PROPER SUBSET OF A SET

About "Proper subset of a set"

Proper subset of a 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

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

The figure given below illustrates this.

Proper subset of a 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.

Apart from the stuff "Proper subset of a 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

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"

Power set

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

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

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)]  =  2

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.

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

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