Set Theory: 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 labelled 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.

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