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 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}

2.3 – 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 = {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 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 – 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