1 Identify sets.
2 Identify and use inequalities.
3 Use set builder notation.
4 Determine the union and intersection of sets.
5 Identify important sets of numbers.
For instance, if t = the time, in hours, that a car is traveling, then t is a variable since the time is constantly changing as the car is traveling. We often use the letters x, y, z, and t to represent variables. However, other letters may be used.
If a letter represents one particular value it is called a constant. For example, if s = the number of seconds in a minute, then s represents a constant because there are always 60 seconds in a minute. The number of seconds in a minute does not vary. In this book, letters representing both variables and constants are italicized.
The term algebraic expression, or simply expression, will be used often in the text.
An expression is any combination of numbers, variables, exponents, mathematical sym- bols (other than equals signs), and mathematical operations.
1 Identify Sets
A set is a collection of objects. The objects in a set are called elements of the set. Sets are indicated by means of braces, 5 6, and are often named with capital letters. When the elements of a set are listed within the braces, as illustrated below, the set is said to be in roster form.
Set Number of Elements
A = 5a, b, c6 3
B = 5yellow, green, blue, red6 4
C = 51, 2, 3, 4, 56 5
The symbol ∈ is used to indicate that an item is an element of a set. Since 2 is an element of set C we may write
2 ∈ C
2 is an element of C This is read “2 is an element of the set C.”
A set may be finite or infinite. Sets A, B, and C each have a finite number of elements and are therefore finite sets. In some sets it is impossible to list all the elements. These are infinite sets. The following set, called the set of natural numbers or counting numbers, is an example of an infinite set.
N = 51, 2, 3, 4, 5,c6
The three dots after the last comma are called an ellipsis. They indicate that the set con- tinues on and on in the same manner.
Another important infinite set is the integers. The set of integers follows.
I = 5c, -4, -3, -2, -1, 0, 1, 2, 3, 4,c6
Notice that the set of integers includes both positive and negative integers and the number 0.
Variable
When a letter is used to represent various numbers it is called a variable.
⁄ ⁄ ⁄
Understanding Algebra
The positive integers are 1, 2, 3, 4, 5, 6, c The negative integers are
-1, -2, -3, -4, -5, -6, c
If we write
D = 51, 2, 3, 4, 5,c, 1636
we mean that the set continues in the same manner until the number 163. Set D is the set of the first 163 natural numbers. D is therefore a finite set.
A special set that contains no elements is called the null set, or empty set, written 5 6 or ∅. For example, the set of students in your class under 3 years of age is the null or empty set.
2 Identify and Use Inequalities
Inequality Symbols
7 is read “is greater than.”
Ú is read “is greater than or equal to.”
6 is read “is less than.”
… is read “is less than or equal to.”
≠ is read “is not equal to.”
Inequalities can be explained using the real number line (Fig. 1.1).
5 6 4 3 2 1 0 24
25
26 232221
FIGURE 1.1
The number a is greater than the number b, a 7 b, when a is to the right of b on the number line (Fig. 1.2). We can also state that the number b is less than a, b 6 a, when b is to the left of a on the number line. The inequality a ≠ b means either a 6 b or a 7 b.
a a . b or b , a b
Lesser Greater
FIGURE 1.2
EXAMPLE 1 Insert either 7 or 6 in the shaded area between the numbers to make each statement true.
a) 2 6 b) 1 -7 c) -5 -4
Solution Indicate the location of the numbers in parts a), b), and c) on a number line as shown in Figure 1.3.
5 6 7 4 3 2 1 0 24
25 26
27 232221
FIGURE 1.3
a) Because 2 is to the left of 6 on the number line, 2 is less than 6, and we write 2 6 6.
b) Because 1 is to the right of -7 on the number line, 1 is greater than -7, and we write 1 7 -7.
c) Because -5 is to the left of -4 on the number line, -5 is less than -4, and we write -5 6 -4.
Now Try Exercise 15
HELPFUL HINT
Remember that the symbol used in an inequality, if it is true, always points to the smaller of the two numbers.
⁄
Notation Means
x 7 2 x is any real number greater than 2.
x … -3 x is any real number less than or equal to -3.
-4 … x 6 3 x is any real number greater than or equal to -4 and less than 3.
In the inequalities x 7 2 and x … -3, the 2 and the -3 are called endpoints. In the in- equality -4 … x 6 3, the -4 and 3 are the endpoints. The solutions to inequalities that use either 6 or 7 do not include the endpoints, but the solutions to inequalities that use either … or Ú do include the endpoints. This is shown as follows:
Endpoint not
included Endpointincluded Below are three illustrations.
Inequality Inequality Indicated on the Number Line x 7 2
5 6 4 3 2 1 0 24
25
26 232221
x … -1 262524232221 0 1 2 3 4 5 6
-4 … x 6 3
5 6 4 3 2 1 0 24
25
26 232221
The word between indicates that the endpoints are not included in the answer. For exam- ple, the set of natural numbers between 2 and 6 is 53, 4, 56. If we wish to include the end- points, we can use the word inclusive. For example, the set of natural numbers between 2 and 6 inclusive is 52, 3, 4, 5, 66.
3 Use Set Builder Notation
A second method of describing a set is called set builder notation. An example of set builder notation is
E = 5xx is a natural number greater than 76
This is read “Set E is the set of all elements x, such that x is a natural number greater than 7.” In roster form, this set is written
E = 58, 9, 10, 11, 12,c6 The general form of set builder notation is
5 x | x has property p 6
The set of ⁄ ⁄ ⁄
all
elements x such that
x has the given property
We often will use the variable x when using set builder notation, although any variable can be used.
Two condensed ways of writing set E = 5xx is a natural number greater than 7} in set builder notation follow.
E = 5xx 7 7 and x∈N6 or E = 5x0x Ú 8 and x∈N6
The set A = 5x-3 6 x … 4 and x∈I6 is the set of integers greater than -3 and less than or equal to 4. The set written in roster form is 5-2, -1, 0, 1, 2, 3, 46. Notice that the endpoint -3 is not included in the set but the endpoint 4 is included.
How do the sets B = 5xx 7 2 and x∈N6 and C = 5xx 7 26 differ? Set B con- tains only the natural numbers greater than 2, that is, 53, 4, 5, 6,c6. Set C contains not only the natural numbers greater than 2 but also fractions and decimal numbers greater than 2. Since there is no smallest number greater than 2, this set cannot be written in ros- ter form. We illustrate these two sets on the number line on the top of the next page. We have also illustrated two other sets.
Set Set Indicated on the Number Line 5xx 7 2 and x∈N6
5 6 4 3 2 1 0 24
25
26 232221
? ? ?
5xx 7 26 262524232221 0 1 2 3 4 5 6
5x-1 … x 6 4 and x∈I6 262524232221 0 1 2 3 4 5 6 5x-1 … x 6 46 262524232221 0 1 2 3 4 5 6
Another method of indicating inequalities, called interval notation, will be discussed in Section 2.5.
4 Determine the Union and Intersection of Sets
Just as operations such as addition and multiplication are performed on numbers, opera- tions can be performed on sets. Two set operations are union and intersection.
Union of Two Sets
The union of set A and set B, written A∪B, is the set of elements that belong to either set A or set B.
Because the word or, as used in this context, means belonging to set A or set B or both sets, the union is formed by combining, or joining together, the elements in set A with those in set B. If an item is an element in either set A, or set B, or in both sets, then it is an element in the union of the sets, A∪B. If an element appears in both sets, we list it only once when we write the union of two sets.
Examples of Union of Sets
A = 51, 2, 3, 4, 56, B = 53, 4, 5, 6, 76, A∪B = 51, 2, 3, 4, 5, 6, 76 A = 5a, b, c, d, e6, B = 5x, y, z6, A∪B = 5a, b, c, d, e, x, y, z6 In set builder notation we can express A∪B as
Union
A∪B = 5xx∈A or x∈B6
Intersection of Two Sets
The intersection of set A and set B, written A¨B, is the set of all elements that are common to both set A and set B.
Because the word and, as used in this context, means belonging to both set A and set B, the intersection is formed by using only those elements that are in both set A and set B. If an item is an element in only one of the two sets, then it is not an element in the intersec- tion of the sets.
Examples of Intersection of Sets
A = 51, 2, 3, 4, 56, B = 53, 4, 5, 6, 76, A¨B = 53, 4, 56 A = 5a, b, c, d, e6, B = 5x, y, z6, A¨B = 5 6 or ∅
Note that in the last example, sets A and B have no elements in common. Therefore, their intersection is the empty set. In set builder notation we can express A¨B as
Intersection
A¨B = 5xx∈A and x∈B6
5 Identify Important Sets of Numbers
In the box below, we describe different sets of numbers and provide letters that are often used to represent these sets of numbers.
Important Sets of Numbers
Real numbers ℝ= 5xx is a point on the number line6 Natural or counting numbers N= 51, 2, 3, 4, 5,c6
Whole numbers W = 50, 1, 2, 3, 4, 5,c6
Integers I = 5c, -3, -2, -1, 0, 1, 2, 3,c6
Rational numbers Q = ep
q2p and q are integers, q≠ 0f Irrational numbers H= 5xx is a real number that is not rational6
A rational number is any number that can be represented as a quotient of two integers, with the denominator not 0.
Examples of Rational Numbers 3
5, -2
3, 0, 1.63, 7, -17, 14
Notice that 0, or any other integer, is also a rational number since it can be written as a fraction with a denominator of 1. For example, 0 = 0
1 and 7 = 7 1. The number 1.63 can be written 163
100 and is thus a quotient of two integers. Since 14 = 2 and 2 is an integer, 14 is a rational number. Every rational number when written as a decimal number will be either a repeating or a terminating decimal number.
Examples of Repeating Decimals Examples of Terminating Decimals 2
3 = 0.6666c 6 repeats.
1 2 = 0.5 1
7 = 0.142857142857c 142857 repeats.
9 4 = 2.25 Understanding Algebra
A rational number can be expressed as the quotient of two integers:
integer integer
where the denominator is not zero.
Understanding Algebra
A rational number whose decimal representation ends is a terminating decimal number.
A rational number whose decimal representation repeats is a repeating decimal number.
To show that a digit or group of digits repeats, we can place a bar above the digit or group of digits that repeat. For example, we may write
2
3 = 0.6 and 1
7 = 0.142857
An irrational number is a real number that is not a rational number. Some irrational numbers are 12, 13, 15, and 16. Another irrational number is pi, p. When we give a decimal value for an irrational number, we are giving only an approximation of the value of the irrational number. The symbol ≈ means “is approximately equal to.”
p ≈ 3.14 12 ≈ 1.41 13 ≈ 1.73 110 ≈ 3.16
The real numbers are formed by taking the union of the rational numbers and the irrational numbers. Therefore, any real number must be either a rational number or an irrational number. The symbol ℝ is often used to represent the set of real numbers.
Figure 1.4 illustrates various real numbers on the number line.
5 6
4 3 2 1 0
0 2 p 4.3
24 25
26 23 22 21
23.62 2285 212
2 207
Some Examples of Real Numbers
√2
√23
FIGURE 1.4 HELPFUL HINT
Remember in writing an approximation, use the symbol ≈.
Subset
The set A is a subset of the set B when every element of A is also an element of B and we write A⊆B.
For example, the set of natural numbers, 51, 2, 3, 4,c6, is a subset of the set of whole numbers, 50, 1, 2, 3, 4,c6, because every element in the set of natural numbers is also an element in the set of whole numbers. Figure 1.5 illustrates the relationships between the various subsets of the real numbers. In Figure 1.5a, you see that the set of natural numbers is a subset of the set of whole numbers, the set of integers, and the set of rational numbers. Therefore, every natural number must also be a whole number, an integer, a rational number, and a real number.
Every natural number is also
• a whole number,
• an integer,
• a rational number, and
• a real number.
Using the same reasoning, we can see that the set of whole numbers is a subset of the in- tegers, the set of rational numbers, and the set of real numbers, and that the set of integers is a subset of the set of rational numbers and the set of real numbers.
Looking at Figure 1.5b, we see that the positive integers, 0, and the negative integers form the integers, that the integers and noninteger rational numbers form the rational numbers, and so on.
Rational numbers Irrational numbers
Integers 25, 29, 2103 Whole numbers
0 Natural numbers 1, 4, 92
Real Numbers
(a) ,
5, 28
273
2p3 2 1.4, 22.35
p 2
√2
√3
√5
√29
FIGURE 1.5
Irrational numbers Positive integers Integers Zero
Negative integers Rational
numbers
Noninteger rational numbers Real
numbers
(b)
EXAMPLE 2 Consider the following set:
e-8, 0, 5
9, 12.25, 17, -111, 22
7, 5, 7.1, -54, pf List the elements of the set that are
a) natural numbers. b) whole numbers. c) integers.
d) rational numbers. e) irrational numbers. f) real numbers.
Solution
a) Natural numbers: 5 b) Whole numbers: 0, 5 c) Integers: -8, 0, 5, -54 d) Rational numbers can be written in the form p
q , q ≠ 0. Each of the following can be written in this form and is a rational number.
-8, 0, 5
9, 12.25, 22
7, 5, 7.1, -54
e) Irrational numbers are real numbers that are not rational. The following numbers are irrational.
17, -111, p
f) All of the numbers in the set are real numbers. The union of the rational numbers and the irrational numbers forms the real numbers.
-8, 0, 5
9, 12.25, 17, -111, 22
7, 5, 7.1, -54, p
Now Try Exercise 39 Not all numbers are real numbers. Some numbers that we discuss later in the text that are not real numbers are complex numbers and imaginary numbers.