2.1 – Symbols and Terminology

1. 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. 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

3. 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 • True • -5 ∉ {5, 10, 15, , }

4. 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

5. 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 • Not countable = Infinite set • {1, 2, 3, …}

6. 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

7. 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

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

9. 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}

10. 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.

11. 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} • {1} • 22 = 4 • {2} • Subsets: • {1,2} •  • Proper subsets: • 22 – 1= 3 • {1} • {2} • 

12. 2.2 – Venn Diagrams and Subsets • 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 • 