Lecture Notes For September 21, 1998

I.  Hand Out Syllabus

II.  Introductions

III.  Set Definitions

Students will pair up to discus the definition of a set.

Definition:  A set is a well defined collection of objects.   Students will give three examples each.

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 6 e C  (notation not supported by the web)  to mean that "6 is an element of C"

horse !e B  "Horse is not an element of B"

IV.  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:  3/4, 2/5, -3/2, 7, 0

Are there other numbers?  

What numbered squared equals 2?

sqrt(2) is not a rational number.  The class will find other examples.

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

V.  Set Notation

E = {x in N | x > 2} =  {3,4,5,6,...}

We read, "The set of all x in the natural numbers such that x is greater than 2"

We will try several examples.

VI.  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?

i)  A C B  

ii)  B C A

iii) C C B

iv)  A C C

Note that we have the following chain:

N C W C Z C Q C R

VII.  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}

VIII.  Intersections

We define A int B "A intersection B" to be the set of all elements that are in both A and B.

A int B = {2,3}  from the previous example.

We can write

A U B = {x | x in A or x in B}

A int B = {x | x in A and x in B}

Next we will discus Venn Diagrams

One point of extra credit if you put the words "I Love Math" on the top of the homework that is due on September 23.