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.)

1. A C B

2. B C A

3. C C B

4. A C C

Note that we have the following chain:

 N C W C Z C Q C R

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