If A is the given set and it contains 'n' number of elements, then we can use the formula given below to find the number of subsets for A.

**Number of subsets = 2 ^{n}**

**And also, we can use the formula given below to find the number of proper subsets. **

**Number of proper subsets = ****2 ^{n}-1**

Let us consider the set A.

A = {a, b, c}

Here, A contains 3 elements.

So, n = 3.

Then, the number of subsets is

= 2^{3}

= 8

The subsets are

{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, { }

In the above list of subsets, the subset {a , b, c} is equal to the given set A.

The subset which is equal to the given set can not be considered as proper subset.

The remaining 7 subsets are proper subsets.

Proper subsets of A :

{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, { }

Improper subset of A :

{a, b, c}

**Note : **

A subset which is not a proper subset is called as improper subset.

Null set is a proper subset for any set which contains at least one element.

For example, let us consider the set A = { 1 }

It has two subsets. They are { } and { 1 }.

Here, null set is proper subset of A. Because null set is not equal to A.

Let us consider null set or empty set given blow.

{ }

Here, the above null set contains zero elements

So, n = 0.

Then, the number of subsets is

= 2^{0}

= 1

The subset of null set is

{ }

The above subset { } is equal to the given null set.

So, null set has only one subset which is equal to it.

So it is improper subset.

Therefore, null set has no proper subset.

**Note :**

1. Null set is the only set which has no proper subset.

2. A set which contains only one subset is called null set.

**Subset of a Set :**

A set X is a subset of set Y if every element of X is also an element of Y.

In symbol, we write

**x ⊆ y**

Read ⊆ as "X is a subset of Y" or "X is contained in Y"

Read ⊈ as "X is a not subset of Y" or "X is not contained in Y".

**Proper Subset : **

A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.

In symbol, we write X ⊂ Y

Read X ⊂ Y as "X is proper subset of Y"

The figure given below illustrates this.

**Power Set : **

The set of all subsets of A is said to be the power set of the set A.

The power set of A is denoted by P(A).

We already know that the set of all subsets of A is said to be the power set of the set A and it is denoted by P(A).

If A contains "n" number of elements, then the formula for cardinality of power set of A is given by

n[P(A)] = 2ⁿ

If the set X is said to be a subset of set Y, then

Y is called the super set of X

**Problem 1 :**

Let A = {1, 2, 3, 4, 5}. Find the number of proper subsets of A.

**Solution : **

The given set A contains 5 elements.

Then, n = 5.

Formula to find number of proper subsets is

= 2^{n }- 1

Substitute n = 5.

= 2^{5} - 1

= 32 - 1

= 31

So, the given set A has 31 proper subsets.

**Problem 2 :**

Let A = {a, e, i, o, u}. Find the number of subsets of A.

**Solution : **

The given set A contains 5 elements.

Then, n = 5.

Formula to find number of subsets is

= 2^{n}

Substitute n = 5.

= 2^{5}

= 32

So, the given set A has 32 subsets.

**Problem 3 :**

Let A = {a, b, c, d}. Find the cardinality of power set of A

**Solution : **

The given set A contains 4 elements.

Then, n = 4.

The formula to find the cardinality of power set of A is

n[P(A)] = 2^{n}

Substitute n = 4.

n[P(A)] = 2^{4}

n[P(A)] = 16

So, the cardinality of the power set of A is 16.

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