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

**Related Pages**

More Lessons On Sets

Subsets

Venn Diagrams

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.

**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}

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.

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