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:.
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Definitions:
Set Designations (3 types):
Definitions:
{ }
Symbols:
True or False:
Sets of Numbers and Cardinality
Cardinal Number or Cardinality:
The number of distinct elements in a set.
Notation
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
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.
Definitions:
A
A
U
Definitions:
Definitions:
=
Definitions:
both
both
Number of Subsets
Intersection of Sets: The intersection of sets A and B is the set of elements common to both A and B.
Union of Sets: The union of sets A and B is the set of all elements belonging to each set.
Find each set.
Find each set.
Difference of Sets: The difference of sets A and B is the set of all elements belonging set A and not to set B.
Find each set.
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.
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.
Locating Elements in a Venn Diagram
7
1
A
B
4
2
3
8
6
5
U
9
10
Shade a Venn diagram for the given statement.
Work with the parentheses.
(A B)
Shade a Venn diagram for the given statement.
Work with the parentheses.
(A B)
Work with the remaining part of the statement.
(A B) C
Shade a Venn diagram for the given statement.
Work with the parentheses.
(A B)
Work with the remaining part of the statement.
(A B) C
Surveys and Venn Diagrams
G
P
16
15
12
8
20
10
5
(PC) – (GPC) 28 – 8 = 20
43 – (10 + 8 +20) = 5
C
U
14
(GC) – (GPC) 18 – 8 = 10
55 – (15 + 8 + 20) = 12
(GP) – (GPC) 23 – 8 = 15
49 – (15 + 8 + 10) = 16
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,
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)
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
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
Summary:
Line AB or BA
AB
BA
B
A
Half-line AB
AB
A
B
Half-line BA
BA
A
B
Ray AB
AB
A
B
Ray BA
BA
A
B
Segment AB or Segment BA
BA
A
B
AB
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.
Intersecting
Skew
Parallel
Definitions:
Parallel planes never meet.
Two distinct intersecting planes meet and form a straight line.
Parallel
Intersecting
Definitions:
An angle is the union of two rays that have a common endpoint.
A
Side
1
Vertex
B
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.
Angles are measured by the amount of rotation in degrees.
Classification of an angle is based on the degree measure.
Between 0° and 90°
Acute Angle
90°
Right Angle
Greater than 90° but less than 180°
Obtuse Angle
Straight Angle
180°
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.
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
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°.
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
1 2
Parallel Lines cut by a Transversal line create 8 angles
3 4
5 6
7 8
Alternate interior angles
5 4
Angle measures are equal.
(also 3 and 6)
1
Alternate exterior angles
Angle measures are equal.
8
(also 2 and 7)
1 2
3 4
5 6
7 8
Same Side Interior angles
4
Angle measures add to 180°.
6
(also 3 and 5)
2
Corresponding angles
6
Angle measures are equal.
(also 1 and 5, 3 and 7, 4 and 8)
Find the measure of each marked angle below.
(3x – 80)°
(x + 70)°
Alternate interior angles.
x + 70 =
x + 70 = 3x – 80
75 + 70 =
2x = 150
145°
x = 75
Find the measure of each marked angle below.
(4x – 45)°
(2x – 21)°
Same Side Interior angles.
4(41) – 45
4x – 45 + 2x – 21 = 180
2(41) – 21
164 – 45
6x – 66 = 180
82 – 21
119°
61°
6x = 246
x = 41
180 – 119 = 61°