Download Symbols and Sets of Numbers

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
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}
P = {∅}
n(R) = 5
n(P) = 1
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}
2.2 – 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
{3, 4, 5, 6} both {3, 4, 5, 6, 8}
{1, 2, 6}
both
{5, 6, 7, 8}

{1, 2, 4, 6, 8}
{5, 6, 7, 8}
The empty set () is a subset and a proper subset of
every set except itself.
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}
22 = 4
Subsets: {1} {2} {1,2} 
Proper subsets:
{1}
{2}

22 – 1= 3
2.2 – Venn Diagrams and Subsets
List the subsets and proper subsets
{a, b, c}
{a} {b} {c}
Subsets:
{a, b}
{a, c}
{b, c}
23 = 8
{b, c}
23 – 1 = 7

{a, b, c}
Proper subsets:
{a} {b} {c}
{a, b}
{a, c}
