1.8 Universal Set

Universal set

A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol ‘ \(U\) ‘. A universal set can be either a finite or an infinite set. The set of natural numbers is a typical example of an infinite universal set.

For Example, let’s consider an example with three sets, \(A, B\), and \(C\). Here, \(A=\{2,4\), \(6\}, B=\{1,3,7,9,11\}\), and \(C=\{4,8,11\}\). 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 \(U=\{1,2,3,4,6,7,8,9,10,11,12\}\).

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 \(U\) )
\(\mathrm{C} \subset \mathrm{U}\) ( \(C\) is the subset of \(U\) )

Complement of Universal Set

For a subset \(A\) of the universal set ( \(U\) ), its complement is represented as \(\mathrm{A}^{\prime}\) which includes the elements of the universal set but not the elements of set \(A\). The universal set consists of a set of all elements of all its related subsets, whereas the empty set contains no elements of the subsets. Thus, the complement of the universal set is an empty set (null set), denoted by ‘ \(\{ \}\) ‘ or the symbol ‘ \(\phi\) ‘.

Example 1: Let us say, there are three sets named as \(A, B\) and \(C\). The elements of all sets \(A, B\) and \(C\) is defined as;
\(
\begin{aligned}
&A=\{1,3,6,8\} \\
&B=\{2,3,4,5\} \\
&C=\{5,8,9\}
\end{aligned}
\)
Find the universal set for all the three sets \(A, B\) and \(C\).

Solution: By the definition we know, the universal set includes all the elements of the given sets. Therefore, the Universal set for sets A, B, and C will be, \(\mathrm{U}=\{1,2,3,4,5,6,8,9\}\)

From the above example, we can see that the elements of sets \(A, B\), and \(C\) are altogether available in Universal set ‘U’. Also, if you observe, no elements in the universal set are repeated and all the elements are unique.