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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.

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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.


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}


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.


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}


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.

universal set

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?

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

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

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