Types of Subsets

Types of Subsets

Subsets are classified as

  • Proper Subset
  • Improper Subsets

A proper subset is one that contains a few elements of the original set whereas an improper subset, contains every element of the original set along with the null set.

For example, if set A = {2, 4, 6}, then,

Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.

Proper Subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}

Improper Subset: {2,4,6}

There is no particular formula to find the subsets, instead, we have to list them all, to differentiate between proper and improper one. The set theory symbols were developed by mathematicians to describe the collections of objects.

Proper Subsets

Set A is considered to be a proper subset of Set B if Set B contains at least one element that is not present in Set A.

Example: If set A has elements as {12, 24} and set B has elements as {12, 24, 36}, then set A is the proper subset of B because 36 is not present in the set A.

Proper Subset Symbol

A proper subset is denoted by and is read as ‘is a proper subset of’. Using this symbol, we can express a proper subset for set A and set B as;  A B

Proper Subset Formula

If we have to pick n number of elements from a set containing N number of elements, it can be done in NCnumber of ways.

Therefore, the number of possible subsets containing n number of elements from a set containing N number of elements is equal to NCn.

Number of subsets and proper subsets in a set

If a set has “n” elements, then the number of subset of the given set is 2n and the number of proper subsets of the given subset is given by 2n-1. 

Consider an example, If set A has the elements, A = {a, b}, then the proper subset of the given subset are { }, {a}, and {b}.

Here, the number of elements in the set is 2. 

We know that the formula to calculate the number of proper subsets is 2n – 1. 

= 22 – 1

= 4 – 1

= 3

Thus, the number of proper subset for the given set is 3 ({ }, {a}, {b}).

Improper Subset

A subset which contains all the elements of the original set is called an improper subset. It is denoted by .

For example: Set P ={2,4,6} Then, the subsets of P are;

{}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6} and {2,4,6}.

Where, {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6} are the proper subsets and {2,4,6} is the improper subsets. Therefore, we can write {2,4,6} P.

Note: The empty set is an improper subset of itself (since it is equal to itself) but it is a proper subset of any other set.

Properties of Subsets

Some of the important properties of subsets are:

  • Every set is considered as a subset of the given set itself. It means that X ⊂ X or Y ⊂ Y, etc
  • We can say, an empty set is considered as a subset of every set. 
  • X is a subset of Y. It means that X is contained in Y
  • If a set X is a subset of set Y, we can say that Y is a superset of X