## 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=16Hence, 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

## Frequently Asked Questions (FAQ)

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