0738_hsm07a1_se_ny_1003.qxd
6/12/07
NY-3
7:08 PM
Page 738
Working With Sets
Learning Standards for
Mathematics
GO for Help
Check Skills You’ll Need
A.A.29 Use set-builder
notation and/or interval
notation to illustrate the
elements of a set, given
the elements in roster
form.
1. List the factors of 12.
Lesson 1-3
2. List the first five prime numbers.
For Exercises 3–5, match each number in the first column with the
corresponding letter in the second column.
3. natural numbers
4. whole numbers
5. integers
A.A.30 Find the complement
of a subset of a given set,
within a given universe.
A. whole numbers and their opposites
B. the nonnegative integers
C. the counting numbers
New Vocabulary • set • roster form • set-builder notation • subset
• null set • universal set • complement of a set
• interval notation
1
2
Part
Graphingwith
y ≠ ax
Sets
1 1 Working
A set is a collection of elements or members. There are various ways to describe sets.
Use braces, { }, to denote a set.
Vocabulary Tip
The word set is often used
to describe a group of
things such as a set of
dishes or a set of luggage.
Roster form lists elements of a set within braces. For example, you can write the
set consisting of 1, 2, and 3 as {1, 2, 3}. Use “ . . . ” to describe an infinite set, such as
the set of natural numbers {1, 2, 3, . . .}.
Set-builder notation describes elements of a set. It uses a variable and limits,
or conditions, on the variable. For example, you read the set-builder notation
{x | x is a factor of 12} as “the set of all values of x such that x is a factor of 12.”
1
EXAMPLE
Using Roster Form and Set-Builder Notation
a. Write “P is the set of whole numbers less than 5” in roster form.
“P is” can be written as “P ⴝ.“
List all whole numbers less than 5.
P ⫽ {0, 1, 2, 3, 4}
b. Write “P is the set of whole numbers less than 5” using set-builder notation.
Use a variable.
Describe limits on the variable.
P ⫽ {x | x is a whole number and x ⬍ 5}
Read “{x |“ as ”the set of all values of x such that.“
Quick Check
NY 738
1 M is the set of odd whole numbers less than 12.
a. Write set M in roster form.
b. Write set M in set-builder notation.
Chapter NY New York Additional Topics
0738_hsm07a1_se_ny_1003.qxd
B
6/12/07
7:08 PM
Page 739
A subset consists of elements from any given set. If B ⫽ {1, 2, 3, 4, 5, 6, 7} and
A {1, 2, 5}, then A is a subset of B. That means A is contained in B and is
indicated by A 債 B.
A
The null set , or empty set, is a set that contains no elements. The null set is a
subset of every set. Use { } or Ø to represent the null set.
2
EXAMPLE
Subsets
List all possible subsets of the set {1, 2, 3}.
{}
Start with the null set.
{1}, {2}, {3}
List the subsets with one element.
{1, 2}, {1, 3}, {2, 3}
List the subsets with two elements.
{1, 2, 3}
List the original set. It is always considered
a subset.
The 8 subsets of {1, 2, 3} are { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}.
Quick Check
2 List the subsets of the set U ⫽ {x, y}.
When working with sets, the largest set you are using is sometimes called the
universal set , or universe. The complement of a set contains the elements
of a universal set not contained in a given set. The complement of set A is
indicated by A⬘.
In the figures below, set U represents the universal set. Notice that
A 債 U and A⬘債 U.
Set A is shaded.
U
The complement of set A is shaded.
U
(4, 3); A
max.
3
EXAMPLE
A’
(–3, –3); min.
Finding the Complement
Given A 債 W, W ⫽ {days of the week}, and A ⫽ {Saturday, Sunday}. Find the
complement of set A.
Elements of W not included in A are Monday, Tuesday, Wednesday, Thursday, and
Friday.
So, A⬘ ⫽ {Monday, Tuesday, Wednesday, Thursday, Friday}.
Quick Check
3 Given P 債 T, T ⫽ {1, 2, 3, 4, 5, 6, 7, 8}, and P ⫽ {2, 4, 6, 8}. Find P⬘.
Lesson NY-3 Working With Sets
NY 739
0738_hsm07a1_se_ny_1003.qxd
2
1
6/12/07
7:08 PM
Page 740
Interval Notation
An inequality such as x ⱕ -2 describes a portion of a number line called an
interval. You can also describe intervals using interval notation. Interval notation
includes three special symbols:
parentheses:
Use ( or ) when endpoints of the interval are not included (⬍ or ⬎).
brackets:
Use [ or ] when endpoints of the interval are included (ⱕ and ⱖ).
infinity:
Use ∞ when the interval continues forever in a positive
direction.
Use -∞ when the interval continues forever in a negative
direction.
Inequality
2≤x≤6
0 1 2 3 4 5 6 7
2<x<6
0 1 2 3 4 5 6 7
2<x≤6
0 1 2 3 4 5 6 7
2≤x<6
0 1 2 3 4 5 6 7
x≥3
0 1 2 3 4 5 6 7
x<3
4
Interval
Notation
Graph
0 1 2 3 4 5 6 7
[2, 6]
(2, 6)
(2, 6]
[2, 6)
[3, ∞)
(–∞, 3)
Using Interval Notation
EXAMPLE
a. Sketch the graph of [-2, 8) and write the interval as an inequality.
–4 –2 0
2
4
6
8 10
The closed dot shows that –2 is included.
The open dot shows that 8 is not included.
Shade between –2 and 8.
The interval notation [⫺2, 8) represents the inequality ⫺2 ⱕ x ⬍ 8.
b. Sketch the graph of x ⱕ 8 and write the inequality as an interval.
–4 –2 0
2
4
6
8 10
The closed dot shows that 8 is included.
Shade to the left of 8.
The interval notation (⫺∞, 8] represents the inequality x ⱕ 8.
Quick Check
NY 740
4 a. Sketch the graph of x ⬍ 5 and write the inequality as an interval.
b. Sketch the graph of (⫺2, 5] and write the interval as an inequality.
Chapter NY New York Additional Topics
0738_hsm07a1_se_ny_1003.qxd
6/12/07
7:08 PM
Page 741
EXERCISES
For more exercises, see Extra Skill and Word Problem Practice.
Practice and Problem Solving
A
Practice by Example
Example 1
GO for
Help
(page NY 738)
Write in roster form.
1. M ⫽ {x | x is a positive multiple of 2 and x ⬍ 18}
2. T ⫽ {x | x is an integer and x ⱖ 12}
3. N ⫽ {n | n is an odd integer, n ⬎ 0}
4. X ⫽ {x | x is an even prime}
Write in set-builder notation.
Example 2
(page NY 739)
Example 3
(page NY 739)
Example 4
(page NY 740)
5. B ⫽ {11, 12, 13, 14, . . .}
6. M ⫽ {2, 3, 5, 7, 11, 13, 17, 19}
7. S ⫽ {1, 2, 3, 4, 6, 12}
8. G ⫽ { . . . , -2, -1, 0, 1, 2, . . . }
List all subsets for the each set.
9. {a, b}
10. {0, 1, 2}
12. Given A 債 B, B ⫽ {1, 2, 3, 4, 5}, and A ⫽ {2, 3}. Find A⬘.
13. Given P 債 U, U ⫽ {1, 2, 3, 4, 5, 6, 7, 8, 9}, and P ⫽ {2, 4, 6, 8}. Find P⬘.
Write interval notation for each graph.
14.
3
2
15.
0 1 0 2to 30
3 feet
4
1Domain:
0 1 02 to 3about 1.5 seconds;1 Range:
3
2
1
16.
B
Apply Your Skills
11. {1, 2, z}
17.
0
1
5
2
4
3
2
1
0
Write each situation in interval notation.
18. Let t be truck weight in tons.
19. Let s be speed in mi/h.
(–3, –2); min.
Write the following inequalities in interval notation.
20. 0 ⱕ x ⱕ 7
21. -4 ⱕ z ⬍ -2
22. -6 ⱕ a ⬍ 5
23. -3 ⱕ x ⬍ 2
24. -3 ⱕ y ⱕ 5
25. -4 ⬍ a ⬍ 2
26. 25 ⬍ z ⱕ 30
27. 15 ⬍ x ⬍ 55
28. -75 ⱕ p ⱕ 150
Suppose U ⫽ {0, 1, 2, 3, 4, 5, 6}, A ⫽ {2, 4, 6}, and B ⫽ {1, 2, 3}. Identify each
statement as true or false. Justify your answer.
29. A 債 U
30. U 債 B
31. B 債 A
32. Ø 債 B
Lesson NY-3 Working With Sets
NY 741
0738_hsm07a1_se_ny_1003.qxd
6/12/07
7:08 PM
Page 742
Nutrition For Exercises 33–35, use the following data. Answer in roster form.
Rosalie’s breakfast consists of peanut butter, toast, grapefruit, and orange juice.
Miguel’s breakfast consists of orange juice, cereal, milk, and toast.
33. Rosalie and Miguel are eating breakfast at the same table. Write the set of their
breakfast foods.
34. Write the set of foods Rosalie and Miguel have in common.
35. Open-Ended Write the set of breakfast foods you eat that are not in Rosalie’s
set or Miguel’s set.
For Exercises 36–39, refer to universal set U {a, b, c, d, e, f, g, 1, 2, 3, 4, 5, 6, 7}.
36. Let P {numbers}. Find P.
37. Let V {letters}. Find V.
38. Let W {even numbers}. Find W.
39. Let M {odd numbers and letters}. Find M.
C
Challenge
If the given set A has n elements, then set A has 2n subsets. How many subsets
are there for each finite set?
40. R {even numbers less than 20}
41. Q { }
REGENTS
Test Prep
Multiple Choice
42. Set U {natural numbers} and set E {2, 4, 6, 8, . . .}. What is E?
A. {1, 3, 5, 7, . . .}
B. {0, 2, 4, 6, 8, . . .}
C. {all positive integers}
D. {2, 4, 6, 8, . . .}
43. Which of the following is interval notation for “all numbers less than 10”?
H. (∞, 10)
J. (∞, 10]
F. { . . . , 10}
G. (∞, 10)
Short Response
44. Write 8 x 4 in interval notation and sketch the graph.
Mixed Review
Lesson 4-5
For Exercises 45–48, solve each compound inequality.
45. 2b 1 7 or 3b 2 26
46. 4 q 3 or 5q 30
47. 2k 6 1 or 2k 6 6
48. 3c 5 10 or 23c 2 2 3
5
49. Write the compound inequality that represents “all real numbers that are less
than -4 or greater than 8.”
Lesson NY-2
For Exercises 50–53 use the data below.
55, 50, 60, 65, 65, 35, 40, 40, 45, 65, 55, 75, 30, 35, 55, 60, 45, 55, 35, 55
50. Make a box-and-whisker plot.
51. Label the median, first quartile, third quartile, and outliers.
52. What are the values for the five number summary?
53. What is the value at the fiftieth percentile?
NY 742
Chapter NY New York Additional Topics