Intersection and Union of Sets

Now we will include binary operations for sets. Binary operations are those operations which need two operands and generate a third new resultant. This lesson will introduce you to the first two basic operations that we can do on sets, taking unions and intersections.

Union of Two Sets

The union of two sets $A$ and $B$is denoted by $A \cup B$ is the set which contains the elements of both sets. Consider this simple example.

We can see that the union simply combines the elements of both sets. All other rules of sets apply such as no duplicates. So we can intuit that;

In the above example we would not get two $1$'s.

Properties of Union

You should also intuit the following behaviors from the definition of union.

• The union of a set and the empty set is just the same set. $A \cup \phi = A$
• The union of a set and the universal set is the universal set. $A \cup U = U$

Symbolic Representation of Union

To practice our set builder notation, we should be able to write the definition of Union in the set builder notation.

This is read as "$x$ such that $x$ is in $A$ or $x$ is in $B$".

Intersection of Two Sets

We often see sets that have some elements common to both sets. Intersection of two sets $A$ and $B$ denoted by $A \cap B$ is the set containing only the elements that exist in both sets.

This should be intuitive since $1$ is the only element that is found in both sets.

It is naturally possible that two sets have no elements in common. In this case, the intersection results in an empty set.

Disjoint Sets

Two sets $A$ and $B$ are said to be disjoint sets if $A \cap B = \phi$. Simply put, disjoint sets do not have any numbers in common. The rational and irrational number sets are disjoint sets since they are mutually exclusive. The negative and positive integer set are also disjoint (but only if zero is ignored for both sets).

Similarly, the set of US Presidents and the set of all British Kings are disjoint sets.

Overlapping Sets

Two sets $A$ and $B$ are said to be overlapping sets if $A \cap B \ne \phi$. Simply put, there is at least one element common between the two sets. The natural and whole number sets are overlapping sets. The rational and the real number set are overlapping sets.

Similarly, the set of US Presidents and the set of reality TV show hosts are overlapping sets because of Donald Trump being part of both sets.

Properties of Intersection

You should also intuit the following behaviors from the definition of intersection.

• The intersection of a set and the empty set is the empty set. $A \cap \phi = \phi$
• The intersection of a set and the universal set is the same set. $A \cap U = A$

Symbolic Representation of Intersection

To practice our set builder notation, we can write;

This is read as "$x$ such that $x$ is in $A$ and $x$ is in $B$.

Venn Diagram Visualization

In this graph, we will visualize the shaded area for the given symbolic expressions. Click on the given buttons on the chart to toggle the expressions to see how shaded areas change. This chart shows overlapping 2 sets.

Union and Intersection Exercises