CARDINAL NUMBER OF POWER SET

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 five elements. So, n = 5. 

Then, 

n[P(A)]  =  2⁵

n[P(A)]  =  32

Cardinality of the power set of A is 32. 

Example 2 :

If the cardinal 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

Number of elements of A is 4. 

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 =  2ⁿ

Formula to find the number of proper subsets :

Number of proper subsets =  2ⁿ-1

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  =  {5}

It has two subsets. They are {  } and {5}. 

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

What 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.  

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.

So, 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.

So, 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 five elements. 

Number of proper subsets  =  2⁵ - 1

=  32 - 1

=  31

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}, { }}

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