# Set Notation

## Basic Concept

##### Intersection and Union

The intersection of two sets A and B is the set of elements common to both A and B. The symbol is "$$\cap$$"

###### $$\text{A}\cap\text{B}$$

The union of two sets A and B is the set of elements in A or B (or both). The symbol is "$$\cup$$"

##### Complement Set

Given a set A, the complement of A is the set off all elements in the universal set U, but not in A. We can write $$\text{A}^{\text{c}}$$

##### Properties of Set Operation

A. Commutative

$$\text{A}\cap\text{B} = \text{B}\cap\text{A}$$
$$\text{A}\cup\text{B} = \text{B}\cup\text{A}$$

B. Distributive

$$\text{A}\cup(\text{B}\cap\text{C}) = (\text{A}\cup \text{B})\cap (\text{A}\cup\text{C})$$

$$\text{A}\cap(\text{B}\cup\text{C}) = (\text{A}\cap \text{B})\cup (\text{A}\cap\text{C})$$

C. De'Morgan Theorem

$$(\text{A}\cup\text{B})^{\text{c}} = \text{A}^{\text{c}} \cap \text{B}^{\text{c}}$$
$$(\text{A}\cap\text{B})^{\text{c}} = \text{A}^{\text{c}} \cup \text{B}^{\text{c}}$$