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2.1 – Symbols and Terminology
Definitions:
Set: A collection of objects.
Elements: The objects that belong to the set.
Set Designations (3 types):
Word Descriptions:
The set of even counting numbers less than ten.
Listing method:
{2, 4, 6, 8}
Set Builder Notation:
{x | x is an even counting number less than 10}
2.1 – Symbols and Terminology
Definitions:
Empty Set: A set that contains no elements. It is
also known as the Null Set. The symbol is 
List all the elements of the following sets.
The set of counting numbers between six and
thirteen.
{7, 8, 9, 10, 11, 12}
{5, 6, 7,…., 13}
{5, 6, 7, 8, 9, 10, 11, 12, 13}
{x | x is a counting number between 6 and 7}
{}
Null set

Empty set
2.1 – Symbols and Terminology
Symbols:
∈: Used to replace the words “is an element of.”
∉: Used to replace the words “is not an element of.”
True or False:
3 ∈ {1, 2, 5, 9, 13}
False
0 ∈ {0, 1, 2, 3}
True
-5 ∉ {5, 10, 15, , }
True
2.1 – Symbols and Terminology
Sets of Numbers and Cardinality
Cardinal Number or Cardinality:
The number of distinct elements in a set.
Notation
n(A): n of A; represents the cardinal number of a
set.
K = {2, 4, 8, 16}
n(K) = 4
∅
n(∅) = 0
R = {1, 2, 3, 2, 4, 5}
P = {∅}
n(R) = 5
n(P) = 1
2.1 – Symbols and Terminology
Finite and Infinite Sets
Finite set: The number of elements in a set are countable.
Infinite set: The number of elements in a set are not
countable
{2, 4, 8, 16}
Countable = Finite set
{1, 2, 3, …}
Not countable = Infinite set
2.1 – Symbols and Terminology
Equality of Sets
Set A is equal to set B if the following conditions are met:
1. Every element of A is an element of B.
2. Every element of B is an element of A.
Are the following sets equal?
{–4, 3, 2, 5} and {–4, 0, 3, 2, 5}
Not equal
{3} = {x | x is a counting number between 2 and 5}
Not equal
{11, 12, 13,…} = {x | x is a natural number greater than 10}
Equal
2.2 – Venn Diagrams and Subsets
Definitions:
Universal set: the set that contains every object of interest
in the universe.
Complement of a Set: A set of objects of the universal set
that are not an element of a set inside the universal set.
Notation: A
Venn Diagram: A rectangle represents the universal set and
circles represent sets of interest within the universal set
A
A
U
2.2 – Venn Diagrams and Subsets
Definitions:
Subset of a Set: Set A is a Subset of B if every
element of A is an element of B. Notation: AB
Subset or not?
{3, 4, 5, 6}

{3, 4, 5, 6, 8}
{1, 2, 6}

{2, 4, 6, 8}
{5, 6, 7, 8}

{5, 6, 7, 8}
Note: Every set is a subset of itself.
BB
2.2 – Venn Diagrams and Subsets
Definitions:
Set Equality: Given A and B are sets, then A = B if
AB and BA.
=
{1, 2, 6}
{1, 2, 6}
{5, 6, 7, 8}

{5, 6, 7, 8, 9}
2.2 – Venn Diagrams and Subsets
Definitions:
Proper Subset of a Set: Set A is a proper subset of
Set B if AB and A  B. Notation AB
What makes the following statements true?
, , or both
{3, 4, 5, 6} both {3, 4, 5, 6, 8}
{1, 2, 6}
both
{5, 6, 7, 8}

{1, 2, 4, 6, 8}
{5, 6, 7, 8}
The empty set () is a subset and a proper subset of
every set except itself.
2.2 – Venn Diagrams and Subsets
Number of Subsets
The number of subsets of a set with n elements is:
2n
Number of Proper Subsets
The number of proper subsets of a set with n
elements is:
2n – 1
List the subsets and proper subsets
{1, 2}
22 = 4
Subsets: {1} {2} {1,2} 
Proper subsets:
{1}
{2}

22 – 1= 3
2.2 – Venn Diagrams and Subsets
List the subsets and proper subsets
{a, b, c}
{a} {b} {c}
Subsets:
{a, b}
{a, c}
{b, c}
23 = 8
{b, c}
23 – 1 = 7

{a, b, c}
Proper subsets:
{a} {b} {c}
{a, b}
{a, c}

2.3 – Set Operations and Cartesian Products
Intersection of Sets: The intersection of sets A and B
is the set of elements common to both A and B.
A  B = {x | x  A and x  B}
{1, 2, 5, 9, 13}  {2, 4, 6, 9}
{2, 9}
{a, c, d, g}  {l, m, n, o}

{4, 6, 7, 19, 23}  {7, 8, 19, 20, 23, 24}
{7, 19, 23}
2.3 – Set Operations and Cartesian Products
Union of Sets: The union of sets A and B is the set of
all elements belonging to each set.
A  B = {x | x  A or x  B}
{1, 2, 5, 9, 13}  {2, 4, 6, 9}
{1, 2, 4, 5, 6, 9, 13}
{a, c, d, g}  {l, m, n, o}
{a, c, d, g, l, m, n, o}
{4, 6, 7, 19, 23}  {7, 8, 19, 20, 23, 24}
{4, 6, 7, 8, 19, 20, 23, 24}
2.3 – Set Operations and Cartesian Products
Find each set.
U = {1, 2, 3, 4, 5, 6, 9}
A = {1, 2, 3, 4}
B = {2, 4, 6}
C = {1, 3, 6, 9}
AB
{1, 2, 3, 4, 6}
A  B
A = {5, 6, 9}
B  C
B = {1, 3, 5, 9)}
{6}
C = {2, 4, 5}
{1, 2, 3, 4, 5, 9}
B  B

2.3 – Set Operations and Cartesian Products
Find each set.
U = {1, 2, 3, 4, 5, 6, 9}
A = {1, 2, 3, 4}
B = {2, 4, 6}
C = {1, 3, 6, 9}
A = {5, 6, 9} B = {1, 3, 5, 9)} C = {2, 4, 5}
(A  C)  B
A  C {2, 4, 5, 6, 9}
{2, 4, 5, 6, 9}  B
{5, 9}
2.3 – Set Operations and Cartesian Products
Difference of Sets: The difference of sets A and B is the
set of all elements belonging set A and not to set B.
A – B = {x | x  A and x  B}
U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 2, 3, 4, 5, 6} B = {2, 3, 6}
C = {3, 5, 7}
A = {7}
B = {1, 4, 5, 7}
C = {1, 2, 4, 6}
Find each set.
A–B
{1, 4, 5}
B–A
Note: A – B  B – A
(A – B)  C
{1, 2, 4, 5, 6, }

2.3 – Set Operations and Cartesian Products
Ordered Pairs: in the ordered pair (a, b), a is the first
component and b is the second component. In
general, (a, b)  (b, a)
Determine whether each statement is true or false.
(3, 4) = (5 – 2, 1 + 3) True
{3, 4}  {4, 3} False
(4, 7) = (7, 4)
False
2.3 – Set Operations and Cartesian Products
Cartesian Product of Sets: Given sets A and B, the
Cartesian product represents the set of all ordered
pairs from the elements of both sets.
A  B = {(a, b) | a  A and b  B}
Find each set. A = {1, 5, 9}
B = {6,7}
AB
{ (1, 6), (1, 7), (5, 6), (5, 7), (9, 6), (9, 7) }
BA
{ (6, 1), (6, 5), (6, 9), (7, 1), (7, 5), (7, 9) }
2.3 – Venn Diagrams and Subsets
Shading Venn Diagrams:
AB
A
B
U
A
U
A
B
U
B
2.3 – Venn Diagrams and Subsets
Shading Venn Diagrams:
AB
A
B
U
A
U
A
B
U
B
2.3 – Venn Diagrams and Subsets
Shading Venn Diagrams:
A  B
A
B
U
A
A
U
B
A
B
U
A  B in yellow
2.3 – Venn Diagrams and Subsets
Locating Elements in a Venn Diagram
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 3, 4, 5, 6}
B = {4, 6, 8}
Start with A  B
7
1
Fill in each subset of U.
A
B
4
2
3
Fill in remaining elements
of U.
8
6
5
U
9
10
2.3 – Venn Diagrams and Subsets
Shade a Venn diagram for the given statement.
(A  B)  C
Work with the parentheses.
(A  B)
A
B
C
U
2.3 – Venn Diagrams and Subsets
Shade a Venn diagram for the given statement.
(A  B)  C
Work with the parentheses.
(A  B)
Work with the remaining part of
the statement.
(A  B)  C
A
B
C
U
2.3 – Venn Diagrams and Subsets
Shade a Venn diagram for the given statement.
(A  B)  C
Work with the parentheses.
(A  B)
Work with the remaining part of
the statement.
(A  B)  C
A
B
C
U
2.4 –Surveys and Cardinal Numbers
Surveys and Venn Diagrams
Financial Aid Survey of a Small College (100 sophomores).
49 received Government grants
55 received Private scholarships
43 received College aid
G
23 received Gov. grants & Pri. scholar.
16
P
15
18 received Gov. grants & College aid
12
8
28 received Pri. scholar. & College aid
20
10
8 received funds from all three
(PC) – (GPC)
28 – 8 = 20
5
C
43 – (10 + 8 +20) = 5
(GC) – (GPC)
18 – 8 = 10
55 – (15 + 8 + 20) = 12
(GP) – (GPC)
23 – 8 = 15
49 – (15 + 8 + 10) = 16
U
14
100 – (16+15 + 8 + 10+12+20+5) = 14
2.4 –Surveys and Cardinal Numbers
Cardinal Number Formula for a Region
For any two sets A and B,
n  A B   n( A)  n( B)  n( A B).
Find n(A) if n(AB) = 78, n(AB) = 21, and n(B) = 36.
n(AB) = n(A) + n(B ) – n(AB)
78 = n(A) + 36 – 21
78 = n(A) + 15
63 = n(A)
9.1 – Points, Line, Planes and Angles
Definitions:
A point has no magnitude and no size.
A line has no thickness and no width and it extends
indefinitely in two directions.
A plane is a flat surface that extends infinitely.
m
A
D

E
9.1 – Points, Line, Planes and Angles
Definitions:
A point divides a line into two half-lines, one on each side of
the point.
A ray is a half-line including an initial point.
A line segment includes two endpoints.
N
E
D

F
G
9.1 – Points, Line, Planes and Angles
Summary:
Name
Figure
Line AB or BA
Half-line AB
Ray AB
A
A
A
BA
B
AB
B
A
BA
AB
B
A
Ray BA
Segment AB or
Segment BA
AB
B
A
Half-line BA
Symbol
BA
B
B
AB
BA
9.1 – Points, Line, Planes and Angles
Definitions:
Parallel lines lie in the same plane and never meet.
Two distinct intersecting lines meet at a point.
Skew lines do not lie in the same plane and do not meet.
Parallel
Intersecting
Skew
9.1 – Points, Line, Planes and Angles
Definitions:
Parallel planes never meet.
Two distinct intersecting planes meet and form a straight line.
Parallel
Intersecting
9.1 – Points, Line, Planes and Angles
Definitions:
An angle is the union of two rays that have a common endpoint.
A
Vertex
B
1
C
An angle can be named using the following methods:
– with the letter marking its vertex, B
– with the number identifying the angle, 1
– with three letters, ABC.
1) the first letter names a point one side;
2) the second names the vertex;
3) the third names a point on the other side.
9.1 – Points, Line, Planes and Angles
Angles are measured by the amount of rotation in degrees.
Classification of an angle is based on the degree measure.
Measure
Name
Between 0° and 90°
Acute Angle
90°
Right Angle
Greater than 90° but less
than 180°
180°
Obtuse Angle
Straight Angle
9.1 – Points, Line, Planes and Angles
When two lines intersect to form right
angles they are called perpendicular.
Vertical angles are formed when two lines intersect.
A
D
B
E
C
ABC and DBE are one pair of vertical angles.
DBA and EBC are the other pair of vertical angles.
Vertical angles have equal measures.
9.1 – Points, Line, Planes and Angles
Complementary Angles and Supplementary Angles
If the sum of the measures of two acute angles is 90°, the
angles are said to be complementary.
Each is called the complement of the other.
Example: 50° and 40° are complementary angles.
If the sum of the measures of two angles is 180°, the angles
are said to be supplementary.
Each is called the supplement of the other.
Example: 50° and 130° are supplementary angles
9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(5x – 10)°
(3x + 10)°
Vertical angels are equal.
3x + 10 = 5x – 10
2x = 20
x = 10
Each angle is 3(10) + 10 = 40°.
9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(2x + 45)°
(x – 15)°
Supplementary angles.
2x + 45 + x – 15 = 180
3x + 30 = 180
3x = 150
x = 50
2(50) + 45 = 145
50 – 15 = 35
35° + 145° = 180
9.1 – Points, Line, Planes and Angles
1
Parallel Lines cut by a Transversal
line create 8 angles
5
4
3
5 6
7 8
Alternate interior angles
Angle measures are equal.
(also 3 and 6)
1
Alternate exterior angles
Angle measures are equal.
8
(also 2 and 7)
2
4
9.1 – Points, Line, Planes and Angles
1
2
3 4
5 6
7 8
Same Side Interior angles
Angle measures add to 180°.
4
6
(also 3 and 5)
2
Corresponding angles
6
Angle measures are equal.
(also 1 and 5, 3 and 7, 4 and 8)
9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(3x – 80)°
(x + 70)°
Alternate interior angles.
x + 70 = 3x – 80
2x = 150
x = 75
x + 70 =
75 + 70 =
145°
9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(4x – 45)°
(2x – 21)°
Same Side Interior angles.
4x – 45 + 2x – 21 = 180
6x – 66 = 180
6x = 246
x = 41
4(41) – 45
2(41) – 21
164 – 45
82 – 21
119°
61°
180 – 119 = 61°