Sets

Set Definition:

A collection of objects.

Examples include letters, number, fruits.

Set Notation

{1, 2, 3, 4, 5}
Use braces {} to enclose a set

Use 3 dots for infinity
{1, 2, 3, ...}

Use capital letters for sets
S = {1, 2, 3}

Examples of Sets:

Letters of the alphabet {a, b, c}
Coins: {penny, nickel, dime}
Counting Numbers {1, 2, 3, ...}
Positive Even Numbers {2, 4, 6, ...}

Elements of sets:

Each element separated by commas
Elements use the element symbol ∈
In the set S = {1, 2, 3}
1 ∈ S, 2 ∈ S, 3 ∈ S

Cardinality of a set:

Measures how many elements
Given a set S, cardinality = |S|
With S = {1, 2, 3}, |S| = 3
since S has 3 elements

Special Type of Sets:

The Universal Set U has every element
The empty set ∅ has no elements

Complement of a set:

Everything not in the set but in U
Write this as S' or S^C
Given U = {1, 2, 3, 4, 5} and S = {1, 2, 3}
S^C = {4, 5} since they are in U not in S

Finite sets:

Finite sets have countable elements.
{1, 2, 3} for example has 3 elements

Infinite sets:

Infinite sets have uncountable elements
They go on forever using ...symbol
{1, 2, 3, ...}

Set Notes:

Element order does not matter
{1, 2, 3} = {3, 2, 1}