 # What Is Union and Intersection of Sets?

Sets are central in probability theory. When considering probabilities within the context of set theory, it will be easier to structure exercises.

Theory

### Union

$A$ union $B$” is all the elements in $A$, $B$ or both. Mathematically, this is written as follows:

 $A\cup B.$

You read $\cup$ as “or”.

In other words, $A\cup B$ is a new compound set consisting of all the outcomes unique to $A$, all outcomes unique to $B$, as well as all the outcomes that $A$ and $B$ have in common. You can visualize it with this Venn diagram: Example 1

Let $A$ be the set $\phantom{\rule{-0.17em}{0ex}}\left\{1,3,5\right\}$ and $B$ be the set $\phantom{\rule{-0.17em}{0ex}}\left\{3,5,6\right\}$. The union $A\cup B$ is the set of outcomes $\phantom{\rule{-0.17em}{0ex}}\left\{1,3,5,6\right\}$.

Theory

### Intersection

The “intersection of $A$ and $B$” is all the elements that are in both $A$ and $B$, and can be written:

 $A\cap B$

You read $\cap$ as “and”.

In other words, $A\cap B$ is composed of the outcomes that are in both $A$ and $B$. It’s visualized with this Venn diagram: The intersection in the Venn diagram represents the overlap between $A$ and $B$.

Example 2

You roll a die. Let $A$ be the event “even number of dots”, which is the set $\phantom{\rule{-0.17em}{0ex}}\left\{2,4,6\right\}$, and $B$ be the event “more than three dots”, which is the set $\phantom{\rule{-0.17em}{0ex}}\left\{4,5,6\right\}$. That means the intersection $A\cap B$ is the set $\phantom{\rule{-0.17em}{0ex}}\left\{4,6\right\}$.