2 1 Symbols and Terminology Definitions Set A
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} {} Null set Empty set
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 -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} 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} {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} 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 U B
2. 3 – Venn Diagrams and Subsets Shading Venn Diagrams: A B U B
2. 3 – Venn Diagrams and Subsets Shading Venn Diagrams: A B A B U A A U B A B 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 4 2 3 Fill in remaining elements of U. B 8 6 5 U 9 10
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