**Number of proper subsets of a set :**

If A is the given set and it contains "n" number of elements, we can use the following formula to find the number of subsets.

**Number of subsets = 2ⁿ**

**Formula to find the number of proper subsets :**

**Number of proper subsets = ****2ⁿ****⁻¹**

**Example 1 :**

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

**Solution : **

Let the given set contains "n" number of elements.

Then, the formula to find number of proper subsets is

**= ****2ⁿ****⁻¹**

The value of "n" for the given set A is "5".

Because the set A = {1, 2, 3, 4, 5} contains "5" elements.

Number of proper subsets = 2⁵⁻¹

= 2⁴

= 16

**Hence, the number of proper subsets of A is 16.**

**Example 2 :**

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

**Solution : **

Let the given set contains "n" number of elements.

Then, the formula to find number of subsets is

**= ****2ⁿ**

The value of "n" for the given set A is "5".

Because the set A = {a, e, i, o, u} contains "5" elements.

Number of proper subsets = 2⁵ = 32

**Hence, the number of subsets of A is 32.**

**Example 3 :**

Let A = {a, b, c, d, e} find the cardinality of power set of A

**Solution : **

The formula for cardinality of power set of A is given below.

**n[P(A)] = 2ⁿ**

Here "n" stands for the number of elements contained by the given set A.

The given set A contains "5" elements. So n = 5.

Then, we have

n[P(A)] = 2⁵

n[P(A)] = 32

**Hence, the cardinality of the power set of A is 32. **

Apart from the stuff "Number of proper subsets of a set ", let us come to know some other important stuff about subsets 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**

**Reading Notation :**

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"

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

**Reading notation :**

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

The figure given below illustrates this.

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

**Reading notation :**

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

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

Here,

**Y is called super set of X **

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

**n[P(A)] = 2ⁿ**

**Note :**

Cardinality of power set of A and the number of subsets of A are same.

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.

If null set is a super set, then it has only one subset. That is { }.

More clearly, null set is the only subset to itself. But it is not a proper subset.

Because, { } = { }

Therefore, A set which contains only one subset is called null set.

After having gone through the stuff given above, we hope that the students would have understood "Number of proper subsets of a set".

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