Universal Set

The universal set is a set that contains all the elements related to a specific context. The universal set is denoted by $U$ which is a superset of all sets with respect to a given context.

Definition

The universal set is the set of all elements or members of all related sets. It is usually denoted by the symbol $U$. For example, in human population studies, the universal set is the set of all the people in the world. The set of all people in each country can be considered as a subset of this universal set.

• A universal set can be either a finite or infinite set.
• The set of natural numbers is a typical example of an infinite universal set.

Let's consider an example with three sets, $A$,$B$ and $C$. Here,$A=\{1,2,3\}$, $B=\{4,5,6,7,8,9\}$, and $C=\{9,11,12\}$. We need to find the universal set for all three sets $A$,$B$ and $C$. All the elements of the given sets are contained in the universal set. Thus, the universal set $U$ of $A$,$B$ and $C$ can be given by $U=\{1,2,3,4,5,6,7,8,9,10,11,12\}$

We can see that all the elements of the three sets are present in the universal set without any repetition. Thus, we can say that all the elements in the universal set are unique. The sets $A$,$B$ and $C$ are contained in the universal set, then these sets are also called subsets of the Universal set.

• $A\subset U$ ($A$ is the subset of $U$)
• $B\subset U$ ($B$ is the subset of $U$)
• $C\subset U$ ($C$ is the subset of $U$