When working with sets, a reference superset that contains all the sets and their subsets in context (i.e. the sets which are currently under consideration) is called a universal set. It provides a base set from which all of the sets of interest can be derived or deduced.

The universal set concept is needed at times in solving computational problems related to sets and providing certain axioms and in the application of Venn diagrams.

When dealing with multiple collections of the same objects, then a set that contains all the possible objects and all sets and subsets formed from these elements is called a superset of every other set or universal set. It is generally denoted by the U.

There are no laws or rules that govern the composition of the universal set. It is up to one’s wish and simplicity that one decides the choice of a universal set. One can also define his or her own universal sets as per his need and usage. 

For example, if we are dealing with positive numbers less than 100, then pre-defining a set U that contains all numbers between 1 & 100 { x ∈ N | 1 ≤ x ≤ 100 } inclusive can serve the purpose of universal sets.

In practical applications of sets, we generally deal with objects or things of a particular kind. For instance, when dealing with numbers, we might be limited to integers within a range of natural numbers or real numbers and so on. When dealing with the performance of students, we would generally be limited to students of a class or a school or in general of a city, etc. In a tournament, our sample data would only be limited to the teams or players participating.

The complement of a set is defined as everything else except the members contained in the set. When you combine a set and its complement, you get what is called a universal set.

The universal set point has been stressed a bit because, from a philosophical view, a universal set cannot exist.

There exists a paradox called Russel’s paradox regarding the universal set definition and the basic definition that contradicts the existence of such a set. This will be elaborated on in a separate section.