# Set Theory: Universal Set

Related Pages
More Lessons On Sets
Subsets
Venn Diagrams

In these lessons, we will learn what is a universal set and how it may be represented in a Venn Diagram.

### What Is A Universal Set?

The following diagram explains what is a Universal Set and gives an example of a Universal Set. Scroll down the page if you need more explanations and examples about Universal Sets. A universal set is the set of all elements under consideration, denoted by capital U or sometimes capital E.

Example:
Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.
a) A = {x : x is a factor of 60}
b) B = {x : x is a prime number}

Solution:
The elements of sets A and B can only be selected from the given universal set U.
a) A = {5, 6, 10, 12}
b) B = {5, 7, 11}

In Venn diagrams, the universal set is usually represented by a rectangle and labeled U.

Example:
Draw a Venn diagram to represent the following sets:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 5, 6}, B = {3, 9}

Solution:
Step 1: Draw a rectangle and label it U to represent the universal set.

Step 2: Draw circles within the rectangle to represent the other sets. Label the circles and write the relevant elements in each circle.

Step 3: Write the remaining elements outside the circles but within the rectangle. #### The Universal Set And Set Complements

Example:
Let the universal set, U = {a, e, i, o, u}
Let the subset A = {a, e}
Then the complement of set A, A’ = {i, o, u}

#### Intersection, Union And Complement Of Sets

A ∩ B (read as A intersection B) are members that are common to both set A and set B.
A ∪ B (read as A union B) are members that are in set A or set B or both.
A’ (read as A complement) are members that are not in set A.

#### What Is Universal Set And Absolute Complement?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 