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What Is Union and Intersection of Sets?

Sets are central in probability theory. When considering probabilities within the context of set theory, it becomes easier to structure exercises and solve problems.

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:

Venn diagram for all outcomes unique to A and B

Example 1

Let $A$ be the set ${1, 3, 5}$ and $B$ be the set ${3, 5, 6}$ . The union $A \cup B$ is the set of outcomes ${1, 3, 5, 6}$ .

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:

Venn diagram showing outcome unique to both A and B

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

{2, 4, 6}

, and

B

be the event “more than three dots”, which is the set

{4, 5, 6}

. That means the intersection

A \cap B

is the set

{4, 6}

.

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