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# 2.1 – Symbols and Terminology - PowerPoint PPT Presentation

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: 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}

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

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, , }

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

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, …}

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

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

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.

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}

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.

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}

• 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

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}

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}

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

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}

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, }

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

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)

• }

• A  B

A

B

U

A

B

A

B

U

U

• A  B

A

B

U

A

A

B

B

U

U

• A B

A

B

U

A

A

B

A

B

U

U

• A B in yellow

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}

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

Shade a Venn diagram for the given statement.

• (A  B)  C

Work with the parentheses.

(A  B)

• A

• B

• C

• U

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

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

Surveys and Venn Diagrams

• Financial Aid Survey of a Small College (100 sophomores).

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

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 – 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 – 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 – 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 – 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 – 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

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 – 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 – 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.

(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 – 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 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 – Points, Line, Planes and Angles

1 2

3 4

5 6

7 8

Same Side Interior angles

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 =

x + 70 = 3x – 80

75 + 70 =

2x = 150

145°

x = 75

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°