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 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 { }

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

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 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

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 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 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} Note: Every set is a subset of itself. BB

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 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 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} Subsets: {1} {2} {1,2}  22 = 4 Proper subsets: {1} {2}  22 – 1= 3

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 

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 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 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 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 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 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 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) }

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 Subsets Shading Venn Diagrams: A  B A B U U A B A B U

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 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

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 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

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

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 (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 – 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: 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

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

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 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.

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

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 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)

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. 4x – 45 + 2x – 21 = 180 4(41) – 45 2(41) – 21 6x – 66 = 180 164 – 45 82 – 21 6x = 246 119° 61° x = 41 180 – 119 = 61°