Universal Set: Class 11 Maths Notes, Definition, Formula, Properties & Solved Examples

We define universal set as a collection that holds all the elements of the other related sets without any repetition. It provides a comprehensive boundary within which all subsets exist.

For example, if $X=\left\{\mathrm{1,3}\right\}$  and $Y=\{1,a,c\}$ , then the universal set that includes these sets would be $U=\{\mathrm{1,3},a,c\}$ .

It ensures that every element of any subset is also an element of the universal set.

Illustrative Example of Universal Set

Let's consider two sets. 

• Set $P=\{\mathrm{2,4},\mathrm{6,8},10,\dots \}$ , consisting of even numbers.
• Set $Q=\{\mathrm{1,3},\mathrm{5,7},9,\dots \}$ , consisting of odd numbers.

In this case, the universal set $U$  would contain all natural numbers:

$\mathrm{U}=\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6},\mathrm{7,8},\mathrm{9,10},\dots \}$

Both even and odd numbers are part of natural numbers. So, U encompasses all elements of P and Q.

Importance of Universal Set (for Engineering exams)

1. The JEE Mains exam usually tests on universal sets with Venn diagrams, where you may have to identify elements within various regions.
2. Some questions on universal sets could be related to finite and infinite sets, or it could be around related concepts in the sets chapter of your NCERT textbook. Try going through our page on Sets and their representations, too!    

Section 1.7 of your NCERT textbook touches upon universal sets. It briefly describes with an example. 

"Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers. For another example, in human population studies, the universal set consists of all the people in the world."

 

 

For every set, a complement can be identified. That includes all the elements not in the given set but within the universal set.

But, the complement of the universal set itself is an empty set because there are no elements outside of it.

The empty set is also known as a null set. It contains no elements and is denoted by ' $\}$  '. This reflects that the universal set covers all possible elements under discussion.

Universal sets can be finite or infinite. When dealing with a vast or unending number of elements, ellipses (three dots '...') are often used for representation.
For example:
$-U=\{\mathrm{1,2},3,\dots \}$ (infinite)
$-U=\{\mathrm{1,2},3,\dots ,4000\}$ (finite)

Solved Examples of Universal Sets

Example 1:
Given three sets:

$\begin{array}{rr}& A=\{\mathrm{2,4},6\}\\ & B=\{\mathrm{3,7},8\}\\ & C=\{\mathrm{4,8},10\}\end{array}$

To find the universal set, combine all distinct elements:

$U=\{\mathrm{2,3},\mathrm{4,6},\mathrm{7,8},10\}$

Example 2:
Given:
$P=\{\mathrm{1,3},9,\dots \}$ (odd numbers)
$Q=\{\mathrm{2,4},6,\dots \}$ (even numbers)
The universal set $U$  includes all natural numbers:

$\mathrm{U}=\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6},\mathrm{7,8},\mathrm{9,10},\dots \}$

Universal Set Representation Through Venn Diagram

A Venn diagram visually represents the relationships between sets. In such diagrams, sets are shown as circles, and the universal set is depicted as a rectangle enclosing all the circles. This makes it easy to visualize the common, distinct, and complementary elements among sets.
Example of universal set.
$\mathit{\xi }$  represents the universal set.
A represents set A and B represents set B, usually a capital letter is used.

Consider U as the universal set and $\mathrm{X},\mathrm{Y}$ , and Z as any three finite sets. The key properties include:

1. $\bar{n}(X-Y)=n\left(X\right)-\bar{n}(X\cap Y)$

2. $\bar{n}(\mathrm{X}\cup \mathrm{Y})=\mathrm{n}\left(\mathrm{X}\right)+\mathrm{n}\left(\mathrm{Y}\right)-\bar{\mathrm{n}}(\mathrm{X}\cap \mathrm{Y})$

3. If $X$ and  $Y$ are disjoint non-void sets, then $\bar{n}(X\cup Y)=n\left(X\right)+n\left(Y\right)$

4. $\bar{n}\left(\right(X-Y)\cup (Y-X\left)\right)=n\left(X\right)+n\left(Y\right)-2\bar{n}(X\cap Y)$

5. The union of a set and its complement equals the universal set: $\mathrm{X}\cup {\mathrm{X}}^{\text{'}}=\mathrm{U}$

6. The complement of the universal set is the empty set, and vice-versa: ${\mathrm{\varnothing }}^{\text{'}}=\mathrm{U}$ and ${\mathrm{U}}^{\text{'}}=\mathrm{\varnothing }$