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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.