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} {x | x is a counting number between 6 and 7} {5, 6, 7, 8, 9, 10, 11, 12, 13} Empty set Null set { }

3. 2.1 – Symbols and Terminology Symbols: ∈ : Used to replace the words “is an element of.” 3 ∈ {1, 2, 5, 9, 13} False 0 ∈ {0, 1, 2, 3} -5 ∉ {5, 10, 15, ,  } True ∉ : Used to replace the words “is not an element of.” True or False: True

4. 2.1 – Symbols and Terminology Sets of Numbers and Cardinality n(A): n of A; represents the cardinal number of a set. K = {2, 4, 8, 16} n(K) = 4 ∅ R = {1, 2, 3, 2, 4, 5} n(R) = 5 n( ∅ ) = 0 Cardinal Number or Cardinality: The number of distinct elements in a set. Notation P = { ∅ } n(P) = 1

5. 2.1 – Symbols and Terminology Finite and Infinite Sets {2, 4, 8, 16}Countable = Finite set {1, 2, 3, …} Not countable = Infinite set Finite set: The number of elements in a set are countable. Infinite set: The number of elements in a set are not countable

6. 2.1 – Symbols and Terminology Equality of Sets {–4, 3, 2, 5} and {–4, 0, 3, 2, 5} Are the following sets equal? Equal Not equal 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. {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}

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 U A A Venn Diagram: A rectangle represents the universal set and circles represent sets of interest within the universal set

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 {3, 4, 5, 6} {3, 4, 5, 6, 8} BBBB Subset or not? Note: Every set is a subset of itself.  {1, 2, 6} {2, 4, 6, 8}  {5, 6, 7, 8} 

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} = {5, 6, 7, 8} {5, 6, 7, 8, 9} 

10. 2.2 – Venn Diagrams and Subsets Definitions: The empty set (  ) is a subset and a proper subset of every set except itself. Proper Subset of a Set: Set A is a proper subset of Set B if A  B and A  B. Notation A  B {3, 4, 5, 6} {3, 4, 5, 6, 8} both {1, 2, 6} {1, 2, 4, 6, 8} both {5, 6, 7, 8}  What makes the following statements true? , , or both

11. 2.2 – Venn Diagrams and Subsets Number of Subsets The number of subsets of a set with n elements is: 2 n {1} List the subsets and proper subsets Number of Proper Subsets The number of proper subsets of a set with n elements is: 2 n – 1 {1, 2} {2} {1,2}  {1} {2}  Subsets: Proper subsets: 2 2 = 4 2 2 – 1= 3

12. 2.2 – Venn Diagrams and Subsets {a} List the subsets and proper subsets {a, b, c} {b} {c}  {a, b}{a, c} Subsets: Proper subsets: 2 3 = 8 2 3 – 1 = 7 {b, c} {a, b, c} {a} {b} {c}  {a, b}{a, c}{b, c}