NUMBER OF PROPER SUBSETS OF A SET

If A is the given set and it contains 'n' number of elements, then we can use the formula given below to find the number of subsets for A.

Number of subsets = 2ⁿ

And also, we can use the formula given below to find the number of proper subsets.

Number of proper subsets = 2ⁿ - 1

Difference between Subsets and Proper Subsets

Let us consider the set A.

A = {a, b, c}

Here, A contains 3 elements.

So, n = 3.

Then, the number of subsets is

= 2³

= 8

The subsets are

{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, {  }

In the above list of subsets, the subset {a , b, c} is equal to the given set A.

The subset which is equal to the given set can not be considered as proper subset.

The remaining 7 subsets are proper subsets.

Proper subsets of A :

{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {  }

Improper subset of A :

{a, b, c}

Note :

A subset which is not a proper subset is called as improper subset.

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.

Subsets of Null Set

Let us consider null set or empty set given blow.

{  }

Here, the above null set contains zero elements.

So, n = 0.

Then, the number of subsets is

= 2⁰

= 1

The subset of null set is

{  }

The above subset {  } is equal to the given null set.

So, null set has only one subset which is equal to it.

So it is improper subset.

Therefore, null set has no proper subset.

Note :

1. Null set is the only set which has no proper subset.

2. A set which contains only one subset is called null set. 

Reading Notation

Subset of a 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).

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 given by

n[P(A)] = 2ⁿ

Super Set

If the set X is said to be a subset of set Y, then

Y is called the super set of X

Practice Problems

Problem 1 :

Let A = {1, 2, 3, 4, 5}. Find the number of proper subsets of A.

Solution :

The given set A contains 5 elements.

Then, n = 5.

Formula to find number of proper subsets is

= 2ⁿ - 1

Substitute n = 5.

= 2⁵ - 1

= 32 - 1

= 31

So, the given set A has 31 proper subsets.

Problem 2 :

Let A = {a, e, i, o, u}. Find the number of subsets of A.

Solution :

The given set A contains 5 elements.

Then, n = 5.

Formula to find number of subsets is

= 2ⁿ

Substitute n = 5.

= 2⁵

= 32

So, the given set A has 32 subsets.

Problem 3 :

Let A = {a, b, c, d}. Find the cardinality of power set of A

Solution :

The given set A contains 4 elements.

Then, n = 4.

The formula to find the cardinality of power set of A is

n[P(A)] = 2ⁿ

Substitute n = 4.

n[P(A)] = 2⁴

n[P(A)] = 16

So, the cardinality of the power set of A is 16.

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