Universal set

Universal Set Definition

Before diving into formulas and symbols, think of something simple. Imagine your entire classroom — students, desks, bags, books, everything inside it. Now, if you start forming smaller groups like “only students” or “only desks,” those smaller groups all come from that one big collection. That complete collection is exactly what mathematics calls the universal set.

In set theory and discrete mathematics, the universal set is defined as the set that contains all elements under consideration, including the elements of every related subset. It acts as the master or reference set from which all other sets are formed. The universal set is commonly denoted by $U$, $S$, or $\Omega$ and its subsets are represented by symbols such as $A, B, C,$ and so on.

Universal Set Examples

Understanding the universal set becomes much easier with everyday and mathematical examples.

Mathematical Example

If we consider the set of all integers, the universal set may be chosen as the set of rational numbers or even the set of real numbers. Since every integer belongs to these larger sets, they naturally act as the universal set.

Real-Life Example: Animals in a Jungle

Suppose $A$ = set of all tigers
$B$ = set of all deer

Then the universal set can be

$U$ = set of all animals in the jungle

Here, both $A$ and $B$ are subsets of $U$.

Real-Life Example: Types of Flowers

Let $A$ = set of all roses
$B$ = set of all tulips

Then the universal set could be

$U$ = set of all flowers

Again, $A$ and $B$ are subsets of the larger collection.

Alphabet Example

Consider all English letters:

$U = \{a,b,c,\dots,x,y,z\}$

If we focus only on vowels:

$A = \{a,e,i,o,u\}$

Here, the set of vowels is clearly a subset of the universal set of all letters.

Venn Diagram of Universal Set

Let $U = \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$ and $A = \{10,4,2,6,1,9,3\}$, $B=\{4,13,1,9,3,15,14\}$ and $C =\{2,6,1,9,3,14,15,11,12\}$.

The Venn diagram to represent the universal set and its subsets of the above example is

Properties of the Universal Set

Think of the universal set like the “big box” that already contains everything. No matter what smaller sets you mix, remove, or compare, everything ultimately stays inside that one master collection. Because of this, the universal set plays a central role in set operations, complements, probability, and discrete mathematics.

Let’s go through the key properties of the universal set in a simple and exam-focused way.

Universal Set Contains All Elements

The universal set includes every element under consideration in a particular problem.
All other sets are simply subsets of the universal set.

So, for any set $A$,

$A \subseteq U$

Union with Universal Set

When we take the union of any set with the universal set, the result is always the universal set.

$U \cup A = U$

This happens because $U$ already contains all elements of $A$.

Note: If $A \subseteq B$, then

$A \cup B = B$

Complement of Universal Set

Complement of a Set

Let $U$ be the universal set and $A$ be a subset of $U$.
The complement of $A$ contains all elements that are in $U$ but not in $A$.

It is denoted by $A'$ or $A^C$

Mathematically, $A' = \{x : x \in U \text{ and } x \notin A\}$

or $A' = U - A$

Complement of Universal Set Itself

If we take the complement of the entire universal set, nothing remains. $U' = \phi$

So, the complement of the universal set is always the empty set.

Cardinality of Universal Set

The cardinality of the universal set refers to the total number of elements it contains.
It is denoted by $|U|$ or $n(U)$

Finite Example

If $U = \{1, 2, 3, 4, 5\}$ then $|U| = 5$

Infinite Example

If $U = \mathbb{N} = \{1, 2, 3, \ldots\}$ then the cardinality is infinite.

Understanding the size of the universal set is important in probability, counting problems, and combinatorics.

Universal Set and Union of Sets

The universal set may contain all elements of multiple subsets and sometimes even more elements beyond them.

Example

Let

$A = \{a, b, c\}$
$B = \{e, f, g\}$

Then $A \cup B = \{a, b, c, e, f, g\}$

A possible universal set could be $U = \{a, b, c, d, \ldots, z\}$

Here, the union includes only elements of $A$ and $B$, but the universal set includes everything under consideration.

Relationship with Subsets and Complement

Every set in a problem is defined relative to the universal set, and complements are always calculated with respect to it.

Example

If $U = \{a, b, c, d\}$
$A = \{a, b\}$

then $A' = U - A = {c, d}$

This clearly shows why the universal set is essential for defining complements, De Morgan’s Laws, and logical operations.

Universal Set vs Other Types of Sets

Understanding how the universal set compares to other types of sets helps clarify its unique role in set theory. This section highlights the key differences between the universal set and subsets, empty sets, and both finite and infinite sets.

Difference Between Universal Set and Subset

A universal set contains all elements under discussion, while a subset includes some or all of those elements.
If $U = \{1, 2, 3, 4\}$ and $A = \{2, 3\}$, then clearly:

$A \subseteq U \quad \text{but} \quad U \not\subseteq A$

This comparison highlights the hierarchical relationship between a universal set and its subsets.

Difference Between Universal Set and Empty Set

The empty set (or null set) contains no elements:

$\emptyset = \{\}$

On the other hand, the universal set contains all possible elements under the context.
For instance, if $U = \{a, b, c\}$, then:

$\emptyset \subseteq U \quad \text{but} \quad \emptyset \ne U$

This shows the extreme contrast between the null set and the universal set in terms of cardinality and content.

Comparison with Finite and Infinite Sets

A universal set can be either finite or infinite, depending on the context.

• Finite example: $U = \{1, 2, 3, 4, 5\}$
• Infinite example: $U = \{1, 2, 3, 4, \ldots\}$
  In contrast, a finite set contains a countable number of elements, while an infinite set goes on endlessly.
  Thus, the universal set may overlap with or include both finite and infinite sets, depending on how it is defined.

Difference Between Universal Set and Union of Sets

In set theory and discrete mathematics, students often confuse the universal set with the union of sets. The trick is simple: the universal set is the complete collection, while the union is just a combination of specific sets.

Here’s a clean, side-by-side comparison to lock it in your memory.

Solved Examples Based On the Universal Set

Example 1: If $A$ is any set, then which of the following is not a property of a Universal Set?
1) $A \cup A^c=U$
2) $A \cap A^c=U$
3) $U-A=A^c$
4) $U-A^c=A$

Solution
There is no common portion between $A$ and $A^{\prime}$,
$A\cap A^{c}=\phi$. So $(2)$ is wrong.
Hence, the answer is option 2.

Example 2: If $\mathrm{G}=\{-9,-8,-7,-6\}$ and $\{8,2,7,4\}$, then which of the following MAY BE a universal set?
1) Set of all whole numbers
2) Set of all irrational numbers
3) Set of all integers
4) All of the above

Solution
We know the elements of both sets $G$ and $H$ are there in the set of all integers, hence option (3) can be a universal set.
Hence, the answer is option 3.

Example 3: If $U=\{a, b, c, d, e, f, h, i, j\}$ which of the following is not a subset of $U$ ?
1) $\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$
2) \{c.d.g\}
3) $\{\mathrm{h}, \mathrm{b}, \mathrm{j}\}$
4) $\{\mathrm{a}, \mathrm{i}, \mathrm{d}\}$

Solution
As we learnt
A set that contains all sets in a given context is called the "Universal Set". The universal set is usually denoted by $U$, and all its subsets by the letters $A, B, C$, etc.
wherein
R is the universal set of $\mathrm{N}, \mathrm{W}$ and Z .
Set U has no element ' g ' in it.
Hence, the answer is option 2.

Example 4: If $\mathrm{n}(\mathrm{U})=100, \mathrm{n}(\mathrm{A})=30, \mathrm{n}(\mathrm{B})=40$ and $n(A \cap B)=10$. Find $n\left(A^c \cap B^c\right)$
1) 30
2) 40
3) 50
4) 60

Solution
$\begin{aligned} & n(A \cup B)=n(A)+n(B)-n(A \cap B) \\ & n(A \cup B)=30+40-10=60 \\ & n(A \cup B)^{\prime}=n\left(A^{\prime} \cap B^{\prime}\right)=n(U)-n(A \cap B) \\ & =100-60=40\end{aligned}$
Hence, the answer is option 2.

Example 5: The set which contains those elements of the Universal set which are not contained in set $A$ is called the complement of set $A$. (True/False)

Solution
Complement of a Set:
If $U$ is a universal set and $A$ is a subset of $U$, then the complement of $A$ is the set which contains those elements of $U$ which are not contained in $A$. Symbolically, we write $A′$ to denote the complement of $A$ with respect to $U$.
The statement is true.

List of Topics Related to Universal Set

To understand the concept of a universal set thoroughly, it is helpful to explore the related foundational topics in set theory. These concepts support the definition, application, and operations involving universal sets. This section lists all key topics connected to universal sets.

NCERT Resources

Explore essential NCERT study materials for Sets, with comprehensive solutions, concise revision notes, and curated exemplar problems. These resources are designed to enhance your conceptual clarity and prepare you effectively for board and competitive exams.

NCERT Solutions for Class 11 Chapter 1 Sets

NCERT Notes for Class 11 Chapter 1 Sets

NCERT Exemplar for Class 11 Chapter 1 Sets

Practice Questions

Mastering any mathematical concept comes through continuous practice. To help strengthen your understanding of the topic, we have given below some practice questions that will test your knowledge of formulas, important properties and general application of knowledge.