# Set Theory: Universal Set

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

Related Topics:
More Topics on Sets

What is a Universal Set?

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?

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