44
2.1 – Symbols and TerminologyDefinitions:
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 TerminologyDefinitions:
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}
{x | x is a counting number between 6 and 7}{5, 6, 7, 8, 9, 10, 11, 12, 13}
Empty set Null set { }
2.1 – Symbols and TerminologySymbols:
∈: Used to replace the words “is an element of.”
3 ∈ {1, 2, 5, 9, 13} False
0 ∈ {0, 1, 2, 3}
-5 ∉ {5, 10, 15, , } True
∉: Used to replace the words “is not an element of.”
True or False:
True
2.1 – Symbols and TerminologySets of Numbers and Cardinality
n(A): n of A; represents the cardinal number of a set.K = {2, 4, 8, 16} n(K) = 4
∅R = {1, 2, 3, 2, 4, 5}
n(R) = 5
n(∅) = 0
Cardinal Number or Cardinality:The number of distinct elements in a set.
Notation
P = {∅} n(P) = 1
2.1 – Symbols and TerminologyFinite and Infinite Sets
{2, 4, 8, 16} Countable = Finite set
{1, 2, 3, …} Not countable = Infinite set
Finite set: The number of elements in a set are countable.
Infinite set: The number of elements in a set are not countable
2.1 – Symbols and TerminologyEquality of Sets
{–4, 3, 2, 5} and {–4, 0, 3, 2, 5}
Are the following sets equal?
Equal
Not equal
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.
{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}
2.2 – Venn Diagrams and SubsetsDefinitions:
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
U
A
A
Venn Diagram: A rectangle represents the universal set and circles represent sets of interest within the universal set
2.2 – Venn Diagrams and SubsetsDefinitions:
Subset of a Set: Set A is a Subset of B if every element of A is an element of B. Notation: AB
{3, 4, 5, 6} {3, 4, 5, 6, 8}
BB
Subset or not?
Note: Every set is a subset of itself.
{1, 2, 6} {2, 4, 6, 8}
{5, 6, 7, 8} {5, 6, 7, 8}
2.2 – Venn Diagrams and SubsetsDefinitions:
Set Equality: Given A and B are sets, then A = B if AB and BA.{1, 2, 6} {1, 2, 6}=
{5, 6, 7, 8} {5, 6, 7, 8, 9}
2.2 – Venn Diagrams and SubsetsDefinitions:
The empty set () is a subset and a proper subset of every set except itself.
Proper Subset of a Set: Set A is a proper subset of Set B if AB and A B. Notation AB
{3, 4, 5, 6} {3, 4, 5, 6, 8}both{1, 2, 6} {1, 2, 4, 6, 8}both
{5, 6, 7, 8} {5, 6, 7, 8}
What makes the following statements true? , , or both
2.2 – Venn Diagrams and SubsetsNumber of Subsets
The number of subsets of a set with n elements is: 2n
{1}
List the subsets and proper subsets
Number of Proper SubsetsThe number of proper subsets of a set with n elements is: 2n – 1
{1, 2}{2} {1,2}
{1} {2}
Subsets:
Proper subsets:
22 = 4
22 – 1= 3
2.2 – Venn Diagrams and Subsets
{a}
List the subsets and proper subsets{a, b, c}
{b} {c}
{a, b} {a, c}
Subsets:
Proper subsets:
23 = 8
23 – 1 = 7
{b, c}
{a, b, c}
{a} {b} {c}
{a, b} {a, c} {b, c}
2.3 – Set Operations and Cartesian ProductsIntersection 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 ProductsUnion 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 ProductsFind each set.
A B
U = {1, 2, 3, 4, 5, 6, 9} A = {1, 2, 3, 4} B = {2, 4, 6} C = {1, 3, 6, 9}
{1, 2, 3, 4, 6}
{6}
{1, 2, 3, 4, 5, 9}
A B A = {5, 6, 9}
B C
C = {2, 4, 5}B = {1, 3, 5, 9)}
B B
2.3 – Set Operations and Cartesian ProductsFind each set.
(A C) B
U = {1, 2, 3, 4, 5, 6, 9} A = {1, 2, 3, 4} B = {2, 4, 6} C = {1, 3, 6, 9}
{2, 4, 5, 6, 9}
{5, 9}
A = {5, 6, 9}
A C
C = {2, 4, 5}B = {1, 3, 5, 9)}
{2, 4, 5, 6, 9} B
2.3 – Set Operations and Cartesian ProductsDifference 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}
Note: A – B B – A{1, 4, 5}
{1, 2, 4, 5, 6, }
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} C = {1, 2, 4, 6}B = {1, 4, 5, 7}Find each set.A – B B – A
(A – B) C
2.3 – Set Operations and Cartesian ProductsOrdered Pairs: in the ordered pair (a, b), a is the first
component and b is the second component. In general, (a, b) (b, a)
True(3, 4) = (5 – 2, 1 + 3)
{3, 4} {4, 3}
False
(4, 7) = (7, 4)
Determine whether each statement is true or false.
False
2.3 – Set Operations and Cartesian ProductsCartesian Product of Sets: Given sets A and B, the
Cartesian product represents the set of all ordered pairs from the elements of both sets.
(1, 6),
A = {1, 5, 9}A B
Find each set.A B = {(a, b) | a A and b B}
B = {6,7}
(1, 7), (5, 6), (5, 7), (9, 6), (9, 7){ }
(6, 1),B A
(6, 5), (6, 9), (7, 1), (7, 5), (7, 9){ }
2.3 – Venn Diagrams and SubsetsShading Venn Diagrams:
A B
U
A B
U
A B
U
A B
2.3 – Venn Diagrams and SubsetsShading Venn Diagrams:
A B
U
A B
U
A B
U
A B
2.3 – Venn Diagrams and SubsetsShading Venn Diagrams:
A B
U
A B
U
A B
U
A B
A B in yellow
A
2.3 – Venn Diagrams and SubsetsLocating Elements in a Venn Diagram
Start with A B
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {2, 3, 4, 5, 6} B = {4, 6, 8}
A B
U
6
4
3
5
82
Fill in each subset of U.
Fill in remaining elements of U.
7
9 10
1
2.3 – Venn Diagrams and SubsetsShade a Venn diagram for the given statement.
(A B) CWork with the parentheses. (A B)
A B
CU
2.3 – Venn Diagrams and SubsetsShade a Venn diagram for the given statement.
(A B) CWork with the parentheses. (A B)
BA
CU
Work with the remaining part of the statement.
(A B) C
2.3 – Venn Diagrams and SubsetsShade a Venn diagram for the given statement.
(A B) CWork with the parentheses. (A B)
BA
CU
Work with the remaining part of the statement.
(A B) C
2.4 –Surveys and Cardinal NumbersSurveys and Venn DiagramsFinancial Aid Survey of a Small College (100 sophomores). 49 received Government grants55 received Private scholarships43 received College aid23 received Gov. grants & Pri. scholar.18 received Gov. grants & College aid28 received Pri. scholar. & College aid8 received funds from all three
G
C
P
U
8
(PC) – (GPC) 28 – 8 = 20
20
(GC) – (GPC) 18 – 8 = 10
10
(GP) – (GPC) 23 – 8 = 15
15
43 – (10 + 8 +20) = 55
55 – (15 + 8 + 20) = 12
12
49 – (15 + 8 + 10) = 16
16
100 – (16+15 + 8 + 10+12+20+5) = 14
14
For any two sets A and B,
Cardinal Number Formula for a Region
( ) ( ) ( ).n A B n A n B n A B
Find n(A) if n(AB) = 78, n(AB) = 21, and n(B) = 36.
n(AB) = n(A) + n(B ) – n(AB)
78 = n(A) + 36 – 21
78 = n(A) + 15
63 = n(A)
2.4 –Surveys and Cardinal Numbers
9.1 – Points, Line, Planes and AnglesDefinitions:
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.
AD
Em
9.1 – Points, Line, Planes and AnglesDefinitions: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.
D
E
N
F
G
Name Figure Symbol
9.1 – Points, Line, Planes and AnglesSummary:
A B AB BA
AB
BA
AB
BA
A B
A B
Line AB or BA
Half-line AB
Half-line BA
Ray AB
Ray BA
Segment AB or Segment BA
A B
A B
A B AB BA
9.1 – Points, Line, Planes and AnglesDefinitions:
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 AnglesDefinitions:Parallel planes never meet.
Parallel Intersecting
Two distinct intersecting planes meet and form a straight line.
9.1 – Points, Line, Planes and AnglesDefinitions:An angle is the union of two rays that have a common endpoint.
Vertex BSide
Side
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.
C
A
1
9.1 – Points, Line, Planes and AnglesAngles are measured by the amount of rotation in degrees. Classification of an angle is based on the degree measure.
Measure NameBetween 0° and 90° Acute Angle
90° Right Angle
Greater than 90° but less than 180° Obtuse Angle
180° Straight Angle
9.1 – Points, Line, Planes and AnglesWhen two lines intersect to form right angles they are called perpendicular.
Vertical angles are formed when two lines intersect.A
CB
D
E
Vertical angles have equal measures.
ABC and DBE are one pair of vertical angles.DBA and EBC are the other pair of vertical angles.
9.1 – Points, Line, Planes and AnglesComplementary 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 AnglesFind the measure of each marked angle below.
(3x + 10)° (5x – 10)°
3x + 10 = 5x – 10
Each angle is 3(10) + 10 = 40°.
Vertical angels are equal.
2x = 20x = 10
9.1 – Points, Line, Planes and AnglesFind the measure of each marked angle below.
(2x + 45)° (x – 15)°
2x + 45 + x – 15 = 180
35° + 145° = 180
Supplementary angles.
3x + 30 = 180 3x = 150
x = 50
2(50) + 45 = 14550 – 15 = 35
9.1 – Points, Line, Planes and Angles1 2
3 4
5 67 8
Alternate interior angles
Alternate exterior angles
Angle measures are equal.
Angle measures are equal.
1
5 4
8
(also 3 and 6)
(also 2 and 7)
Parallel Lines cut by a Transversal line create 8 angles
9.1 – Points, Line, Planes and Angles1 2
3 45 6
7 8
Same Side Interior angles
Corresponding angles
Angle measures are equal.
Angle measures add to 180°.4
6
2
6
(also 3 and 5)
(also 1 and 5, 3 and 7, 4 and 8)
9.1 – Points, Line, Planes and AnglesFind the measure of each marked angle below.
(x + 70)°(3x – 80)°
Alternate interior angles.
x + 70 = 3x – 80 2x = 150x = 75 145°
x + 70 =75 + 70 =
9.1 – Points, Line, Planes and AnglesFind the measure of each marked angle below.
(2x – 21)°
(4x – 45)°
Same Side Interior angles.
4x – 45 + 2x – 21 = 180 6x – 66 = 1806x = 246 119°
4(41) – 45 164 – 45
x = 41180 – 119 = 61°
61°
2(41) – 21 82 – 21
