Universal Set

The universal set, represented by the symbol ‘U,’ is a set that contains all elements related to a particular subject without any repetition. It is sometimes also denoted by the symbols ‘V,’ ‘Ω,’ or ‘ξ.’

A universal set of real numbers includes all rational numbers, integers, whole numbers, and natural numbers.

Example

If A = {3, 6, 9, 12} and B = {2, 4, 6, 8, 10, 12}.

The universal set U of A and B is:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Since sets A and B belong to the universal set U, they are subsets of the universal set.

Here, 

A is the proper subset of U, A ⊂ U

B is the proper subset of U, B ⊂ U 

Since every set is a subset of itself, the universal set is also a subset.  

U ⊆ U

Finite Universal Set

It is a type of universal set that contains a countable number of elements. For example, the set of vowels in the English alphabet {a, e, i, o, u} is a finite universal set. 

Infinite Universal Set

It is a type of universal set that contains an uncountable number of elements. For example, the set of positive natural numbers {1, 2, 3, …} is an infinite universal set. 

Universal Set and Venn Diagram 

When representing a universal set in a Venn diagram, a rectangular box denotes the universal set. Circles within this box represent the subsets, ensuring that all subsets are contained within the boundaries of the universal set.

Let us consider another example: 

A =  {p, q, r, s}, B = {m, n, o, p, q}, C = {e, f, g, h, k}, and U = {b, c, d, e, f, g, h, k, m, n, o, p, q, r, s, t}

On representing them on a Venn diagram:

Complement

The complement of a set contains those elements from the universal set that are absent in the given set. Thus, if an element is present in a complement of a set, it is not present in that set.

The complement of the universal set is thus an empty set. It is represented as U’ = ɸ

Difference between Universal Set and Union of Sets

Universal Set	Union of Sets
It is a set containing all elements related to a particular subject without repetition.	It is a set operation involving two sets where the resultant set contains all the elements of the given sets.
It is denoted by the symbol U, V, Ω, or ξ.	It is denoted by the symbol ∪. The union of two sets, A and B, is written as A ∪ B (read as A union B)
It is possible for a universal set to contain elements that are not included in its subsets.	It is impossible to have extra elements in a union of two sets other than those of the two sets given.
Example: Set A = {1, 3, 5} and Set B = {2, 4, 6}Thus, the universal set U is {0, 1, 2, 3, 4, 5, 6}	Example: Set A = {1, 3, 5} and Set B = {2, 4, 6}Thus, the union of sets A and B is A ∪ B = {1, 2, 3, 4, 5, 6}

Solved Examples