

Solution.

The truth set is the set of all integers whose square is less than or equal to 9. The truth set is ${- 3, - 2, - 1, 0, 1, 2, 3} .$

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(c)

Are the truth sets in Task 2.18.a and Task 2.18.b equal? Explain.

Solution.

The truth sets in Parts (1) and (2) equal are not equal. One purpose of this progress check is to show that the truth set of a predicate depends on the predicate and on the universal set.

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So far, our standard form for set builder notation has been

{x \in U ∣ P (x)} .

It is sometimes possible to modify this form and put the predicate first. For example, the set

A = {3 n + 1 ∣ n \in N}

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describes the set of all natural numbers of the form

3 n + 1

for some natural number. By substituting 1, 2, 3, 4, and so on, for

n,

we can use the roster method to write

A = {3 n + 1 ∣ n \in N} = {4, 7, 10, 13, \dots} .

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We can sometimes “reverse this process” by starting with a set specified by the roster method and then writing the same set using set builder notation.

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Example 2.19. Set Builder Notation.

Let $B = {\dots, - 11, - 7, - 3, 1, 5, 9, 13, \dots} .$ The key to writing this set using set builder notation is to recognize the pattern involved. We see that once we have an integer in $B,$ we can obtain another integer in $B$ by adding 4. This suggests that the predicate we will use will involve multiplying by 4.

Since it is usually easier to work with positive numbers, we notice that $1 \in B$ and $5 \in B .$ Notice that

1 = 4 \cdot 0 + and 5 = 4 \cdot 1 + 1 .

This suggests that we might try ${4 n + 1 ∣ n \in Z} .$ In fact, by trying other integers for $n,$ we can see that

B = {\dots, - 11, - 7, - 3, 1, 5, 9, 13, \dots} = {4 n + 1 ∣ n \in Z} .

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Progress Check 2.20. Set Builder Notation.

Each of the following sets is defined using the roster method.

\begin{aligned} A & = {1, 5, 9, 13, \dots} \\ B & = {\dots, - 8, - 6, - 4, - 2, 0} \\ C & = {\sqrt{2}, {(\sqrt{2})}^{3}, {(\sqrt{2})}^{5}, \dots} \\ D & = {1, 3, 9, 27, \dots} \end{aligned}

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(a)

Determine four elements of each set other than the ones listed using the roster method.

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(b)

Use set builder notation to describe each set.

Solution.

\begin{aligned} A & = {4 n - 3 ∣ n \in N} = {x \in N ∣ x = 4 n - 3 for some natural number n} . \\ B & = {- 2 n ∣ n is a nonnegative integer} . \\ C & = {{(\sqrt{2})}^{2 m - 1} ∣ m \in N} = {{(\sqrt{2})}^{n} ∣ n is an odd natural number} . \\ D & = {3^{n} ∣ n is a nonnegative integer} . \end{aligned}

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Subsection The Empty Set

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When a set contains no elements, we say that the set is the empty set. For example, the set of all rational numbers that are solutions of the equation

x^{2} = - 2

is the empty set since this equation has no solutions that are rational numbers.

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In mathematics, the empty set is usually designated by the symbol

\emptyset .

We usually read the symbol

\emptyset

as “the empty set” or “the null set.” (The symbol

\emptyset

is actually the next to last letter in the Danish-Norwegian alphabet.)

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Subsection When the Truth Set Is the Universal Set

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The truth set of a predicate can be the universal set. For example, if the universal set is the set of real numbers

R,

then the truth set of the predicate “

x + 0 = x

” is

R .

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Notice that the sentence “

x + 0 = x

” has not been quantified and a particular element of the universal set has not been substituted for the variable

x .

Even though the truth set for this sentence is the universal set, we will adopt the convention that unless the quantifier is stated explicitly, we will consider the sentence to be a predicate or open sentence. So, with this convention, if the universal set is

R,

then

$x + 0 = x$ is a predicate;
For each real number $x,$ $(x + 0 = x)$ is a statement.

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Exercises Exercises

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1.

Use the roster method to specify the elements in each of the following sets and then write a sentence in English describing the set.

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(a)

${x \in R ∣ 2 x^{2} + 3 x - 2 = 0}$

Answer.

The set of all real number solutions of the equation $2 x^{2} + 3 x - 2 = 0,$ which is ${\frac{1}{2}, - 2} .$

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(b)

${x \in Z ∣ 2 x^{2} + 3 x - 2 = 0}$

Answer.

The set of all integer solutions of the equation $2 x^{2} + 3 x - 2 = 0,$ which is ${- 2} .$

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(c)

${x \in Z ∣ x^{2} < 25}$

Answer.

The set of all integers whose square is less than 25, which is ${- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4} .$

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(d)

${x \in N ∣ x^{2} < 25}$

Answer.

The set of all natural numbers whose square is less than 25, which is ${1, 2, 3, 4}$

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(e)

${y \in Q ∣ | y - 2 | = 2.5}$

Answer.

The set of all rational numbers that are 2 units from 2.5 on the number line, which is ${- 0.5, 4.5} .$

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(f)

${y \in Z ∣ | y - 2 | \leq 2.5}$

Answer.

The set of all integers that are less than or equal to 2.5 units from 2 on the number line, which is ${0, 1, 2, 3, 4} .$

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2.

Each of the following sets is defined using the roster method.

\begin{aligned} A & = {1, 4, 9, 16, 25, \dots} & B & = {\dots, - π^{4}, - π^{3}, - π^{2}, - π, - 1} \\ C & = {3, 9, 15, 21, 27, \dots} & D & = {0, 4, 8, \dots, 96, 100} \end{aligned}

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(a)

Determine four elements of each set other than the ones listed using the roster method.

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(b)

Use set builder notation to describe each set.

Answer.

Possible answers:

\begin{aligned} A & = {n^{2} ∣ n \in N} \\ B & = {- π^{n} ∣ n ~is a nonnegative integer} \\ C & = {6 n + 3 ∣ n ~is a nonnegative integer} = {6 n - 3 ∣ n \in N} \\ D & = {4 n ∣ n \in Z ~and~ 0 \leq n \leq 25} \end{aligned}

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3.

Let $A = {x \in R ∣ x {(x + 2)}^{2} (x - \frac{3}{2}) = 0} .$ Which of the following sets are equal to the set $A$ and which are subsets of $A ?$

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(a)

${- 2, 0, 3}$

🔗

(b)

${\frac{3}{2}, - 2, 0}$

Answer.

This set is equal to the given set.

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(c)

${- 2, - 2, 0, \frac{3}{2}}$

Answer.

This set is equal to the given set.

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(d)

${- 2, \frac{3}{2}}$

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4.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $Z .$

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(a)

$n + 7 = 4 .$

Answer.

${- 3}$

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(b)

$n^{2} = 64 .$

Answer.

${8, 8}$

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(c)

$\sqrt{n} \in N$ and $n$ is less than 50.

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(d)

$n$ is an odd integer that is greater than 2 and less than 14.

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(e)

$n$ is an even integer that is greater than 10.

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5.

Use set builder notation to specify the following sets:

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(a)

The set of all integers greater than or equal to 5.

Answer.

${x \in Z ∣ x \geq 5}$

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(b)

The set of all even integers.

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(c)

The set of all positive rational numbers.

Answer.

${x \in Q ∣ x > 0}$

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(d)

The set of all real numbers greater than 1 and less than 7.

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(e)

The set of all real numbers whose square is greater than 10.

Answer.

${x \in R ∣ x^{2} > 10}$

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6.

For each of the following sets, use English to describe the set and when appropriate, use the roster method to specify all of the elements of the set.

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(a)

${x \in R | - 3 \leq x \leq 5}$

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(b)

${x \in Z | - 3 \leq x \leq 5}$

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(c)

${x \in R | x^{2} = 16}$

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(d)

${x \in R | x^{2} + 16 = 0}$

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(e)

${x \in Z | x is odd}$

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(f)

${x \in R | 3 x - 4 \geq 17}$

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Activity 7. Closure Explorations.

In Section 1.1, we studied some of the closure properties of the standard number systems. (See Closure Properties of Number Systems.) We can extend this idea to other sets of numbers. So we say that:

A set $A$ of numbers is closed under addition provided that whenever $x$ and $y$ are in the set $A,$ $x + y$ is in the set $A .$
A set $A$ of numbers is closed under multiplication provided that whenever $x$ and $y$ are are in the set $A,$ $x \cdot y$ is in the set $A .$
A set $A$ of numbers is closed under subtraction provided that whenever $x$ and $y$ are are in the set $A,$ $x - y$ is in the set $A .$

For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations.

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(a)

The set of all odd natural numbers

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(b)

The set of all even integers

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(c)

$A = {1, 4, 7, 10, 13, \dots}$

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(d)

$B = {\dots, - 6, - 3, 0, 3, 6, 9, \dots}$

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(e)

$C = {3 n + 1 | n \in Z}$

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(f)

$D = {\frac{1}{2^{n}} | n \in N}$

Section 2.3 Open Sentences and Sets

Beginning Activity Beginning Activity 1: Sets and Set Notation A4US

1.

(a)

(b)

(c)

2.

(a)

(b)

(c)

(d)

Beginning Activity Beginning Activity 2: Variables A4US

1.

(a)

(b)

Definition.

2.

3.

4.

5.

6.

7.

Subsection Some Set Notation

Definition.

Progress Check 2.14. Set Notation.

(a)

(b)

Subsection Variables and Open Sentences

Definition.

Notation.

Example 2.15. Open Sentences.

Progress Check 2.16.

(a)

(i)

(ii)

(iii)

(b)

(i)

(ii)

Definition.

Subsection Set Builder Notation

Example 2.17. Truth Sets.

Progress Check 2.18. Working with Truth Sets.

(a)

(b)

(c)

Example 2.19. Set Builder Notation.

Progress Check 2.20. Set Builder Notation.

(a)

(b)

Subsection The Empty Set

Subsection When the Truth Set Is the Universal Set

Exercises Exercises

1.

(a)

(b)

(c)

(d)

(e)

(f)

2.

(a)

(b)

3.

(a)

(b)

(c)

(d)

4.

(a)

(b)

(c)

(d)

(e)

5.

(a)

(b)

(c)

(d)

(e)

Beginning Activity Beginning Activity 1: Sets and Set Notation
A4 US

Beginning Activity Beginning Activity 2: Variables
A4 US