Subsets and types of subsets

What is a Subset in Maths?

In daily life, we often create smaller groups from a bigger collection. For example, from a class of students, you might select only the football team members. This smaller group still belongs to the main class. You’re not creating a new collection — you’re simply choosing some members from the original group.

Mathematics follows the same logic, and this is exactly what we call a subset.

Definition of a Set in Mathematics

Before learning about subsets, it’s important to understand what a set is.

In set theory, a set is a well-defined collection of distinct objects called elements. These elements can be:

• numbers
• letters
• symbols
• people
• or any clearly identifiable items

Sets are usually represented using capital letters such as $A, B, C, S, U,$ and $V$.

What is a Subset?

A subset in maths is a set formed by selecting elements from another set.

If every element of one set is also present in another set, then the first set is called a subset of the second set.

In simple words: A subset is a smaller group completely contained inside a larger set.

For example, if $A = \{1,2,3,4\}$ and $B = \{1,2\}$

then $B \subseteq A$ because all elements of $B$ are inside $A$.

Why are Subsets Important in Set Theory?

Understanding subsets is very important because they form the foundation of many mathematical concepts. Subsets are widely used in:

• set theory fundamentals
• algebra and relations
• relations, functions and mappings
• probability and statistics
• logical reasoning
• data grouping and classification

Without subsets, many advanced maths topics become harder to understand.

Examples of Subsets

In mathematical terms, if $A$ and $B$ are two sets, then $A$ is a subset of $B$, written as $A \subseteq B$, if every element of $A$ is also an element of $B$. This relationship is formally defined as:

$A \subseteq B \Leftrightarrow \forall x(x \in A \Rightarrow x \in B)$

Let’s take a few simple examples to understand this:

Example 1:
Let $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4, 5\}$
Since every element of $A$ is present in $B$, we can say:
$A \subseteq B$

Example 2:
Let $P = \{a, b\}$ and $Q = \{a, b, c\}$
Then:
$P \subseteq Q$

Example 3:
Let $X = \{7, 8\}$ and $Y = \{6, 7, 8, 9\}$
Then clearly:
$X \subseteq Y$

Symbol and Notation of Subsets in Set Theory

In set theory, the notation for subsets plays a crucial role in expressing mathematical relationships. The standard symbols include:

• $A \subseteq B$: $A$ is a subset of $B$ (possibly equal).
• $A \subset B$: $A$ is a proper subset of $B$ (all elements of $A$ are in $B$, but $A \ne B$).
• $A \not\subseteq B$: $A$ is not a subset of $B$.

These notations help represent inclusion relationships between sets clearly and concisely.

What is the difference between $\subseteq$ and $\subset$? The symbol $\subseteq$ represents a subset while $\subset$ represents a proper subset.

For example:

If $M = \{2, 4\}$ and $N = \{2, 4, 6\}$, then:
$M \subset N$

However, if $M = \{2, 4, 6\}$ and $N = \{2, 4, 6\}$, then:
$M \subseteq N$ but not $M \subset N$

Real-Life Examples

Subsets are not just abstract concepts; they are used frequently in real-world contexts. Consider the following examples:

Example 1: Subjects in a Class
If Set $A = \{\text{Math, Science}\}$ and Set $B = \{\text{Math, Science, History, English}\}$, then:
$A \subseteq B$
Here, the subjects Math and Science are part of a larger curriculum.

Example 2: Team Members
If $A = \{\text{Riya, Karan}\}$ and $B = \{\text{Riya, Karan, Mitali, Aarav}\}$, then:
$A \subseteq B$
This shows a smaller group of team members inside a larger team.

Example 3: Fruits in a Basket
Let $A = \{\text{Apple, Banana}\}$ and $B = \{\text{Apple, Banana, Mango, Orange}\}$. Then:
$A \subseteq B$
This represents a subset of fruits taken from a bigger fruit basket.

These examples show how subsets help in organising and analysing grouped data or categories in daily life, education, and business logic.

Types of Subsets in Set Theory

In set theory, subsets are not all the same. Depending on how a smaller set relates to the original or larger set, we can classify it into different categories. Understanding the types of subsets in mathematics helps us analyse relationships between sets more clearly and solve problems in algebra, probability, logic, and data grouping.

The main classifications of subsets include:

• proper subsets
• improper subsets
• empty set
• universal set
• finite and infinite subsets

Let’s understand each type one by one in a simple and practical way.

Proper Subset – Definition, Symbol, and Examples: What is a Proper Subset?

A proper subset is a subset that contains some but not all elements of the original set.

In easy words, the smaller set is completely inside the larger set, but both sets are not equal.

If set $A$ is a proper subset of set $B$, then every element of $A$ belongs to $B$, but at least one element of $B$ is missing in $A$.

Proper Subset Symbol

$A \subset B$

This symbol $\subset$ is read as
“A is a proper subset of B”.

Example of Proper Subset

Let $A = \{1,2\}$ and $B = \{1,2,3\}$.

Since all elements of $A$ are in $B$ and $A \ne B$, we have $A \subset B$. So, $A$ is a proper subset of $B$.

Total Number of Proper Subsets (Formula)

If a set contains $n$ elements:

Total subsets $= 2^n$
Proper subsets $= 2^n - 1$

Example

Let $X = \{1,2\}$

Number of elements $= 2$

Total subsets $= 2^2 = 4$
Proper subsets $= 2^2 - 1 = 3$

Proper subsets are: $\{\}, \{1\}, \{2\}$

The set $\{1,2\}$ itself is not proper because it is equal to $X$.

Improper Subset – Meaning and Examples: What is an Improper Subset?

An improper subset is a subset that is exactly equal to the original set.

This means no elements are removed. The subset and the parent set are identical.

Important idea: Every set is always an improper subset of itself.

Example

Let $A = \{a,s,c\}$

Subsets: $\{\}, \{a\}, \{s\}, \{c\}, \{a,s\}, \{a,c\}, \{s,c\}, \{a,s,c\}$

Here, $\{a,s,c\}$ is the improper subset, and all others are proper subsets.

Special Types of Subsets in Mathematics

Some subsets are very important in basic set theory concepts and appear frequently in exams and problem-solving.

Empty Set (Null Set)

The empty set contains no elements at all.

Symbol: $\emptyset$ or $\{\}$

Key properties:

• subset of every set
• always exists
• considered a proper subset of any non-empty set

Example

Let $A = \{x,y\}$

Then $\emptyset \subset A$

Universal Set

The universal set contains all elements under consideration in a particular problem or discussion.

Symbol: $U$

Every other set in that context is a subset of the universal set.

Example

Let $U = \{1,2,3,4,5\}$
$A = \{2,3\}$

Then $A \subseteq U$

Finite and Infinite Subsets

Subsets can also be classified based on the number of elements they contain.

Finite Subsets

A finite subset has a limited or countable number of elements.
All elements can be listed clearly.

Example

$A = \{2,4,6\}$
$B = \{2,4,6,8,10\}$

$A \subset B$

Here, $A$ is a finite subset.

Infinite Subsets

An infinite subset has unlimited or infinitely many elements.
These are common in number systems like natural numbers or real numbers.

Example

Let $A = \{2,4,6,8,10,\dots\}$ (even numbers)
$B = \mathbb{N}$ (natural numbers)

Then $A \subset B$

So, $A$ is an infinite proper subset of the infinite set $B$.

Properties of Subsets

Understanding the properties of subsets is crucial for working efficiently with set operations and relations in mathematics. These properties form the basis for how sets interact and are often used in proving statements or solving problems in algebra, logic, and number theory.

Basic Rules and Characteristics of Subsets

Below are the fundamental rules and characteristics that apply to all subsets:

• Reflexive Property: Every set is always a subset of itself.
  For any set $A$, $A \subseteq A$
• Empty Set Property: The empty set is a subset of every set.
  For any set $A$, $\emptyset \subseteq A$
• Total Number of Subsets: If a set has $n$ elements, then the total number of possible subsets is given by: $2^n$
• Proper Subsets Count: The number of proper subsets of a set with $n$ elements is: $2^n - 1$
• Singleton Set: A set with a single element is called a singleton set and is a subset of any set that contains that element.

Relationship Between Sets and Subsets

The relationship between sets and subsets is hierarchical and directional. It is based on element inclusion:

• If $A \subseteq B$, every element of $A$ is in $B$, but $B$ may contain additional elements.
• If $A \subset B$, then $A$ is a proper subset of $B$, meaning $A$ is not equal to $B$.
• If $A = B$, both sets contain exactly the same elements; hence, $A$ is an improper subset of $B$.

Example:

Let $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then:

• $A \subset B$ (proper subset)
• $A \subseteq B$ (subset)
• $A \ne B$

Also, note:

• $\emptyset \subseteq A$
• $A \subseteq A$

Important Theorems on Subsets

Several important theorems govern how subsets behave and relate to one another:

Theorem 1:

If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.
(This is known as the transitive property of subsets.)

Theorem 2:

The empty set is a subset of every set:
$\emptyset \subseteq A$

Theorem 3:

If $A$ has $n$ elements, then its power set, the set of all subsets, is denoted by $P(A)$ and contains:
$|P(A)| = 2^n$ where $|\cdot|$ denotes cardinality (number of elements).

Example:

Let $A = \{x, y\}$, then:

• Subsets: $\emptyset$, $\{x\}$, $\{y\}$, $\{x, y\}$
• Total subsets: $2^2 = 4$

Power set of $A$:
$P(A) = \{\emptyset, \{x\}, \{y\}, \{x, y\}\}$

These properties and theorems help in building a strong foundation in understanding types of subsets, set operations, and logical proofs in higher mathematics.

Power Set of a Set

The power set of a set is defined as a set of all the subsets (along with the empty set and the original set). The power set of a set Y is denoted by P(Y). If Y has n elements, then P(Y) has 2ⁿ elements. For example,

• If $E = \{x, p\}$, then $P(E) = \{ \{ \}, \{x\}, \{p\}, \{x, p\} \}$
  Total number of elements = 4.
• If $A = \{a, b, c\}$, then $P(A) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{c,a\}, \{a,b,c\}\}$

Observe that A has 3 elements and P(A) has $2^3 = 8$ elements.

Intervals as Subsets of Real Numbers

In set theory and real analysis, intervals are special types of subsets of real numbers. They represent all real numbers lying between two endpoints and are widely used in functions, calculus, and inequalities. These intervals can be bounded or unbounded and are always subsets of the real number line $ \mathbb{R} $.

Open, Closed, and Half-Open Intervals

Intervals are classified based on whether their endpoints are included or not:

Open Interval $(a, b)$:
Includes all real numbers strictly between $a$ and $b$ (excluding endpoints).
$(a, b) = \{x \in \mathbb{R} \mid a < x < b\}$

Closed Interval $[a, b]$:
Includes all real numbers between $a$ and $b$, including both endpoints.
$[a, b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}$

Half-Open Interval

$[a, b)$:
Includes $a$ but not $b$.
$[a, b) = \{x \in \mathbb{R} \mid a \leq x < b\}$

$(a, b]$:
Includes $b$ but not $a$.
$(a, b] = \{x \in \mathbb{R} \mid a < x \leq b\}$

Graphical Representation of Intervals

On a number line:

• Closed dots represent inclusion of endpoints (used for $[a, b]$).
• Open dots represent exclusion of endpoints (used for $(a, b)$).

Examples:

• $[2, 5]$: solid dots at $2$ and $5$
• $(2, 5)$: open dots at $2$ and $5$
• $[2, 5)$: solid dot at $2$, open dot at $5$

Difference Between Intervals and General Subsets

While intervals are continuous subsets of $\mathbb{R}$, general subsets can be discrete or non-continuous.

• Interval Example:
  $(1, 5) = \{x \in \mathbb{R} \mid 1 < x < 5\}$
• General Subset Example:
  $A = \{1, 2, 3, 5\}$
  (Not an interval, as $4$ is missing.)
• Notation: Intervals use (, ), [, ]; subsets use curly braces $\{\}$.

Infinite Intervals

Intervals may extend infinitely:

• Left-Unbounded:
  $(-\infty, b) = \{x \in \mathbb{R} \mid x < b\}$
• Right-Unbounded:
  $(a, \infty) = \{x \in \mathbb{R} \mid x > a\}$
• Entire Real Line:
  $(-\infty, \infty) = \mathbb{R}$

Solved Examples based on Subsets and Types of Subsets

Example 1: Let $A$ and $B$ be two sets containing four and two elements, respectively. Then the number of subsets of the set $A \times B$, each having at least three elements, is:

Solution: $A$ set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$.
wherein
It is represented by $\subset$. eg. $A \subset B$ if $A=\{2,4\}$ and $B=\{1,2,3,4,5\}$
Let $A$ having elements $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$ and $B$ having $\{\mathrm{e}, \mathrm{f}\}$
Then $A \times B$ having 8 elements and no. of subsets $=2^8=256$
No. of subsets $=\phi,(a e), \ldots \ldots(d f)$ and (ae, be) $\ldots \ldots \ldots=256$

Now $8$ subsets have only one element

$\{ae\},\{af\},\{be\},.......\{df\}$

Similarly, No. of the sets having two elements

$\{ae,af\}, \{ae,be\}, .....\{ae,df\}=7$ elements

$\{af,be\},\{af,bf\}..........= 6$ elements

$7 + 6 + 5 + 4 + 3 + 2 + 1 = 28$ (having two elements)

and a subset having a single element $\phi$

$28 + 8+1 = 37$

At least three elements $= 256 - 37 = 219 $

Hence, the answer is $219.$

Example 2: If $A=\{1,2,3,5,7\}$ and $B=\{3,5\}$, then :
1) $A \subset B$
2) $B \not \subset A$
3) $A=B$
4) $B \subset A$

Solution: All elements of $B$ are present in $A$, thus $B \subset A$.
Hence, the answer is option 4.

Example 3: If a set $A$ has $8$ elements, then the number of proper subsets of the $\operatorname{set} A$ is
Solution:
Number of proper subsets $=2^8-1=256-1=255$
Hence, the answer is $255$.

Example 4: Let $S=\{1,2,3, \cdots, 100\}$. The number of non-empty subsets A of S such that the product of elements in A is even is:
1) $2^{50}\left(2^{50}-1\right)$
2) $2^{100}-1$
3) $2^{50}-1$
4) $2^{50}+1$

Solution: Number of subsets of a set -
If a set has n elements, then it has $2^{\mathrm{n}}$ subsets.
In this Question,
Subsets $\boldsymbol{=}$ Total Subsets $\boldsymbol{-}$ Number of subsets which have only odd numbers
$
\begin{aligned}
& =2^{100}-2^{50} \\
& =2^{50}\left(2^{50}-1\right)
\end{aligned}
$
Hence, the answer is option 1.

Example 5: If a set has $32$ subsets. How many elements does it have?
Solution: As we know, if a set has $n$ elements, it will have $2^n$ subsets.
Thus $2^n=32$

$\Rightarrow n=5$

Hence, the answer is $5$.

List of related topics to Subsets

To strengthen your grasp of Subsets and their types, it's important to study other related concepts in set theory. Topics like Equal and Equivalent Sets, Finite and Infinite Sets, Singleton Set, Power Set, and Universal Set are closely linked and often used together in problems.

NCERT Useful Resources

This section provides a collection of valuable NCERT study materials for Class 11 Mathematics Chapter 1 – Sets. It includes comprehensive notes, solved NCERT textbook questions, and exemplar problems to help you strengthen your understanding and master the concepts effectively.

NCERT Maths Class 11 Chapter 1 Sets Notes

NCERT Maths Class 11 Chapter 1 Sets Solutions

NCERT Maths Exemplar Problems Class 11 Chapter 1 Sets

Practice Questions based on Subsets

This section offers a carefully designed collection of practice questions on subsets, created to help you strengthen your understanding of important set theory concepts in a simple and structured way. After learning the definitions and types of subsets, solving questions is the best way to build confidence and improve problem-solving speed.

Subsets, Proper Subset, Improper Subset, Intervals - Practice Question MCQ

We have given below the practice questions based on related topics of sets: