Download 3.1 Types of Sets and Set Notation

UNIT #3 SET THEORY AND LOGIC.
3.1 Types of Sets and Set Notation
A set is a collection of distinct objects. The objects can be called elements or members of the
set.
A set does not list an element more than once.
Examples:
Word Description
a) the set of whole numbers less than 5
b) the set of whole numbers
c) the set of letters in the word “rate”
Set Notation
P = {0, 1, 2, 3, 4}
K = {0, 1, 2, 3, 4,…}
R = {r, a, t, e}
n(P) = 5 ← this is the notation used to indicate the total number of elements in a set (also called
the cardinal number)
universal set – a set of all the elements under consideration for a particular context
(also called a sample space).
Example 1: The universal set of numbers between 0 and 10 is X = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A subset is a set whose elements all belong to another set
Example 2: The set of even numbers between 0 and 10, E = {2, 4, 6, 8} is a subset of X.
In set notation, this relationship is written as E ⊂ X (or E ⊆ X)
Equal sets are two or more sets that have the same members
or elements.
Example 3: A: (1, 2, 3, 4) and C: (4, 1, 3, 2) Set A = Set C
An empty set contains no elements or null set
Example 4: Given the set X = {1, 2, 3, 4, 5, 6, 7, 8, 9} and the set G = { -3 } determine A ⋂ G.
⋂ indicates intersection of sets – the elements that are the same – or where the sets overlaps.
A⋂G={}
or
A⋂G=Ø
The complement of a set is all the elements of a universal set that do not belong to a subset
of it.
Example 5: The set E’ = {1, 3, 5, 7, 9} is the complement of E = {2, 4, 6, 8} a subset of the univer
sal set of X
Two or more sets are disjoint if they have no elements in common. From Example 4, sets X and
G are disjoint.
The set of even numbers and the set of odd numbers are disjoint.
The events that describe disjoint sets are mutually exclusive.
They are two or more events that cannot occur at the same time.
A finite set is a set with a countable number of elements.
Example 6: The set of even numbers less than 10, E = {2, 4, 6, 8} is finite.
An infinite set is a set with an infinite number of elements.
Example 7: The set of natural numbers, N = {1, 2, 3, ...} is infinite.
Example 8: What is the set notation for the set of all natural numbers
greater than 1 and less than or equal to 50?
A = {x | 1 < x < 50, x ε N}
x ε N means that x is an element of the set of natural numbers