# Subset and Superset (Definition, Examples, Symbol, Difference)

## Subset

### What is Subset of a Set?

Set $A$ is called a Subset of set $B$ if all the elements of set $A$ are present in set $B$. Also, the set $B$, in this case, can be called the Superset of the set $A$.

### Subset Symbol

If set $A$ is a Subset of set $B$, then mathematically, it can be written as,

$A\subseteq B$

### Example of Subset

To understand the concept of Subset of a Set, let us consider two sets $A$ and $B$ as given below,

$A = \{1, 2, 3\}$

$B = \{1, 2, 3, 4, 5, 6\}$

From the above two sets $A$ and $B$, it can be observed that all the elements of set $A$ are also a part of set $B$. In other words, set $B$ contains all the elements of $A$ as well as some other elements. Therefore, set $A$, in this case, can be called the Subset of set $B$ and $B$ can be called Superset of the set $A$.

The figure below represents the Venn Diagram for $A$ Subset of $B$ for the above example.

From the above figure, it is clear that all the elements of set $A$ are also the elements of set $B$. Therefore, set $A$ can be called a Subset of set $B$. Mathematically it can be written as,

$A \subseteq B$

### Types of Subset

The Subsets can be divided into two types as follows:

• Proper Subsets

For the above example, set $B$ has all the elements of set $A$ as well as some other elements in it. Therefore, set $A$ can be called a Proper Subset of Set $B$.

• Improper subsets

If two sets, say $A$ and $B$, are Equal Sets, that means both the sets have the same number of elements as well as the same elements, then set $A$ is also a Subset of the set $B$. The Subset, in this case, can be called Improper Subset of the set $B$ or vice versa.

### Properties of Subset

If set $A$ is a Subset of Set $B$, then ,

• The Intersection of set $A$ with set $B$ is set $A$ itself. Mathematically,

$A \cap B = A$

• The Union of set $A$ with set $B$ is set $B$.

$A \cup B = B$

• The Cardinality of Intersection of set $A$ with set $B$ is equal to the cardinality of set $A$

$n(A\cap B)=n(A)$

### Maximum Number of Subset of a Set

If a set has $n$ number of elements, then the maximum number of Subsets that the Set can have is $2^n$.

For Example, Let us consider a set $A=\{a, b, c\}$. The set $A$ has $3$ elements. Therefore, the maximum number of Subsets of $A$ is $2^3=8$ and the Subsets of $A$ are,

$\{\},\,\{a\},\,\{b\},\, \{c\},\, \{a,\, b\},\, \{b,\, c\},\, \{c,\, a\},\, \{a,\, b,\, c\}$

### How Do you Find The Subset of a Set

Let us consider a set $B=\{p, q, r, s\}$

Before finding the Subset of a Set we need to find out how many Subsets that the given set can have. The maximum number of Subsets that a set can have is $2^n$, where, $n$ is the number of elements of the given set.

Here, $n=4$

$\therefore 2^n=16$

Hence, the given set $B$ can have a maximum of 16 numbers of Subsets.

We know that, the Empty set $\phi$ written as $\{\}$ is a Subset of every set.

Therefore, the subset of the given set will be,

Sets having a no element or Single element: $\{\},\, \{p\},\, \{q\},\, \{r\},\, \{s\}$

Sets having two elements: $\{p, q\},\, \{p, r\},\, \{p, s\},\, \{q, r\},\, \{q, s\},\, \{s, r\}$

Sets having three elements: $\{p, q, r\},\, \{p, r, s\},\, \{p, q, s\},\, \{q, r, s\}$

Sets having four elements: $\{p, q, r, s\}$

## Superset

### Superset Definition

A set $B$ is called a Superset of set $A$ if all the elements of set $A$ are present in set $B$. Therefore, the Superset is the opposite of a Subset.

### Symbol of Superset

If set $B$ is a superset of set $A$, then mathematically, it can be written as,

$B\supseteq A$

### Example of Superset

To understand the concept of Superset of a Set, let us consider the same sets $A$ and $B$ as given below.

$A = \{1, 2, 3\}$

$B = \{1, 2, 3, 4, 5, 6\}$

The set $B$ has all the elements of set $A$ as well as some extra elements. Therefore, set $B$ is a Superset of set $A$. The figure below shows the Venn Diagram of Superset of a Set.

As shown in the above figure, all the elements of set $A$ are present in set $B$. Therefore, $B$ is a Superset of set $A$.

Mathematically,

$B\supseteq A$