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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:.

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2 1 symbols and terminology
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 terminology1
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 terminology2
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

  • True

  • -5 ∉ {5, 10, 15, , }


2 1 symbols and terminology3
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


2 1 symbols and terminology4
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

  • Not countable = Infinite set

  • {1, 2, 3, …}


2 1 symbols and terminology5
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
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 subsets1
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}

  • BB

  • Note: Every set is a subset of itself.


2 2 venn diagrams and subsets2
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 subsets3
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.


2 2 venn diagrams and subsets4
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}

  • {1}

  • 22 = 4

  • {2}

  • Subsets:

  • {1,2}

  • Proper subsets:

  • 22 – 1= 3

  • {1}

  • {2}


2 2 venn diagrams and subsets5
2.2 – Venn Diagrams and Subsets

  • List the subsets and proper subsets

  • {a, b, c}

  • {a}

  • {b}

  • Subsets:

  • {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


2 3 set operations and cartesian products
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 products1
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 products2
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


2 3 set operations and cartesian products3
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 products4
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 products5
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 products6
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),

  • (5, 6),

  • (1, 7),

  • (5, 7),

  • (9, 6),

  • (9, 7)

  • }

  • B  A

  • {

  • (6, 1),

  • (6, 9),

  • (6, 5),

  • (7, 1),

  • (7, 5),

  • (7, 9)

  • }


2 3 venn diagrams and subsets
2.3 – Venn Diagrams and Subsets

Shading Venn Diagrams:

  • A  B

A

B

U

A

B

A

B

U

U


2 3 venn diagrams and subsets1
2.3 – Venn Diagrams and Subsets

Shading Venn Diagrams:

  • A  B

A

B

U

A

A

B

B

U

U


2 3 venn diagrams and subsets2
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


2 3 venn diagrams and subsets3
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

8

  • Fill in remaining elements of U.

6

5

U

9

10


2 3 venn diagrams and subsets4
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 subsets5
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

(A  B)  C

  • C

  • U


2 3 venn diagrams and subsets6
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

(A  B)  C

  • C

  • U


2 4 surveys and cardinal numbers
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

20

10

  • 8 received funds from all three

5

(PC) – (GPC) 28 – 8 = 20

43 – (10 + 8 +20) = 5

C

U

14

(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

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(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 p oints line planes and angles
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


9 1 p oints line planes and angles1
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


9 1 p oints line planes and angles2
9.1 – Points, Line, Planes and Angles

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


9 1 p oints line planes and angles3
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.

Intersecting

Skew

Parallel


9 1 p oints line planes and angles4
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 p oints line planes and angles5
9.1 – Points, Line, Planes and Angles

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.


9 1 p oints line planes and angles6
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.

Between 0° and 90°

Acute Angle

90°

Right Angle

Greater than 90° but less than 180°

Obtuse Angle

Straight Angle

180°


9 1 p oints line planes and angles7
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 p oints line planes and angles8
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 p oints line planes and angles9
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°.


9 1 p oints line planes and angles10
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 p oints line planes and angles11
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

5 4

Angle measures are equal.

(also 3 and 6)

1

Alternate exterior angles

Angle measures are equal.

8

(also 2 and 7)


9 1 p oints line planes and angles12
9.1 – Points, Line, Planes and Angles

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)


9 1 p oints line planes and angles13
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 + 70 = 3x – 80

75 + 70 =

2x = 150

145°

x = 75


9 1 p oints line planes and angles14
9.1 – Points, Line, Planes and Angles

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°


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