Union

Chapter 3. Probability: Relationships between Events

Union

When considering two events $A$ and $B$ , it could be of interest to investigate the scenario of $A$ OR $B$ happening. That means that either one of the following things happens:

$A$ happens and $B$ does not happen
$B$ happens and $A$ does not happen
$A$ and $B$ both happen

The combined event of $A$ OR $B$ happening is called the union of $A$ and $B$ .

Union

Definition

The union of two events $A$ and $B$ is the set of outcomes that are classified as $A$ OR $B$ .

The union of events $A$ and $B$ is denoted as $A \cup B$ .

All of $A$ is contained in $A \cup B$ and all of $B$ is contained in $A \cup B$ . Additionally, the outcomes that are in both $A$ and $B$ are contained in $A \cup B$ .

Outcomes that are classified as $A$ and $B$ should only be listed once; every outcome is unique.

Consider the following two events:

$A = \{1, 2, 3, 4\}$
$B = \{2, 4, 5, 6\}$

What is the union of $A$ and $B$ ?

The union of $A$ and $B$ is the set of outcomes that are classified as either $A$ OR $B$ :

$A \cup B = \{1, 2, 3, 4, 5, 6\}$

New example