Handout on Sets and Set Notation, Lecture notes of Mathematics

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Introduction to Sets

Basic, Essential, and Important

Properties of Sets

Definitions

A set is a collection of

objects.

Objects in the collection

are called elements of the

set.

Examples - set

The collection of all quadrupeds is a set.  Each quadruped is an element of the set. The collection of all four-legged dogs is a set.  Each four-legged dog is an element of the set.

Examples - set

The collection of counting numbers is a set.  Each counting number is an element of the set. The collection of pencils in your briefcase is a set.  Each pencil in your briefcase is an element of the set.

Notation

The roster method of

specifying a set consists of

surrounding the collection

of elements with braces.

Example – roster method

For example the set of counting

numbers from 1 to 5 would be

written as

Notation

 Set builder notation has the

general form {variable |

descriptive statement }.

The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.

Example – set builder notation

 {x | x < 6 and x is a counting number} is the set of all counting numbers less than

6. Note this is the same set as {1,2,3,4,5}.  {x | x is a fraction whose numerator is 1 and whose denominator is a counting number }.

 Set builder notation will become

much more concise and precise as

more information is introduced.

Venn Diagrams

It is frequently very helpful to depict a set in the abstract as the points inside a circle (or any other closed shape). We can picture the set A as the points inside the circle shown here.

A

Venn Diagrams

To learn a bit more about Venn

diagrams and the man John Venn

who first presented these

diagrams click on the history icon

at the right.

History

Definition

The set with no elements is called the empty set or the null set and is designated with the symbol .

Examples – empty set

The set of all pencils in your briefcase might indeed be the empty set. The set of even prime numbers greater than 2 is the empty set. The set {x | x < 3 and x > 5} is the empty set.

Notation - subset

If A is a subset of B we write

A  B to designate that relationship.

If A is a proper subset of B we write

A  B to designate that relationship.

If A is not a subset of B we write

A  B to designate that relationship.

Example - subset

The set A = {1, 2, 3} is a subset of

the set B ={1, 2, 3, 4, 5, 6}

because each element of A is an

element of B.

We write A  B to designate this

relationship between A and B.

We could also write