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}

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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.

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What Is Universal Set And Absolute Complement?

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