1.8 Universal Set

Universal set: This is a basic set; in a particular context whose elements and subsets are relevant to that particular context. For example, for the set of vowels in English alphabet, the universal set can be the set of all alphabets in English. The universal set is denoted by \(\mathbf{U}\), and all its subsets by the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C} \text {, etc. }\)

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.