2.1 – Symbols and TerminologyDefinitions:

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 TerminologyDefinitions:

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 { }

2.1 – Symbols and TerminologySymbols:

∈: 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

2.1 – Symbols and TerminologySets 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

2.1 – Symbols and TerminologyFinite 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

2.1 – Symbols and TerminologyEquality 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}

2.2 – Venn Diagrams and SubsetsDefinitions:

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

2.2 – Venn Diagrams and SubsetsDefinitions:

Subset of a Set: Set A is a Subset of B if every element of A is an element of B. Notation: AB

{3, 4, 5, 6} {3, 4, 5, 6, 8}

BB

Subset or not?

Note: Every set is a subset of itself.

{1, 2, 6} {2, 4, 6, 8}

{5, 6, 7, 8} {5, 6, 7, 8}

2.2 – Venn Diagrams and SubsetsDefinitions:

Set Equality: Given A and B are sets, then A = B if AB and BA.{1, 2, 6} {1, 2, 6}=

{5, 6, 7, 8} {5, 6, 7, 8, 9}

2.2 – Venn Diagrams and SubsetsDefinitions:

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 AB and A B. Notation AB

{3, 4, 5, 6} {3, 4, 5, 6, 8}both{1, 2, 6} {1, 2, 4, 6, 8}both

{5, 6, 7, 8} {5, 6, 7, 8}

What makes the following statements true? , , or both

2.2 – Venn Diagrams and SubsetsNumber of Subsets

The number of subsets of a set with n elements is: 2n

{1}

List the subsets and proper subsets

Number of Proper SubsetsThe number of proper subsets of a set with n elements is: 2n – 1

{1, 2}{2} {1,2}

{1} {2}

Subsets:

Proper subsets:

22 = 4

22 – 1= 3

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:

23 = 8

23 – 1 = 7

{b, c}

{a, b, c}

{a} {b} {c}

{a, b} {a, c} {b, c}