Sets
Set Definitions
The idea of a set is on of the fundamental concepts of
mathematics. Below, we stat the difinition.
Definition
A set is a well defined collection of objects.
Notation:
A =
{2,7,1} B = {cat, dog,
mouse} C =
{2,4,6,8,10,...100}
D =
{1,3,5,7,9,...} E = {}
The set E is called the empty set.
We say that
to
mean that "6 is an element of C"
and
means "Horse is not an element of B."
Types of Numbers

We define the natural numbers to be
N
= {1, 2, 3, 4, 5, 6, ...}

We define the whole numbers to be
W = {0, 1,
2, 3, 4, 5, ...}

We define the set of integers as
Z = {... 2, 1, 0, 1, 2, 3, ...}
So that the integers consist of the whole numbers and their negatives.

We define the set of rational numbers as the collection of all
fractions. Formally we have
Q = { p/q  p in Z , q in Z  {0}}
Examples of Rational Numbers
3/4, 2/5, 3/2, 7, 0
Are there other numbers? What numbered squared equals?
is not a rational number.
We define the set of real numbers
by
R = {All numbers that can be put on the number
line so that the
number line has no holes}
We define the irrational numbers
as the real numbers that are not rational 
Set Notation
We read, "The set of all x in the natural numbers such that
x is greater
than 2"
Subsets
A is a subset of
B, A
C
B if every element of
A is also an element of
B.
Exercise
Let
A = {1,2,3), B = {0,1,2,3,4} and C = {2,3,4}
Which of the following is true? (let your mouse rest on the yellow box for
a few seconds to see the solution.)

A C B

B C A

C C B

A C C
Note that we have the following chain:
Unions
We define A U B "A
union
B" to be the set of all elements either in
A or B.
Example
Let
A = {1, 2,
3} B = {2, 3, 4}
then
A U B
= {1, 2, 3, 4}
Intersections
We define A I B,
"A
intersection
B" to be the set of all elements that are
in both A and
B.
A I B
= {2, 3}
from the previous example. We can write
A U B
= {x  x in A or x in B}
A I B
= {x  x in A and x in B}
Next we will discuss Venn Diagrams
To further explore with set definitions go to http://mathcsjava.emporia.edu/~godbocat/setOperations.html
Back to Math 152A Home Page
email
Questions and Suggestions
