Probability/Set Theory

Introduction

The overview of set theory contained herein adopts a Template:Colored em point of view. A rigorous analysis of the concept belongs to the foundations of mathematics and mathematical logic. Although we shall not initiate a study of these fields, the rules we follow in dealing with sets are derived from them.

Sets

Template:Colored definition Template:Colored remark Template:Colored example We have different ways to Template:Colored em a set, e.g.

• word description: e.g., a set $S$ is the set containing the 12 months in a year;
• listing: elements in a set are listed within a pair of braces, e.g., $S\stackrel{\text{ def }}{=}\{\text{January, }\text{March, February, }\text{April, May, June, July, August, September, October, November, December}\}$;

the Template:Colored em of the elements is Template:Colored em, i.e. even if the elements are listed in different order, the set is still the same. E.g., $\{\text{January, }\text{February, March, }\text{April, May, June, July, August, September, October, November, December}\}$ is still referring to the same set.

• set-builder notation:$$\underset{\text{The set of}}{\underbrace{\{}}\underset{\text{all elements }x}{\underbrace{x}}\underset{\text{such that }}{\underbrace{:}}\underset{\text{the property }P(x)\text{ holds}}{\underbrace{P(x)}}\}$$

(the closing brace must also be written.)

For example, $S\stackrel{\text{ def }}{=}\{x:x\text{ is a month in a year}\}$.

Subsets

We introduce a Template:Colored em between sets in this section. Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark Illustration of Template:Colored em by Venn diagram:

A ⊆ B (A ≠ B):

*-----------------------*
|                       |
|                       |
|   *----------*        | <---- B
|   |          |        |
|   |    A     |        |
|   |          |        |
|   *----------*        |
*-----------------------*

Set operations

Probability theory makes extensive use of some set operations, and we will discuss them in this section. Template:Colored definition Template:Colored remark Template:Colored example In the following, some basic properties possessed by the union operation: commutative law and associative law, are introduced. Template:Colored proposition Template:Colored remark Template:Colored example Template:Colored definition Template:Colored remark Template:Colored example Template:Colored definition Template:Colored example Template:Colored remark Venn diagram

*-----*       *-----*       *-----*       
|     |       |     |       |     |
|  A  |       |  B  |       |  C  |
*-----*       *-----*       *-----*

(A, B and C are disjoint)
      
*----------------*
|                | <---- D 
| *--*   *-------*--------*               
| |  |   |       |        | 
*-*--*---*-------*        | <--- E
  |  |   |                |
  *--*   *----------------*
   ^
   |
   F

(D, E and F are not disjoint, but E and F are disjoint)

Template:Colored proposition Template:Colored remark Template:Colored example The following result combines the union operation and intersection operation. Template:Colored proposition Template:Colored example Template:Colored definition Template:Colored remark Template:Colored example Template:Colored theorem Template:Colored remark Template:Colored example Template:Colored definition Template:Colored example Template:Colored remark Template:Colored example Template:Colored exercise Template:Colored definition Template:Colored remark Template:Colored example Template:Colored exercise Template:Nav