Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
set
A well-defined collection of items.
Emphasize that there are two parts to this definition
 well-defined means it is clear what is “in” and what is “not in” the collection
 collection of items means a grouping or aggregation of any type of items
Each individual item in a set is called an element of the set.
Question: How many elements of the set of five pointed stars are in the following
picture?
Answer: The set of five pointed stars in this picture has three elements.
Question: How many elements of the set of four pointed stars are in the following
picture?
Answer: The set of four pointed stars in this picture has five elements.
Using Number Lines to Define Sets
Number lines can also be used to indicate sets of numbers.
-3
-2
-1
0
1
2
3
Lesser
Greater
When using a number line to well-define a set of numbers, we use open and closed dots
to show where the set begins or ends.
Examples:
The number line representation of the set of all numbers greater than 1 is as follows:
-3
Lesser
-2
-1
0
1
2
3
Greater
The number line representation of the set of all numbers greater to or equal to 1 is as
follows:
Chapter 1, Section 1.3 (and 1.1)
Page 1
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
-3
Lesser
-2
-1
0
1
2
3
Greater
Note that the dot on the number line where the number 1 is located is open in the first
example and closed in the second example. The open dot indicates that the number 1 is
not an element of the set of all numbers greater than 1. The closed dot on the second
number line indicates that 1 is an element of the set of all number greater than or equal to
1. Whether the dot is open or closed signifies whether the starting number (or ending
number) is “in” or “not in” the set.
Examples:
The set of all numbers greater than 2 and less than 6 can be shown on a number line as:
The set of all numbers greater than or equal to 2 and less than or equal to 6 can be shown
on a number line as:
The set of all numbers less than 2 or greater than 6 can be shown on a number line as:
The set of all numbers less than or equal to 2 or greater than or equal to 6 can be shown
on a number line as:
Chapter 1, Section 1.3 (and 1.1)
Page 2
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
set-builder notation
A notation used to describe the elements of a set.
Example: The set of all positive real numbers in set builder notation is
x : x   and x  0 This is read as "the set of all values of x such that x is a real
number and x is greater than 0."


Symbol
Translates into English
{
beginning of notation
x
the set of all values of x
:
such that
x 
x is an element of the set of real numbers
and
and
x>0
x is greater than zero
}
ending of notation
Note that set builder notation proceeds from general-to-specific levels of abstraction as
follows:
 always begins and ends with special “set” parentheses  { }
 always specifies the general category of elements in the collection of items (in
the above sample, the variable x)
 always includes a colon (:), which is read “such that”
 always includes one or more distinguishing features or patterns that well-define
the specific characteristics of elements of the collection.
 always concludes with special “set” parentheses  { }
Modeling:
Sample Problem: Use set builder notation to define the set of all dogs that are red and
also named Clifford.
Solution: {dogs: color = red and name = Clifford}
Sample Problem: Use set builder notation to define the set of all teachers at our school
that are male and over forty years old.
Solution: {teachers: workplace = our school and gender = male and age > 40}
Sample Problem: Use set builder notation to define the set of all positive odd integers.
Solution: x : x  0 and and x  divisible by 2 and x  integers
Modeling:
Previous Number Line Examples In Interval Notation
Chapter 1, Section 1.3 (and 1.1)
Page 3
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
-3
-2
-1
0
1
2
3
Lesser
Greater
{x : x  R and x  1}
“the set of x such that x is an element of the set of Real Numbers and x is greater than 1”
-3
Lesser
-2
-1
0
1
2
3
Greater
{x : x  R and x  1}
“the set of x such that x is an element of the set of Real Numbers
and x is equal to or greater than 1”
{x : x  R and x  2 and x  6}
“the set of x such that x is an element of the set of Real Numbers
and x is greater than 2
and x is less than 6”
{x : x  R and x  2 and x  6}
“the set of x such that x is an element of the set of Real Numbers
and x is equal to or greater than 2
and x is equal to or less than 6”
Chapter 1, Section 1.3 (and 1.1)
Page 4
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
Complement of a set
The elements of a universe not contained in a given set; the subset that must be added
to any given subset to yield the original set. The complement of set A is indicated by
A’ or AC
Example:
Universe= interior of the square
A’
A
Set A= the circular region
Complement of A is A’ (or AC)
Complements, Unions, and Intersections of Set
Question: In the set of Integers, what is the complement of the subset of odd integers?
Answer: The subset of even integers.
Question: In the set of all dogs, what is the complement of the subset of dogs named
Rover?
Answer: The subset of all dogs not named Rover.
Source: New York State Education Department Glossary
Question: What is the union of the set of all odd Integers and the set of all even
Integers?
Answer: The set of Integers.
Question: What is the union of the set of {a,b,c,d,e,f,g} and the set of {e,f,g,h,i,j,k}?
Answer: The set {a,b,c,d,e,f,g,h,i,j,k}.
Chapter 1, Section 1.3 (and 1.1)
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Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
Question: What is the intersection of the set of {a,b,c,d,e,f,g} and the set of
{e,f,g,h,i,j,k}?
Answer: The set {e,f,g}.
Question: Can you identify yourself in terms of the intersection of several sets? Are
there other elements in the intersection?
Answers Will Vary: Each person might be described as an individual element in the
intersection of:
 the set of all persons of a certain sex,
 the set of all people born on a certain day,
 the set of all people of a certain race or ethnicity,
 The set of all students in a certain grade,
 Etc., etc., etc.
Using Venn Diagrams to Describe Sets
Venn Diagrams are often used to describe sets. Each circle represents a given set, and all
element of the given set are inside the circle. The complement of the set is outside the
circle. The union of sets in a Venn diagram would consist of all elements inside either
circle in the union. The intersections of sets occurs where the circles overlap, as shown
in the following diagram.
AB
Set B
BC
Set A
Set C
A C
A  BC
A  B  C is shown in green.
Chapter 1, Section 1.3 (and 1.1)
Page 6
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
1 Which interval notation represents the set of all numbers from 2 through 7,
inclusive?
(1) (2,7]
(2) (2,7)
(3) [2,7)
(4) [2,7]
2 The set {1,2,3,4} is equivalent to
(1) {x |1  x  4, where x is a whole number}
(2) {x | 0  x  4, where x is a whole number}
(3) {x | 0  x  4, where x is a whole number}
(4) {x |1  x  4, where x is a whole number}
3
The set {11,12} is equivalent to
(1) {x |11  x  12, where x is an integer}
(2) {x |11  x  12, where x is an integer}
(3) {x |10  x  12, where x is an integer}
(4) {x |10  x  12, where x is an integer}
4 Twelve players make up a high school basketball team. The team jerseys are
numbered 1 through 12. The players wearing the jerseys numbered 3, 6, 7, 8, and
11 are the only players who start a game. Using set notation, list the complement
of this subset.
5 Consider the set of integers greater than -2 and less than 6. A subset of this set is
the positive factors of 5. What is the complement of this subset?
(1) {0, 2, 3, 4}
(2) {-1, 0, 2, 3, 4}
(3) {-2, -1, 0, 2, 3, 4, 6}
(4) {-2, -1, 0, 1, 2, 3, 4, 5, 6}
Chapter 1, Section 1.3 (and 1.1)
Page 7
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
Homework Solutions:
REGENTS QUESTIONS
Which interval notation represents the set of
1 all numbers from 2 through 7, inclusive?
(1) (2,7]
(3) [2,7)
(2) (2,7)
(4) [2,7]
SOLUTIONS
(4)
2 The set {1,2,3,4} is equivalent to
(1) {x |1  x  4, where x is a whole number}
(2) {x | 0  x  4, where x is a whole number}
(3) {x | 0  x  4, where x is a whole number}
(4) {x |1  x  4, where x is a whole number}
(3)
3 The set {11,12} is equivalent to
(4)
(1)
(2)
(3)
(4)
{x |11  x  12, where x is an integer}
{x |11  x  12, where x is an integer}
{x |10  x  12, where x is an integer}
{x |10  x  12, where x is an integer}
4 Twelve players make up a high school
basketball team.
The team jerseys are
numbered 1 through 12. The players wearing
the jerseys numbered 3, 6, 7, 8, and 11 are the
only players who start a game. Using set
notation, list the complement of this subset.
5
Consider the set of integers greater than -2
and less than 6. A subset of this set is the
positive factors of 5. What is the complement
of this subset?
(1) {0, 2, 3, 4}
(2) {-1, 0, 2, 3, 4}
(3) {-2, -1, 0, 2, 3, 4, 6}
(4) {-2, -1, 0, 1, 2, 3, 4, 5, 6}
Chapter 1, Section 1.3 (and 1.1)
Page 8
{1,2,4,5,9,10,12}
(2)
The set of integers greater than -2
and less than 6 is {-1, 0, 1, 2, 3,
4, 5}. The subset of this set that
is positive factors of 5 is {1, 5}.
The complement of this subset is
{-1, 0, 2, 3, 4}.
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
1
REGENTS QUESTIONS
SOLUTIONS
fall0704ia
(4)
Which interval notation represents the set of
all numbers from 2 through 7, inclusive?
(1) (2,7]
(3) [2,7)
(2) (2,7)
(4) [2,7]
2
010917ia
(3)
The set {1,2,3,4} is equivalent to
(1) {x |1  x  4, where x is a whole number}
(2) {x | 0  x  4, where x is a whole number}
(3) {x | 0  x  4, where x is a whole number}
(4) {x |1  x  4, where x is a whole number}
3
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(4)
The set {11,12} is equivalent to
(1) {x |11  x  12, where x is an integer}
(2) {x |11  x  12, where x is an integer}
(3) {x |10  x  12, where x is an integer}
(4) {x |10  x  12, where x is an integer}
4
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{1,2,4,5,9,10,12}
Twelve players make up a high school
basketball team.
The team jerseys are
numbered 1 through 12. The players wearing
the jerseys numbered 3, 6, 7, 8, and 11 are the
only players who start a game. Using set
notation, list the complement of this subset.
5
(2)
The set of integers greater than Consider the set of integers greater than -2
2 and less than 6 is {-1, 0, 1, 2,
and less than 6. A subset of this set is the
3, 4, 5}. The subset of this set
positive factors of 5. What is the complement
that is positive factors of 5 is {1,
of this subset?
5}. The complement of this
(1) {0, 2, 3, 4}
subset is {-1, 0, 2, 3, 4}.
(2) {-1, 0, 2, 3, 4}
060818ia
Chapter 1, Section 1.3 (and 1.1)
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Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
(3) {-2, -1, 0, 2, 3, 4, 6}
(4) {-2, -1, 0, 1, 2, 3, 4, 5, 6}
6
fall0710ia
(3)
Given:
Set A  {( 2,1),( 1,0),(1,8)}
Set B  {( 3,4),( 2,1),( 1,2),(1,8)}.
What is the intersection of sets A and B?
(1) {(1,8)}
(2) {(–2,–1)}
(3) {(–2,–1), (1,8)}
(4) {(–3,–4), (–2,–1), (–1,2), (–1,0), (1,8)}
7
060833ia
Maureen tracks the range of outdoor
temperatures over three days. She records the
following information.
Express the intersection of the three sets as an
inequality in terms of temperature, t.
Chapter 1, Section 1.3 (and 1.1)
Page 10
The intersection of these three
sets is 0  t  40.
Chapter 1: Sets, Operations and Algebraic Language- TEACH
1.1: Numbers, Variables, and Symbols and 1.3: Learning More About Sets
Chapter 1, Section 1.3 (and 1.1)
Page 11