2. 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}

3. 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} 
Empty set
Null set
{ }

4. 2.1 – Symbols and Terminology
∈: 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

5. 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}
n(R) = 5
P = {∅}
n(P) = 1

6. 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

7. 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

8. 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

9. 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

10. 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}

11. 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
both
{3, 4, 5, 6} {3, 4, 5, 6, 8}
both
{1, 2, 6} {1, 2, 4, 6, 8} 
{5, 6, 7, 8} {5, 6, 7, 8}
The empty set () is a subset and a proper subset of every set except itself.

12. 2.2 – Venn Diagrams and Subsets
Number of Subsets
The number of subsets of a set with n elements is: n
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}
Subsets:
{1}
{2}
{1,2} 
22 = 4
Proper subsets:
{1}
{2} 
22 – 1= 3

13. 2.2 – Venn Diagrams and Subsets
List the subsets and proper subsets
{a, b, c}
Subsets:
{a}
{b}
{c}
{a, b}
{a, c}
{b, c}
23 = 8
{a, b, c} 
Proper subsets:
{a}
{b}
{c}
{a, b}
{a, c}
{b, c}
23 – 1 = 7 

14. 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}

15. 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}

16. 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}
{6}
B  C
B = {1, 3, 5, 9)}
C = {2, 4, 5}
{1, 2, 3, 4, 5, 9}
B  B 

17. 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}

18. 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, }

19. 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

20. 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) }

21. 2.3 – Venn Diagrams and Subsets
Shading Venn Diagrams:
A  B A B U A B A B U U

22. 2.3 – Venn Diagrams and Subsets
Shading Venn Diagrams:
A  B A B U U A B A B U

23. 2.3 – Venn Diagrams and Subsets
Shading Venn Diagrams:
A  B A B U A A B A B U U
A  B in yellow

24. 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
Fill in remaining elements of U. 3 8 6 5 U 9 10

25. 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

26. 2.3 – Venn Diagrams and Subsets
Shade a Venn diagram for the given statement.
(A  B)  C
Work with the parentheses.
(A  B) B A C U
Work with the remaining part of the statement.
(A  B)  C

27. 2.3 – Venn Diagrams and Subsets
Shade a Venn diagram for the given statement.
(A  B)  C
Work with the parentheses.
(A  B) B A C U
Work with the remaining part of the statement.
(A  B)  C

28. 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 P
23 received Gov. grants & Pri. scholar. 16 15 12
18 received Gov. grants & College aid 8
28 received Pri. scholar. & College aid 10 20
8 received funds from all three 5
(PC) – (GPC) – 8 = 20
43 – ( ) = 5 C U 14
(GC) – (GPC) 18 – 8 = 10
55 – ( ) = 12
(GP) – (GPC) 23 – 8 = 15
49 – ( ) = 16
100 – ( ) = 14

29. 2.4 –Surveys and Cardinal Numbers
Cardinal Number Formula for a Region
For any two sets A and 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)

30. 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 E D

31. 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 G F

32. 9.1 – Points, Line, Planes and Angles
Summary:
Name
Figure
Symbol
Line AB or BA A B AB BA
Half-line AB A B AB
Half-line BA A B BA
Ray AB A B AB
Ray BA A B BA
Segment AB or Segment BA A B AB BA

33. 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

34. 9.1 – Points, Line, Planes and Angles
Definitions:
Parallel planes never meet.
Two distinct intersecting planes meet and form a straight line.
Parallel
Intersecting

35. 9.1 – Points, Line, Planes and Angles
Definitions:
An angle is the union of two rays that have a common endpoint. A
Side
Vertex B 1
Side 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.

36. 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°
Obtuse Angle
180°
Straight Angle

37. 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.

38. 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

39. 9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(3x + 10)°
(5x – 10)°
Vertical angels are equal.
3x + 10 = 5x – 10
2x = 20
x = 10
Each angle is 3(10) + 10 = 40°.

40. 9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(2x + 45)°
(x – 15)°
Supplementary angles.
2x x – 15 = 180
3x + 30 = 180
3x = 150
x = 50
2(50) + 45 = 145
50 – 15 = 35
35° + 145° = 180

41. 9.1 – Points, Line, Planes and Angles
1 2
Parallel Lines cut by a Transversal line create 8 angles
3 4
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)

42. 9.1 – Points, Line, Planes and Angles
1 2
3 4
5 6
7 8
Same Side Interior angles 4 6
Angle measures add to 180°.
(also 3 and 5) 2
Corresponding angles 6
Angle measures are equal.
(also 1 and 5, 3 and 7, 4 and 8)

43. 9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(3x – 80)°
(x + 70)°
Alternate interior angles.
x + 70 =
x = 3x – 80 =
2x = 150
145°
x = 75

44. 9.1 – Points, Line, Planes and Angles
Find the measure of each marked angle below.
(4x – 45)°
(2x – 21)°
Same Side Interior angles.
4x – x – 21 = 180
4(41) – 45
2(41) – 21
6x – 66 = 180
164 – 45
82 – 21
6x = 246
119°
61°
x = 41
180 – 119 = 61°