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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: {x | x is an even counting number less than 10}

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**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 { }

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**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

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**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

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**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

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**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

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**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

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**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: AB 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. BB

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**2.2 – Venn Diagrams and Subsets**

Definitions: 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}

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**2.2 – Venn Diagrams and Subsets**

Definitions: Proper Subset of a Set: Set A is a proper subset of Set B if AB and A B. Notation AB 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.

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**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

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**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

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**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}

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**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}

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**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

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**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}

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**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, }

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**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

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**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) }

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**2.3 – Venn Diagrams and Subsets**

Shading Venn Diagrams: A B A B U A B A B U U

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**2.3 – Venn Diagrams and Subsets**

Shading Venn Diagrams: A B A B U U A B A B U

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**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

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**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

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**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

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**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

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**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

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**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 (PC) – (GPC) – 8 = 20 43 – ( ) = 5 C U 14 (GC) – (GPC) 18 – 8 = 10 55 – ( ) = 12 (GP) – (GPC) 23 – 8 = 15 49 – ( ) = 16 100 – ( ) = 14

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**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)

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**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

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**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

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**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

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**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

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**9.1 – Points, Line, Planes and Angles**

Definitions: Parallel planes never meet. Two distinct intersecting planes meet and form a straight line. Parallel Intersecting

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**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.

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**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

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**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.

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**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

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**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°.

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**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

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**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)

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**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)

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**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

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**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°

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Objective 1.01 Apply the properties and definitions of lines and angles to model and solve problems. Vocabulary Acute: an angle that measure less than.

Objective 1.01 Apply the properties and definitions of lines and angles to model and solve problems. Vocabulary Acute: an angle that measure less than.

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