Set Theory: Universal Set

What Is The Universal Set?

The universal set is the set of all elements under consideration, denoted by capital U or sometimes capital E or ξ.

The following diagram explains what is a Universal Set and gives an example of a Universal Set. Scroll down the page if you need more explanations and examples about Universal Sets.

 

Properties of Universal Sets

1. All-Inclusive:
   Every other set in the discussion is a subset of U.
   Example: If U = {1,2,3,4,5}, then A = {2,4} is a subset of U.
   
2. Complement of a Set:
   The complement of set A (written A′ or A^c) includes all elements not in A but still in U.
   A′ = U − A
   Example: If U = {1,2,3,4} and A = {1,2}, then A′ = {3,4}.
   
3. Union/Intersection:
   The union of any set with U is U: A ∪ U = U.
   The intersection of any set with U is the set itself: A ∩ U = A.
   

Relationship with Venn Diagrams: In Venn diagrams, the universal set is typically represented by a rectangle, and all the subsets under consideration are drawn as circles within this rectangle.

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}

• Show Video Lesson

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.

• Show Video Lesson

What Is Universal Set And Absolute Complement?

• Show Video Lesson

Check out our most popular games!
Fraction Concoction	Fact Family Game	Number Bond Garden
Online Addition Subtraction Game	Penguin Solitaire	Sawayama Solitaire
Ark Solitaire	Eldritch Invasion	Shenzhen Solitaire

Check out more Solitaire games here.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.