# SUPERSET MEANING

Superset meaning :

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

More clearly, every element of X is also an element of Y and X is not equal to Y. That is, number of elements of X is less than the number of elements of Y.

Example :

Let Y  =  {1, 2, 3, 4, 5} and X  =  {1, 3, 5}

In the above two sets, every element of X is also an element of Y and also number of elements of X is less than number of elements of Y.

That is, n(x)  =  3 and n(Y)  =  5 -----> n(x) < n(Y)

Hence, Y is the superset of X.

Apart from the stuff "Superset meaning", let us know some other important stuff about subsets of a set.

## Subset of a set

Subsets of

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

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

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.

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

## 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.

## 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 "Superset meaning".

Apart from the stuff, "Superset meaning", if you need any other stuff in math, please use our google custom search here.

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