SUBSETS 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

Cardinality 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 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 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. 

Solved 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.

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 5 elements. 

Number of proper subsets :

= 2⁵ - 1

= 32 - 1

= 31

So, the number of proper subsets of A is 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.

Number of subsets = 2³ = 8.

Therefore,

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

Example 5 :

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

So, the cardinality of the power set of A is 32. 

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