CARDINAL NUMBER OF POWER SET

About "Cardinal number of power set"

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)]  =  2

Note :

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

Cardinal number of power set - Examples

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)]  =  2

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

2  =  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. 

Subset of a given 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"

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

Reading notation :

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.

Reading notation :

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

Super set

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 

Formula to find number of subsets

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 =  2

Formula to find the number of proper subsets :

Number of proper subsets =  2ⁿ¹

Null set is a subset or proper 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.  

If null set is a super set

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.

Subset of a given set - 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

=  2ⁿ¹

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  =  2¹

=  2

=  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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