Probability/Set Theory

Introduction
The overview of set theory contained herein adopts a 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
We have different ways to 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., $$S\overset{\text{ def }}=\{\text{January, }{\color{darkgreen}\text{March, February, }}\text{April, May, June, July, August, September, October, November, December}\}$$;
 * the of the elements is, i.e. even if the elements are listed in different order, the set is still the same. E.g., $$\{\text{January, }{\color{darkgreen}\text{February, March, }}\text{April, May, June, July, August, September, October, November, December}\}$$ is still referring to the same set.


 * set-builder notation:$$\underbrace{\{}_{\text{The set of}\;}\underbrace{x}_{\text{all elements }x\;}\underbrace{:}_{\text{such that }}\underbrace{P(x)}_{\text{the property }P(x)\text{ holds}}\}$$
 * (the closing brace must also be written.)
 * For example, $$S\overset{\text{ def }}=\{x:x\text{ is a month in a year}\}$$.

Subsets
We introduce a between sets in this section.

Illustration of by Venn diagram: A ⊆ B (A ≠ B):


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

Set operations
Probability theory makes extensive use of some set operations, and we will discuss them in this section.

In the following, some basic properties possessed by the union operation: commutative law and associative law, are introduced.

Venn diagram
 * A |       |  B  |       |  C  |
 * A |       |  B  |       |  C  |
 * A |       |  B  |       |  C  |

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

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

The following result combines the union operation and intersection operation.