**Cardinal number of power set :**

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 cardinal number 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.

**Example 1 :**

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. **

**Example 2 :**

If the cradinal number of the power set of A is 16, then find the number of elements 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.

Then, we have

16 = 2ⁿ

2⁴ = 2ⁿ

4 = n

**Hence, the number of elements of A is 4. **

Apart from the stuff "Cardinal number of power set", let us 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 **

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**

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.

To have better understand on "Subsets of a given set", let us look some examples.

**Example 1 :**

Let A = {1, 2, 3, 4, 5} and B = { 5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A. But B is equal A.

**Hence, B is the subset of A, but not a proper subset. **

**Example 2 :**

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 5}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A.

And also But B is not equal to A.

**Hence, B is a proper subset of A. **

**Example 3 :**

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 4 :**

Let A = {1, 2, 3 } find the power set of A.

**Solution : **

We know that the power set is the set of all subsets.

Here, the given set A contains 3 elements.

Then, the number of subsets = 2³ = 8

Therefore,

**P(A)** = **{**** {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }**** }**

After having gone through the stuff given above, we hope that the students would have understood "Cardinal number of power set".

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